kluedo.ub.uni-kl.de · /01 # , 243 57698;:=

$Page 1: kluedo.ub.uni-kl.de · /01 # , 243 57698;:=<'34>?a@cbedgfihkjmlon pqn rshkjutvfxwyjzn bfxwyrs[ \]n pa^/n-hkjz_zra`yhkjzn-_+bc_zredcn pq`yrewkfxw4g9fira`yn pq`yhsfii]wjn-py_lk$
"!#%$&')(* +,-. /01#,

24357698;:=<'34>

?A@CBEDGFIHKJMLONPQNRSHKJUTVFXWYJZNBFXWYRS[\]NPA^/N-HKJZ_ZRa`YHKJZN-_+bc_ZRedCNPQ`YReWKfXW4g9FIRa`YNPQ`YhSFIi]WjN-PY_lki]P?AN-PYhaNRSJMiZ_]m\]N`1FI[&FI\ZNBnRS`QHKJZN_UoPjFI\ZN-Àpq@C[rWj@IP4\]NP1sqFXWYiZPQtRa`Y`YN-_Z`YHKJZFXuvWjN-_xwypc@C[rWj@CP

PYNPQiZBz_7F&WjiZPYFIhSRaiZB|XpqP-rPYNP-r_7FXW~mCN-_ZNJZBnRSmXWjNpqRa`Y`YN-PQWjFXWjRa@C_

oi]WjFIHKJrWjNP-MPY@IuMCdC'NRak`YfIHK[IN-P o9iWKFIHKJrWjN-PMPY@XuMso9FI_rWjNPW

?A@IhShSk-iZm\ZN-P1PQ@CB@XWjRS@I_; C MIiZhSR ICI

pIC


"! $#%&'(*)+,-/.0&&.0& 1243%5 6. ! .07- 5 & 8:9<;>=@?0ACBED:F:GIHJAEKLD/?M9+9N9+9N9+9N9N9O9N9+9N9+9N9+9N9N9+9N9O9N9+9N9N9+9N9+9N9+9N9N9O9N9+9N9+9N9+9N9 ;P;8:9Q8SROTU?ITUBWVXZY[BEKL?(H K<\]XLT^9+9N9N9O9N9+9N9+9N9+9N9N9+9N9O9N9+9N9N9+9N9+9N9+9N9N9O9N9+9N9+9N9+9N9 ;88:9 _S`ba]TdcefYhg$DPBfijAEV,AEKLD/?IVBEklemK<^nACBEK<o]G:ACK<DP?(^p9N9+9N9O9N9+9N9N9+9N9+9N9+9N9N9O9N9+9N9+9N9+9N9 ;q8:9 r `ba]TdcefYhg$DPBts%uv\IKLBEK<HVXemKw^xAEBCK<o]G:AEKLD/?I^y9N9N9+9N9O9N9+9N9N9+9N9+9N9+9N9N9O9N9+9N9+9N9+9N9 ;z8:9QSi:VPuv\IXLTNY|V,AEacemY 9+9N9N9O9N9+9N9+9N9+9N9N9+9N9O9N9+9N9N9+9N9+9N9+9N9N9O9N9+9N9+9N9+9N9 8~

d.y,&U.jE'.0.0*)+,-P.j&& 2 _]9<;>=@?0ACBED:F:GIHJAEKLD/?M9+9N9+9N9+9N9N9O9N9+9N9+9N9+9N9N9+9N9O9N9+9N9N9+9N9+9N9+9N9N9O9N9+9N9+9N9+9N9 8__]9Q8Si0AEBCD/?]efBEKgA9+9N9+9N9+9N9N9O9N9+9N9+9N9+9N9N9+9N9O9N9+9N9N9+9N9+9N9+9N9N9O9N9+9N9+9N9+9N9 8,r

5 ./, 5 d.jI,.j& 2 1rI9<;"emTBEG]K x? ^t`ba]TDPBETUu 9N9O9N9+9N9+9N9+9N9N9+9N9O9N9+9N9N9+9N9+9N9+9N9N9O9N9+9N9+9N9+9N9 8/rI9Q8?cefYg$D/BBED,t?]KwV?YVACaI^btKLACa^CuVX<X

L2 @ D/BCu9N9N9+9N9+9N9+9N9N9O9N9+9N9+9N9+9N9 8 rI9 _S¡\I\(TBfV?IF¢cD,TBcKLuyKLAE^£9O9N9+9N9+9N9+9N9N9+9N9O9N9+9N9N9+9N9+9N9+9N9N9O9N9+9N9+9N9+9N9S_/8

¤4! $#%&'(&m¥y§¦-U,6 ! ,¨© 1:9<;"ªtTVPHWa]K<?]vACa]Td«K<?IVXZYD/KL?0A¬9N9+9N9+9N9+9N9N9+9N9O9N9+9N9N9+9N9+9N9+9N9N9O9N9+9N9+9N9+9N9S_P :9Q8Si0AWVk0K<?] TVBbAEa]TNs%0G]K<XLK<o]BEKLG]u9N9+9N9+9N9N9+9N9O9N9+9N9N9+9N9+9N9+9N9N9O9N9+9N9+9N9+9N9 P8:9 _S`ba]TdcefYhg$DPBbAEa]TNs[?IF]\(D/KL?0A¬9+9N9+9N9+9N9N9+9N9O9N9+9N9N9+9N9+9N9+9N9N9O9N9+9N9+9N9+9N9Sq/8

®¯ &°-²±('y¦.0± 5 5 ¨[)+,-P.j&&.j& 1 ¤q]9<;"`ba]TdbVkPT^bªtK<^C³´9+9N9+9N9N9O9N9+9N9+9N9+9N9N9+9N9O9N9+9N9N9+9N9+9N9+9N9N9O9N9+9N9+9N9+9N9 q]9Q8f^Ckjuy\:ACDPACKwHdijTU\IVPBEVACK<DP?Dg%µO¡Y%BCD:H T^C^CT^¶9+9N9O9N9+9N9N9+9N9+9N9+9N9N9O9N9+9N9+9N9+9N9 Pq]9 _Sf^Ckjuy\:ACDPACKwHdijTU\IVPBEVACK<DP?Dg%·DP?0ACK<?jG]DPGI^b`bK<uyTd¸¹VBE³PD,ºl·aIVPKL?I^¶9+9N9N9O9N9+9N9+9N9+9N9 ,z

1S»¼(± 5 5 ½m¾Z±%.0'.j& ¿ ¤j9<;"`ba]TNs[GIXLTB ¸¹VBEG]k0VuVÀ¸TUACa]D:F9N9+9N9+9N9N9+9N9O9N9+9N9N9+9N9+9N9+9N9N9O9N9+9N9+9N9+9N9S /j9Q84=@uy\(D/BnAWV?IHUTdi:Vuy\]X<KL?I9N9N9O9N9+9N9+9N9+9N9N9+9N9O9N9+9N9N9+9N9+9N9+9N9N9O9N9+9N9+9N9+9N9S Pqj9 _S`ba]TdªtTnxTH ACK<DP?¸T AEa]D:FÁ9N9O9N9+9N9+9N9+9N9N9+9N9O9N9+9N9N9+9N9+9N9+9N9N9O9N9+9N9+9N9+9N9S P j9 r i:VPuv\IXLK<?]ÂBCKwF:/T^9N9+9N9N9O9N9+9N9+9N9+9N9N9+9N9O9N9+9N9N9+9N9+9N9+9N9N9O9N9+9N9+9N9+9N9Sz];

3 '&UN¨¦-°-M('& Ã ¤ .0¾ Ã1Äv$6( 5 ±° ÃÃ

;

8

b

`ba]TvACaITUDPBEk¢DgX<VPBC/T¼F:Tº0KwV,AEKLD/?I^mK<^NHUDP?IHUTUBE?]TFtKLACa£ACaITy^nACG(F:k¹Dg%ACa]Ty\]BEDPoIVPo]K<XLKLACK<TÔDgºPTUBEkBWVBETTUº/TU?0AE^9:=@?¢ACaIK<^bAETjAfTN\]BET^CTU?0A^nD/uvTNBET^CG]XLAE^ttaIK<HWaHUVP?¢o(TND/o:AEVPKL?]TFo0kV\]\IXLkjK<?]X<VPBC/TOF]TUºjK<VACK<DP?ACTHWa]?]Kw0G]TÂEDyACa]Td^nACGIF]kÂDg|F:K GI^CK<DP?¢\]BEDjHUTÊ^nTÛ9

`ba]T¼oIVP^CK<H¼BET^CG]XAOKL?£ACaIK<^+VBETVÂK<^Ni:HWa]K<XwF:TUB ^fACaITUDPBETUu VPo(D/G:A+X<VPBC/TÀF:TUºjKwV,AEKLD/?I^fDPgÊHUVX<TFMF:D,t?BCD,t?IK<VP?uyDPACK<DP?-9 KAEaACaIK<^fACa]TDPBETUuÁDP?]TvHUVP?¹HUVXwH GIX<VACTNAEa]TÀTj\DP?ITU?0ACKwVXF:THVkBWV,AET^bg$D/Bm\]BEDPo VoIKLX<KAEKLTÔDg%AEa]Tvg$DPBEu

P (√εB ∈ A)

g$DPBε ↓ 0

ta]TBCTBK<^NVBCD,t?IK<VP?¹uyDACK<DP? V?IF

AK<^+V¢^nTUA

Dg\IV,AEaIÛ9Z`ba]KwÔBCT^nGIXA+K<Ô/TU?]TBEVPXLKw^nTFojk¢ACa]Ty«]BETUKwF:X<KL? TU?0ATUX<XACa]TDPBEk¢ACDACa]TyHUV/^nTvDg^nACD:HWaIV/^xAEK<HF:K TBCT?0ACKwVX|T0GIVACK<DP?I^+tKLACa^CuVX<X|?]DPKw^CTP9-`ba]TyAEa]TUD/BCkMF]TÊH BEKLoT^+a]D,^nD/XLG]ACK<DP?I^+DPgbV¢^nACD:HWaIVP^nACKwHF:K TBCT?0ACKwVXZT0GIVACK<DP?XLK<³PTdXt = b(X) dt+

√ε dBDP?¢V:TFACK<uyTNKL?0AETUBEºVPX

[0; t]oTUaIVº/Tmg$D/B^nuVPXLX

ε9

=@?¹AEa]Ty\]BET^CTU?0AmAETjANTy\]XwVPHUT¼D/G]B+uVK<?¹g$D:H GI^+D/?¹AEa]THUVP^CTvDg^nACBEDP?]¢F:BEKgANK<?I^nACTV/F¹DPg^CuyVPXLX?]D/K<^CT:K©9 T/9]DP?¢^CDPX<G:ACK<DP?I^bDPgAEa]Td^xAED:HWaIVP^nACKwH+F:KTUBETU?0AEK<VPXZT0GIV,AEKLD/?

dXϑt = ϑb(Xϑ) dt+ dB

g$DPB+XwVBEPTϑ9-¡m^nK<?]¢VÂAEKLuyTHWaIV?IPTÀD/?]T¼HV?MACBWV?I^ng$DPBEuÁACa]THUV/^nT¼Dg^nACBEDP?IF:BEKgANK<?0ACDACa]THUV/^nT¼Dg

^CuyVPXLXZ?IDPKw^nT:o]G:AG]?:g$D/BnAEG]?IVACTUX<kÂAEa]TNBET^CG]XAEKL?IvT0GIV,AEKLD/?Kw^tF:T(?]TFDP?¢VvACK<uyTNKL?0AETUBEºVPXZta]D/^CT+XLT?]AEaF:T\(T?IF]ÔDP?AEa]Tv\IVBWVuyT AETUB

ε^CDÂACaITv«IBCTK<F:X<K<? TU?0AUTUX<XACa]TDPBEk¢HV?]?IDAmTVP^CKLX<k¢o(TyV\I\]XLK<TFACDÂAEa]TACK<uyT HWaIV?IPTF\]BED:H TÊ^9 TNtKLX<XF:TBCK<ºPTdVyXwVBEPTNF:TºjK<VACK<DP?¢BCT^nGIXAg$D/BtACaITdo(TaIVºjKLD/G]BtDg

Xϑt

g$DPBjTFttaITU?

ϑo(TH DPuyT^

XwVBEPTNojk¢GI^nK<?]lVF:K TBCT?/AmACTHWa]?]Kw/GITP9`ba]Td\]BEDjDg[GI^CTÂCa]TÀgVPHJAmACaIVAmTÀ³j?]D, ACaIT¼F:T?I^nKLAxk¢DgACaITF:Kw^xAEBCK<o]G:AEKLD/?¹Dg

Xϑ tKLACa£BCT^n\TH AACDlACa]T KLT?]TUBOuyTVP^CG]BETdDP?¢AEa]TÀ\(V,ACa£^n\IV/H TNg$BEDPuÁAEa]TvROK<BEÊV?ID,ºg$DPBEuÀGIX<VI9]fÊ^nG]uyK<?]

b = gradΦAEa]K<^fF:TU?I^CKLAxklKw^

ϕ = exp(

ϑF − ϑ2G)

tKLACaF = Φ(0) − Φ(Bt) +

1

2

∫ t

0

∆Φ(Bs) ds

V?(FG =

1

2

∫ t

0

b2(Bs) ds.

«]D/BmXwVBEPTϑAEa]TdAETUBEu

ϑFHV?¹oTv?]TU/XLTHJACTF¹V?(FT¼HUV?MGI^nTÀACa]TyV\I\]BCD:K<uV,ACK<DP?

P (Xϑt ∈ A) ≈

E(

exp(−ϑ2G)1A(Bt)) 9-`baITvBEK<Pa0AOa(V?IF ^nKwF:TyDgACa]Kw^+BETUXwV,ACK<DP? HUVP?£oTH DP?(^nKwF:TUBETF£VP^+VcV\IX<V/H T

ACBWV?(^xg$D/BCuDgGACa]TyXwVBEPT¼F]TUºjK<VACK<DP?¹oTUa(Vº0K<DPGIBmDPg

P (Xϑt ∈ A)

g$D/Bϑ → ∞ HUVP?¹oTvT:\]BCT^C^CTF

K<?¹ACTBCu^mDg[AEa]TdAWVK<X oTUaIVºjK<DPG]BmDg[AEa]K<^+cV\]XwVPHUTNACBWV?I^ng$DPBEu¢9¡m^nK<?]Vl`VG]oTUBEK<VP?ACaITUDPBETUuT¼HV?ACBWV?(^nXwV,AETd/GIT^nACK<DP?I^mVoDPG:AAEa]Kw^tAEVPKLXo(TaIVºjKLD/G]BK<?0ACDl0G]T^xAEKLD/?I^fVPo(D/G:AACaITdo(TaIVºjKLD/G]BDPgAEa]T¼F]K<^nACBEK o]G:AEKLD/?¢Dg

G?]TVBbACaITNDPBEKL/KL?-9

«]D/XLX<D,tK<?]lAEa]Kw^+\]BCD/PBWVuyuyT¼TyaIVºPT¼ACD¢T^xAEKLuV,AET\]BEDPoIVPo]KLX<KLACK<TÔDgACa]Tvg$D/BCuP (G < ε)

g$D/B^CuyVPXLX

ε9(`ba]T¼BEVP?IF:DPu º,VBEK<VPo]XLT

GKw^f^CuVX<Xta]T?ACaIT¼BCD,t?IK<VP?uyDAEKLD/?M^C\(T?IF]ûyD0^xAfDPg|ACa]TÀACK<uyT

?]TVBmVP?MT/GIKLX<KLoIBCK<G]u\(D/KL?0A+Dg|AEa]TvF]BCKLgAb9Z`DlT^xAEKLuV,AETdAEa]T¼\IBCD/oIVo]K<X<KAEKLT^tg$DPBmACaIK<ÔTUº/TU?0AfT¼a(VºPT

ACD^nACGIF]kAxD¢F:K TBCT?0AÀVP^C\(THJAE^ Z«KLBW^xAEXLkACa]T\IBCD:H T^CÔaIV/^mACD¢BCTVPHWaMACa]TyT0G]K<X<KLo]BEK<G]u \DPK<?0A+º/TUBEk0G]KwHW³0X<kP9`ba]Kw^fHUVP?o(TdACBETV,AETF¢tKLACaAEa]Tda]TXL\MDg%i:HWa]K<X<F]TUB ^bAEa]TUD/BCTu¢9(?(F¹^nTH D/?IF:X<kID/?IH TN?ITVBfACa]T

_

r

T0G]K<X<KLo]BEK<G]u¶\DPK<?0AACa]TN\IBCD:H T^C^ba(VP^bAEDÂ^xAWVklH X<D/^CTOACDyACaIK<^f\(D/KL?0AuyD/^nADgACa]TdACK<uvT/9TBCTNTdHUVP?¢GI^nT`Vk0X<DPBmV\]\]BED:KLuVACK<DP?DgACa]TvF:BEKgAmACDÂBETU\]XwVPHUT+ACa]T¼\]BEDjHUTÊ^ttKLACaMVP?£µmBC?I^nACTKL? ¡fa]XLT?jo(THW³l\]BED:H TÊ^ ta]KwHWaKw^tu¼GIHWaTVP^CKLTBbACDyDPBE³tKAEa-9

`ba]Tvo]G]K<X<F]KL?]¢o]X<D:HW³:^fg$D/BOACa]Kw^N\]BEDjDgVPBCTvF:TUº/TUX<DP\TF F:G]BEKL?]ACaIT(BE^nAdHWaIV\:AETUBWÛ9`ba]T I?IVPX|BCT ^CG]XANVoDPG]AfACaIT¼XwVBEPTÀF]TUºjK<VACK<DP?¹oTUa(Vº0K<DPGIBmDPgXϑ

t

g$DPB+^xAEBCD/?]F:BEKLgAOKw^f\IBCT^nT?/AETF¹V/ÂEa]TUD/BCTu :9<;zACD/PTUACa]TBttKAEaAxDyH DPBEDPX<XwVBEKLTÛ9

`ba]TvACTjA+Kw^N^nACBEGIHJAEG]BCTF£V/Ôg$DPX<XLD,^9Z`ba]T IBW^xAdHWaIVP\:ACTBO/KLº/T^+V¢HWaIVBWVPH ACTBCKw^CVACK<DP?¹DPgF:K GI^nK<DP?\]BED:H TÊ^CT^fV/^m^CDPX<G:AEKLD/?I^fDPg%^xAED:HWaIVP^nACKwHÀF:K TUBETU?0ACKwVXT0GIVACK<DP?I^9 TvtKLX<XD/?]X<k¢HUDP?I^CKwF:TUBOT0GIVACK<DP?I^mDgACaIT+g$DPBEu

dXt = b(Xt) dt+ σ(Xt) dBt,ta]KwHWa£X<TV/F¹AED¸¹VBE³PD,ºjKwV?¹^CDPX<G:ACK<DP?I^9Z`ba]THWaIV\]ACTUBN^CG]uyuVBEK<^CT^+^CDPuyTyBCT^nG]XLAE^+VPo(D/G:A+ACa]T^nTy\]BED H T^C^CT^9`ba]T^CTH D/?IF HWaIV\:AETUB+PK<ºPT^+Vº/TUBEk¹^naIDPBCA+K<?0ACBED:F:GIHJAEKLD/?£K<?0ACDACaIT¼AEa]TUD/BCkMDgXwVBEPTyF:Tº0KwV,AEKLD/?I^9

TÀ\]BET^CTU?0Af^CDPuyTNACDjD/X<^g$BCD/u¶AEa]TÀX<KAETUBWV,AEG]BCT]tKLACaM^C\(TH KwVXTUuy\]aIV/^nKw^D/?¢ACTHWaI?]K<0G]T^ta]KwHWatK<XLXo(TGI^CT g$G]Xta]TU?¹VP\]\]X<KLTFACDgVPuyKLX<KLT^DPgF:K GI^nK<DP?M\]BCD:HUTÊ^nTÛ9]THVGI^CTdF:K G(^nK<DP?\]BED:H T^C^CT^tVPBCTdHUDPuy\]X<TDPo]xTHJAW^XwVBEPTÀF]TUºjK<VACK<DP?£BCT^nG]XLAEÔHUVP?MoTV\]\]X<K<TF¹D/?MF:K TBCT?0A+X<TUº/TUXwÛ9 Tv/KLº/TÀBET^CG]XLAEÔVoDPG:AOACa]ToTUaIVºjK<DPG]BtDPg|^nAEVACK<DP?IVPBCkF:K<^nACBEK<o]G:ACK<DP?(^tta]TU?¢AEa]TÀF:BEKLgAfo(TH D/uvT^^nACBEDP?] ]VoDPG:AfTuy\]KLBEKwHUVXF:K<^nACBEK<o]G ACK<DP?(^tta]TU?AEa]TN\]BEDjHUTÊ^bK<^bD/oI^CTUBEºPTFD,ºPTBbXLD/?]vACK<uvTNK<?0ACTBCº,VXw^ ]V?IF¢VoDPG]AtACa]TN\IVACaI^bDPgAEa]Td\]BED:H T^C^ta]T?AEa]TN?]D/K<^CT+K<^t^CuVX<X'9

·aIVP\:ACTBf_yK<^mF:TUº/DAETFACDÂVP?T]Vuy\]X<TIACa]TyµmBC?I^nACTKL? ¡fa]XLT?jo(THW³l\]BED:H TÊ^bKw^tACaIT¼^CDPX<G:ACK<DP?¢DgAEa]T^nACD:HWaIVP^nACKwH+F:K TBCT?/AEK<VPXZT0GIVACK<DP?dXt = −αXt dt+ dBtg$DPBO^nD/uyTN\(D0^nKLACK<ºPTd\(VBWVuyT AETUB

α9THVGI^CTNDgAEa]T¼^CKLuy\]X<TÀ^nACBEGIHJAEG]BETNDgACa]T¼\]BEDjHUTÊ^tKLAfK<^f\(D0^C^CK<o]XLTdACD

HUVPX<HUG]XwV,ACTÀuyVkACaIKL?]0^fT:\]XLKwH KLACX<k¢a]TUBETP9 T¼F:TBCK<ºPTÀVP?¹XwVBEPTdF:Tº0KwV,AEKLD/?¹BET^CG]XAfg$DPBmACa]T¼o(TaIVºjKLD/G]BfDPgXt

g$DPB :TFt ∈ ta]TU?MACa]Tv\IVBWVuyTUACTUB

αo(TH DPuyT^mXwVBEPT/9(`ba]KwÔK<^+V^nK<uy\]XLK ITFMº/TUBW^nK<DP?DPg%DPG]B

uVK<?¢BCT^nG]XLA9=@?HWaIV\]ACTUBtrTN\]BET^CTU?0AfV`VG]oTUBEK<VP?lAEa]TUD/BCTu¶DPgTj\DP?ITU?0ACKwVXZAxkj\TÀVP^tVP?]DAEa]TUBACDjDPXZAEDDPo:AWVK<?

XwVBEPTNF:TUºjKwV,AEKLD/?BCT^nG]XLAE^9I`ba]TdACa]TDPBETUu¶\IBCD,ºjKwF:T^VHUDP?]?ITHJAEKLD/?o(TUAxTTU?ACaITÀoTUaIVºjK<DPG]BtDPg|VÂ\]BCD/oIV o]K<XLKLAxkF:Kw^xAEBCK<o]G:AEKLD/??]TVPBACaITÀDPBEK<PK<?¢VP?IFACaIT¼cVP\]XwVPH T+AEBEVP?I^ng$DPBEu ?]TVBK<? I?IKAxk/9¡m^nK<?]AEa]TdAEa]TUD/BCTuVX<X<D,^G(^bACDF:TF:GIH TNBET^CG]XLAE^bX<KL³/T

limε↓0

ε · logP

(∫ t

0

B2s ds ≤ ε

)

= − t2

8

ta]TBCTBK<^NV¢BED,t?]K<VP?MuyDAEKLD/?-9-f^NVP? VP\]\]X<K<HV,AEKLD/?£TvF]TUBEKLº/TyVXwVBEPTvF:TUºjKwV,ACK<DP?£ACaITUDPBETUu g$DPB

BCD,t?IK<VP?\(V,ACa(^ftKLACaM^CuVX<XL2 ?]DPBEu¢9(`ba]T¼X<V/^xAf\(VBCAmDg[HWaIVP\:ACTBfrF:TUBEKLº/T^VlBCT^nGIXAmVPo(D/G:AmGI\]\(TBV?(FlX<D,TBbXLK<uyKAW^tKL?AEa]Td`VPG]oTUBEK<VP?lAEa]TUD/BCTu9

`ba]TdHUTU?0ACBWVXZ\IVPBnADPgAEa]Kw^tACTjAfK<^fHWaIV\:AETUBf]9TBCTNTdH D/uÀo]K<?]TNuV?jkÂBET^CG]XLAE^bg$BEDPu¶ACa]Td\IBCTº0K<DPG(^HWaIVP\:ACTBE^fACDF:TUBEKLº/TdAEa]T¼XwVBEPT¼F:TUºjKwV,ACK<DP?MBCT^nG]XLAOg$DPBmACa]Tvo(TaIVºjKLD/G]BfDPg[ACaIT¼T?IF:\DPK<?/A+Dg%VF:K G(^nK<DP?G]?IF]TUBf^nACBEDP?]yF:BEKgA9]= AAEBEVP?I^n\IKLBETÂCa(V,AfV¼Axkj\]K<HVX-\(V,ACa¢G]?(F:TUB^nACBEDP?]F:BEKLgAfV?IFtKLACa¢PK<ºPT?T?IF:\DPK<?/ABEG]?I^mACD,bVBWF]^fVP?MT/GIKLX<KLoIBCK<G]u \DPK<?/A+Dg%ACa]TyF:BEKgAd0G]KwHW³0X<k^nAEVk:ÂEa]TUBETvG]?0ACK<X|?]TVPBfAEa]T¼T?IF¹DPg[ACaITACK<uyTÀK<?0ACTUBEº,VX (VP?IF¢DP?]X<kACa]T?¹uyD,ºPT^f/GIK<HW³jX<klAEDÂACaITdPK<ºPT?TU?IF:\DPK<?0A9`ba]TÀK<?]KLACKwVXV?IF (?IVX\]KLTH T^DgAEa]TÀ\IVACa¹HV?oTdACBETVACTFtKLACa i:HWa]K<X<F:TB ^tAEa]TUD/BCTu VPo(D/G:Af\IVACajtK<^CTdXwVBEPTdF:TºjK<VACK<DP?I^bg$D/BmÊHUVPXLTFF:D,t? BED,t?]KwV?£uvDPACK<DP?-9Z`ba]TyuyKwF]F:X<T\]KLTH TyDgTVPHWa£\IV,AEa HV? o(TvAEBCTV,ACTFMtKLACa£ACaITy`VGIo(TBCKwV?ACaITUDPBETUuÁg$BEDPu HWaIVP\:ACTBfrI9 TÀ/KLº/T¼^CTU\IVPBEVACTdBET^CG]XLAE^fg$DPBmACa]TvHUVP^CTÀDPg%V,ACACBWVPHJAEKL?IF:BEKgANV?(F¢g$DPBmACa]THUV/^nT+DPgBETU\TUX<XLK<?]F:BEKgA9

·aIVP\:ACTB+qPK<ºPT^mVP?]DAEa]TUBNVP\]\]X<K<HV,ACK<DP?£DgX<VPBC/T¼F:Tº0KwV,AEKLD/?MBET^CG]XLAE^ ?(VuyTUX<kACD¢F:TUACTUBEuyKL?IT¼AEa]TT:\DP?]T?0ACKwVXF:THVkBWV,AETdg$D/BfAEa]TvbVk/T^BEKw^n³¢ta]T?£^CTU\IVPBEVACK<?]AxDF:K TBCT?0Am\]BED:H TÊ^CT^9ijK<?IHUTÀAEa]TbVkPT^BEKw^n³ÂKw^VyuyTV/^nGIBCT+Dga]D, H X<D/^CTmAEa]Td\]BEDPo(Vo]K<XLKLAxklF:Kw^nACBEKLo]G]ACK<DP?I^tDPgAEa]TNAxDv\IBCD:H T^C^CT^tVPBCTjACa]Kw^BWV,ACTNF:T^CHUBCK<oT^ba]D, gVP^nAbTNHUVP?0VK<?lK<?:g$DPBEuV,AEKLD/?VPo(D/G:AtACa]TN\IBCD:H T^C^CTôjkÂX<DjDP³jKL?IvVAtACa]TN\IVACaI^9

«KL?(VX<XLk¹HWaIVP\:ACTB¼ÂF]TÊH BEKLoT^N^CDPuyT¼ACTHWaI?]K<0G]T^+ta]KwHWa£a]TXL\ AEDT:\(TBCK<uyTU?0AOtKLACa*BWVBET¼TºPT?/AW^g$DPB+F:K G(^nK<DP?£\]BCD:HUTÊ^nTôjkuyTV?(^fDPgH D/uv\IG:ACTBO^CKLu¼G]XwV,ACK<DP?(Û9£TyF:T^CHUBCK<o(T¼ACaIT¼s[GIXLTB ¸¹VBEG]k0VuVuyT AEa]D:FMAED^nK<uÀG]XwV,AET^nD/XLG:AEKLD/?IÔDgb^xAEDjHWa(VP^nACKwHvF]KTUBETU?0ACKwVX|T0GIV,AEKLD/?IÛ9=@?²^CG]oI^CT0G]TU?0Ad^CTHJAEKLD/?IÔDg

HWaIVP\:ACTBdÂTvF:T^CHUBCK<oT¼a]D, ACa]TyK<uv\DPBCAEVP?IH TyÊVuy\]X<KL?]uyT ACaIDjF HUVP?£oTvGI^nTF¹AEDT^xAEKLuVACT^nuVPXLX\]BEDPoIVPo]K<XLKLACK<T^+V?(FMa]D, ACa]TyBETxxTHJACK<DP? uyT AEa]D:F HV?£o(TGI^CTFMACDÊVuy\]X<Tvg$BCD/u H D/?IF:KLACK<DP?IVPXF]K<^nACBEK o]G:AEKLD/?IÔta]TUBETdAEa]TyH DP?(F:KAEKLD/?¹aIV/^fº/TUBEkX<D,´\]BEDPoIVPo]KLX<KLAxkP9«K<?IVPXLX<k¢TvF:TÊH BEKLoTÀaID, ACa]TycVP?]PTº0K<?uyT AEa]D:FHV?o(TNG(^nTFlAED^CVPuv\IXLT+\IVACaI^tDPgVyF:K G(^nK<DP?¢tKLACa¢PK<ºPT?T?IF\(D/KL?0A9

`ba]TNAWVo]X<TÀVAf\IV/Tdz/yT:\]X<VPKL?(^^nD/uyTÀ^nkju¼o(D/X<^VP?IF^CDPuyTd?]DPAEV,AEKLD/?GI^CTFACaIBCD/G]Pa]D/G:AtACaITdACTjA9`ba]TBCT+Kw^tVXw^nDV?K<?IF:Tta]KwHWauyK<Pa0Aa]TUX<\ACDVPHH T^CÂCaIT+ACTjA9=ttKw^naMACDÂAEaIV?I³uÀk^nGI\(TBCºjKw^nD/BfY[BEDg$T^C^CDPB d9º9 TUK ^PHW³/TUBg$D/BOVPXLXa]KwÔa]TUX<\£VP?IF¹^CG]\]\DPBCAmK<?

tBEKAEKL?]vAEa]K<^tACTjAfVP?IF¸MVBCACK<? mVK<BCTBbg$DPBta]Kw^tVPF:ºjKwH T/9

q


> b Á

`ba]Kw^tK<?/AEBCD:F:G(HJACD/BCkÂHWa(V\:AETUBttK<XLX^CG]uyuVBEK<^CTN^nD/uyTOBET^CG]XLAE^tVPo(D/G:AfF:K GI^CK<DP?¢\]BEDjHUTÊ^nT^jta]KwHWa¢T+tK<XLXGI^CTÀXwV,AETUB9¸kuVPKL?¹BET g$TBCT?IH T¼a]TUBETÀKw^mACa]T¼o(DjD/³Dg[º9 £TK ^ /HW³PTUBmV?IF KL?]³jX<TUB z/~ 9µfACa]TBGI^CT g$G]XBCTUg$TUBETU?IHUT^tVBETOACa]TNoDjDP³:^tDPgNVBWV,AV/^VP?IFija]BETUº/T di:zI; (Dg=@³/TF]VyVP?IF VAEV?(VoT = /z V?(FlDPg|i0AEBCDjD:HW³ÂV?(F[VPBEV/F:aIVP?Qi+z 9

`ba]TBCTVBETyuV?jk¹bVkj^mAED¹HWaIVBWVPH ACTUBEKw^nTvF:KGI^CKLD/?*\]BCD:HUTÊ^nTÛ9=@? ACa]Kw^+AEa]T^CK<^dTytK<XLX[G(^nTyAEa]THWaIVPBEV/HJACTBCKwÊV,ACK<DP?lDgVyF:K GI^nK<DP?¢\]BED:H T^C^bVPÂEa]Td^CDPX<G:ACK<DP?DgACa]Td^nACD:HWaIV/^xAEK<HOF:K TBCT?0ACKwVX-T/G(V,ACK<DP?

dXt = b(Xt) dt+ σ(Xt) dBt ;/9L; g$DPBm^nD/uvTNK<?]KLACKwVXºVPXLGIT

X0 ∈ L1 Ita]TUBET B Kw^fV?n F:KLuyT?I^nK<DP?(VX-BCD,t?IK<VP?uyDAEKLD/? b : d → d K<^^CDPuyTNF:BCKLgAtg$G]?IH ACK<DP? (V?IF

σ : d → d×n K<^bAEa]TdF:K G(^nK<DP?H DjT"!HUKLT?0A9`ba]Tdo(VP^CK<HÀBCT^nG]XLAE^mVoDPG:AOT:Kw^xAETU?IHUTÀV?IF¹G]?]Kw/GITU?]T^C^mDg[^CDPX<G:ACK<DP?I^g$DPBmACa]Kw^mT0GIV,AEKLD/? AEa]TUD/BCTuy^=#À9Q8:9 rV?IF=#À9 _]9<;Og$BCD/u$ = Pz% bVBET+VP^g$DPX<X<D,^d.jI,.j '&' )(*,+"-

b : d → d .0/21 σ : d → d×n 3 +45 / -76 /'8 5 89:;9<. -76 9%=> -%?@+BADCE5FG-%?45 /H1 6I-J6K5 /‖σ(x)‖2 + ‖b(x)‖2 ≤ K

(

1 + |x|2) = 50C .LIL x ∈ d ;/9 8M

= 5C 9 5N+ K > 0 :O./H1PL +"- E|X0|2 < ∞ QBR ?S+ / -%?@+T45CUCE+ 9JV 5 /21 6 / AWXZY[? .9\.9 5 L]8 -76K5 / FG6I-%?E|Xt|2 <∞ = 5C .LIL t ≥ 0 Q

d.jI,.j 2&w2(,*^+"-b : d → d ./21 σ : d → d×n 9". -J6 9I=> -%?@+ = 5 LIL 50FG6 / A L 5_4 .L *G6 V'9 4`?a6I-Ib45 /H1 6I-J6K5 /2cd= 50Ce+"f+<C > N ∈ g -%?@+"CE+Z6 9e. KN > 0

FG6I-%?‖σ(x) − σ(y)‖2 + ‖b(x) − b(y)‖2 ≤ KN |x− y|2 = 50Ce+"f+<C > x, y ∈ KN ;/9 _d

F^?@+"CE+KN

6 9 -I?S+h4 L 5 9 + 1i3.0LIL FG6I-I?jC .D1 6 89 N QkR ?S+ / -I?S+h45CUC`+ 9JV 5 /21 6 / APWXZYl? .9h.T8a/ 6Km 8 + 9 -7CE5 / A9 5 L]8 -J6K5 / Q TNtKLX<XuyD/^nACX<kHUDP?I^CKwF:TUBbAEa]TÀHUV/^nTNtaITUBET

BK<^mV

d F:KLuyT?I^nK<DP?(VXBED,t?]K<VP?luyDAEKLD/?V?IF σ(x) =Idg$DPBOVX<X

x ∈ d 9I= gAEa]TU?g$D/BT]VPuv\IXLT b Kw^/XLD/oIVX<X<klc-K<\IÊHWa]KA(K'9 TP9IACa]TBCTdKw^fV c > 0tKLACa |b(x) −

b(y)| < c|x− y| g$D/BVX<X x, y ∈ d :AEa]TU?¢T+aIVºPT‖σ(x)‖2 + ‖b(x)‖2 ≤ 12 +

(

‖b(x) − b(0)‖+ ‖b(0)‖)2

≤ 1 + 2c2|x− 0|2 + 2b2(0)

≤ max(1 + 2b2(0), 2c2)(

1 + |x|2)

V?(F‖σ(x) − σ(y)‖2 + ‖b(x) − b(y)‖2 ≤ 0 + |x− y|2g$DPBmVX<X

x, y ∈ d IK©9 T/9]T0GIVACK<DP?I^d;P9Q8¼VP?IF ;P9 _ya]DPXwF¢V?IFAEa]TNACaITUDPBETUu^t/GIVBWV?0AETUTmACa]TdT:K<^nACT?IH TNDPgVG]?]Kw0G]Td^CDPX<G:ACK<DP?KL?AEa]K<^tHVP^CTP9

ijD/uvTUACK<uyT^bKLAK<^tG(^nTUg$G]XZACDyBETÊHUVPXLTNF:K G(^nK<DP?¢\]BED:H TÊ^CT^9j`baIK<^ba]TXL\(^tGI^bg$D/BtT]Vuy\]X<TjAEDyACBWV?I^C\(D/BnAF:K G(^nK<DP?I^fDP?¢AEKLuyTÀK<?0ACTBCº,VXw^

[0; t]g$DPBOF:K TBCT?/AmºVPXLGIT^DPg

tACDlVÂHUDPuyuyDP?¹ÊVuy\]X<TÀ^C\IVPHUTP9(`baIK<^mHUV?oTdF:DP?]T+tKLACaACaIT+g$DPX<XLD,tK<?]yXLTuyuyVI9

3 .j 5 '& & *^+<-c > 0 .0/21 X 3 + .\9 5 L]8 -J6K5 / 5 = -%?@+BWaXBY

dX = b(X) dt+ σ dB.

X+ / +e-%?@+hCE+ 9 4 .0L + 1BV CE5 4+ 99 Y 3<> Yt = Xct/√c = 50Ch+<f+<C > t ≥ 0 : ./ +<F CE50F / 6 ./ N5-J6K5 / B3"> Bt = Bct/

√c = 50C .LIL t ≥ 0 ./H1P./ +<F 1 CU6 = - O+ L 13<> b(x) =

√c · b(√c · x) = 5C .0LIL x ∈ d Q

R ?@+ / -I?S+ V C`5 4+ 9`9 Y 9 5 L f+ 9 -I?S+BWXZYdY = b(Yt) dt+ σ dB.

)+,-¨ & kACa]T¼oIVP^CK<HÀ^CHVX<KL?]\IBCD/\(TBnAxkDg[BCD,t?]KwV?¢uyDAEKLD/?¢ACa]T¼\]BEDjHUTÊ^BV/^F:TI?]TFMVoD,ºPT

Kw^tVBED,t?]KwV?uvDPACK<DP?-9]«IDPBVP?0kl\IVK<BbDg|^xAEDP\]\]K<?]vACK<uyT^SV?(F

TtKAEa

S < TAEa]T+g$DPX<XLD,tK<?]ya]D/X<F]^9

YT − YS =1√c

(

XcT −XcS

)

=1√c

(

∫ cT

cS

b(Xt) dt+ σBcT − σBcS

)

c · s = t=

1√c

∫ T

S

b(Xcs) c ds+ σ1√cBcT − σ

1√cBcS

=

∫ T

S

√cb(

√cYs) ds+ σ(BT − BS).

`ba]Kw^t\]BED,ºPT^[AEa]TdH XwVK<u¢9 0TF' `ba]T+oIV/^nKwHNHWaIVBWVPH ACTBCKw^CVACK<DP?ÂDPgBETUº/TUBW^nK<o]X<TOF]KGI^CKLD/?I^tKwÂCa]TNg$D/XLX<D,tK<?]vACa]TDPBETUu :ta]KwHWa/DjT^boIV/HW³

ACDVyBET^CG]XLAbDPgmDPX<uyDPPD/BCD,º90´F:T AWVK<XLTF\IBCDjDgK<^b/KLº/TU?g$DPBtT:VPuy\]XLT+K<? |D/z0" 9d.jI,.j '&<;(l*^+<-

b : d → d 3 +e*;6 V'9 4?6I-%bT45 / -J6 /S8 5 89: B . CE50F / 6 .0/ N5-J6K5 / FG6I-I?f .L]8 + 9 6 / d : ./21 X .\9 5 L 8 -76K5 / 5 = -I?S+BWXZYdX = b(X) dt+ dB

FG6I-%?X0 ∈ L2 Q R ?S+ / -%?@+ = 5 LIL 50FG6 / A4`5 /21 6I-76K5 /@9 . CE+ +m 8 6If .0L + / - c R ?@+ V CE5_4`+ 99 X 6 9 CE+"f+<C 9 6 3"L +eFG6I-%? 9 - . -76K5 /2. C >1 6 9 -JCU6 3<8 -J6K5 / µ Q 6 R ?@+"CE+e6 9e. =8a/ 4"-76K5 / Φ: d → FG6I-I?

b = −∇Φ : dµ = exp(−2Φ(x)) dx ./H1∫

d

exp(

−2Φ(x))

dx = 1.

µfg%H D/G]BW^nTÀACa]TyH D/?IF:KLACK<DP?b = − gradΦ

F]T ACTBCuyK<?]T^fACaITv\DAETU?0ACKwVXΦD/?]XLkGI\¹ACD¢VlHUDP?I^nAEVP?0A9

ijDta]T?]TUº/TUBexp(−2Φ)

K<^NK<?0ACTPBWVo]X<TDP?ITÂHUV?*VPFIF V?]D/BCuVPXLKw^nK<?]H D/?I^nAEV?0A+ACDΦZAED¹HWaIV?IPT

exp(−2Φ)K<?0ACDVy\]BEDPoIVPo]K<XLKLAxkÂF:TU?(^nKLAxkP9

`ba]TÀHVP^CTNDg[F:K G(^nK<DP?¹\IBCD:H T^C^CT^btaITUBETNACa]TvF:BEKgAfKw^fVÂPBWVPF:K<TU?0AfV?(FACa]TvF:KGI^CKLD/?¹H DjT<!ÂHUKLT?/AmK<^H D/?I^nAEV?0AfKw^mT^C\(TH KwVX<XLkTV/^nkIoTHUVPGI^CTdT¼HUV?MT:\]X<K<HUKAEXLkHUVPX<HUG]XwV,ACTdACa]TvF:TU?I^CKLAxkDPg|ACa]TvF:K<^nACBEK<o]G:ACK<DP?DgACaITd\]BEDjHUTÊ^nT^btKAEaBCT^n\TH AACD K<TU?ITUBuyTV/^nGIBCT/9I`ba]Tdo]K<ÂVPF]ºVP?0AEV/T+Dgg$DPBEuÀG]XwV ;/9 r o(TXLD, K<^ ACa(V,AKLAfF:DjT^b?]DPAfH DP?0AWVK<?ÂACaITd^xAEDjHWa(VP^nACKwHmK<?0ACTPBWVXg$BCD/u ACaITÀROKLBW^CVP?]D,ºyg$DPBEuÀGIX<VyV?jkuyDPBETP9

3 .j 5¹ 2&w¤ & *^+"-B 3 + . (Ft)

CE5F / 6 ./ N5-J6K5 / .0/21TL +<- X 3 + .T9 5 L]8 -J6K5 / 5 = -%?@+ 9 - 5_4`? .9 -J6K41 6 +<CE+ / -J6 .L +m 8S. -J6K5 /

dXt = b(Xt) dt+ dBt

X0 = 0.

z

8 CU-I?S+<CUN5CE+ L +"- Φ: d → 3 +e-JF 5T-76IN+ 9 45 / -76 /'8 5 89UL]>P1 6 +<CE+ / -J6 .D3"L +hFG6I-I? b = − gradΦ ./H1L +"- b 3 + *;6 V29 4`?a6I-Ib45 / -76 /'8 5 89 QOR ?S+ / -%?@+ 1 6 9 -JCU6 3<8 -J6K5 / 5 = X 5 / Ft

? .9 .P1 + /S9 6I- > ϕtFG6I-%?CE+ 9JV +4"--#5-I?S+6K+ / +"CBNT+ .9U8 CE+ :;1 + / + 13">

ϕt(ω) = exp(

−Φ(ωt) + Φ(ω0) −∫ t

0

v(ωs) ds) = 5C .LIL ω ∈ C

(

[0;∞), d) ;/9 r

F^?@+"CE+v = 1

2

(

(∇Φ)2 − ∆Φ) : 6 Q + Q = 5C +"f+"C > A ∈ Ft

F + ? . f+

P (X ∈ A) =

∫

A

ϕ(ω) d (ω).

)+,-¨ & `ba]TlF]TU?I^CKAxk£Dg L(X)DP? Ft

K<^dHWa(VBWVPHJAETUBEK<^CTFMojk¹ACaITlROKLBW^CVP?]D,ºg$D/BCu¼G]X<V ^nTTTP9 I9^CTHJAEKLD/?*;~]9Q8vDg zP~% J9`ba]TÀcKL\(^CHWa]KLA HUDP?0ACK<?jG]KLAxkÂDg b PK<ºPTÂCa]T¼?]THUTÊ^CVPBCklKL?0ACTPBWVo]K<X<KAxklHUDP?IF]K ACK<DP?(^ ^CTUTIT/9 (9IH D/BCD/XLXwVBEKLTÔ;P9<;dV?IF*;/9 8vDPg |D 0z0"% J9ZijDTdDP?]X<kla(VºPT+ACDHWa]THW³lACaIVA ϕtV/^fF:TI?]TF

K<? ;P9 rd ÊV,ACKw^ IT^ϕt(X) = exp

(

∫ t

0

b(Xs) dXs −1

2

∫ t

0

b2(Xs) ds)

.

= A( ^g$D/BCu¼G]XwVÀ/KLº/T^

dΦ(X) = gradΦ(X) dX +1

2∆Φ(X) dt

= −b(X) dX +1

2∆Φ(X) dt

V?(FloTHVGI^CT+Dg

b(X) dX − 1

2b2(X) dt = −dΦ(X) − 1

2

(

(∇Φ(X))2 − ∆Φ(X)) dt

= −dΦ(X) − v(X) dtTO/T A

ϕt(X) = exp(

−Φ(Xt) + Φ(X0) −∫ t

0

v(Xs) ds)

= exp(

∫ t

0

b(Xs) dXs −1

2

∫ t

0

b2(Xs) ds)

.

ijDyTUº/TUBEk0ACa]K<?]yK<^b\IBCD,º/TFZ9 0TF' ½m¾ 5 ²±. 2& 2& »¼& 5 y'¨© & fTUBETÀTÀaIVº/T¼^CDPuyTdº/THJAEDPB

b ∈ d tKAEa b(x) = bg$DPB+VX<X

x ∈ d 9 KAEaACa]TN?]DPAEVACK<DP?g$BCD/u¶X<TUuyuV;/9 ¼T+PT AΦ(x) = −b · xb(x) = − gradΦ(x) = b

v(x) =1

2

(

(∇Φ)2 − ∆Φ)

(x) =b2

2V?(FHUDP?IHUXLGIF]T

ϕt(ω) = exp(

−Φ(ωt) + Φ(ω0) −∫ t

0

v(ωs) ds)

= exp(

b · (ωt − ω0) −b2

2t)

.

`ba]Kw^HUV/^nTOKw^tT^C\(TH KwVX<XLklTVP^CkjoTHVGI^CTvK<^fH DP?(^xAWV?0AV?IFlACajGI^bAEa]TdF:T?I^nKLAxkÂKw^tVvg$G]?IH ACK<DP?¢DgDP?]X<kACaIT

TU?(F:\(D/KL?0ADPgACaITN\IV,AEa-9

;~

½m¾ 5 ²±. 2&w2 & d.¹y,&.j'-W.j%.j*±,-/.0&& & TUBETdTÀaIVº/Tb(x) = −αx g$D/Bm^CDPuyT

α > 09]¡f^CK<?]yACa]TN?IDAEVACK<DP?DgXLTuvuV¢;/9 ¼V0VK<? :T+/T A

Φ(x) =α

2x2 − d

4log

α

πb(x) = − gradΦ(x) = −αx

v(x) =1

2

(

(∇Φ)2 − ∆Φ)

(x) =1

2(α2x2 − αd).

`ba]TÀ^nACBWV?IPTNH D/?I^xAWV?0AtKL?¢AEa]TÀF:TI?]KLACK<DP?DPgΦuV³/T^

exp(−2Φ)Vy\]BCD/oIVoIKLX<KAxkF:TU?(^nKLAxk]?(VuyTUX<kÂAEa]T

F:T?I^nKLAxklDPgVv?]D/BCuVPX-F:K<^nACBEK<o]G:ACK<DP?¢tKLACa¹HUD,ºVPBCKwV?(H TfuVACBEK1/2α · I 9I`ba]TN\]BED:H TÊ^tH D/BCBET^C\DP?IF:K<?]¼ACDACaIK<^F]BCKLgAK<^bBETUº/TUBW^nK<o]X<TOVP?IFaIVP^Vy^nAEVACK<DP?IVPBCkÂF:Kw^nACBEKLo]G]ACK<DP?µ9I`ba]TNX<TUuyuV¼/KLº/TÂCaITdF:TU?I^CKLAxk

ϕt(ω) = exp(

−α2

(ω2t − ω2

0 − t · d) − α2

2

∫ t

0

ω2s ds

)

.

b >

=@?¹AEa]Kw^mHWaIVP\:ACTBm=fVP?/AfACDBCTº0K<TU´^CDPuyTdAEDjDPXw^ta]KwHWa£VBETÀVº,VK<XwVo]X<TNACD¢^nACGIF:k¢XwVBEPTÀF]TUºjK<VACK<DP?I^mDgF:K G(^nK<DP?£\]BCD:HUTÊ^nTÛ9¸kuVK<?£^CDPG]BWH T¼a]TUBETÀKw^mACa]TyoDjDP³¢DgemTUu¼o(D¢V?(FTKAEDPG]?]K eZzP 9µfACa]TBGI^CT g$G]XBCTUg$TUBETU?IHUT^KL?(H X<GIF:T+ACaITNo(DjDP³:^bDPgemTUGIÊHWa]TX-V?IF¢i0ACBEDjD:HW³ e+i: /zV?(FF]TU? DPX<XwV?IF:TBe fDPXw~P~ 9

! "#`ba]Kw^m^CTH ACK<DP?g$D/BCu¼G]XwV,ACT^bACaIT¼oIV/^nKwHNXwVBEPTÀF]TUºjK<VACK<DP?\]BEK<?IH K<\]X<T¼V?(FPK<ºPT^tACaITÀoIV/^nKwHdF:TI?]KLACK<DP?-9Z«]D/BF:TUAEVK<Xw^t^nTTmAEa]TNBET g$TUBETU?(H T^b/KLº/TU?VoD,ºPTP9

! .%$$'( 2)&' '& 5 .¢¨ Kw^mVyg$GI?IHJAEKLD/?I : X → [0;∞]

DP?¹V mVGIÊF:DPBMAEDP\DPX<DP/K<HVX^C\IVPHUT X ta]K<HWaMK<^fXLD,TUBf^CTUuyK HUDP?0ACK<?jG]DPG(^(K'9 TP9(ta]TUBET¼VPXLXACa]T¼XLTºPTUX^nTUAE^ x ∈ X | I(x) ≤ c g$DPBc ≥ 0

VPBCTNH X<D/^CTFlKL? X 9 BEVACTOg$G]?IH ACK<DP? I : X → [0;∞]K<^tHVX<XLTFV 6-- 5 .¨ :KLg[VX<XACa]T

X<TUºPTX-^nTUAE^ x ∈ X | I(x) ≤ c g$DPB c ≥ 0VBET+H DPuy\IV/HJAtK<? X 9

=@?AEa]K<^mAETjAmAEa]T^n\(VPH T X tK<XLXAxkj\]KwHUVPXLX<k¢o(TvTUKLACa]TBfAEa]Tys[GIHUXLKwF:TVP?£^C\IVPHUT n D/BOVl\IV,AEa£^C\IVPHUTX<KL³/TC(

[0;∞), d) 9

! .%$$'( 2 &w2 & gVPuvK<X<k(µε)ε>0

Dgb\]BEDPoIVPo]KLX<KLAxkuvTVP^CG]BET^+DP?²V fVPGIÊF:DPB²ACD/\(D/XLD/PKwHUVX^C\IVPHUT X Kw^dÊV,ACKw^ IT^mACaIT 5 ,6.M./7- 5 '(±,P±'. D/Bd^na]D/BnAETUB ACa]TÂcefY NtKAEa BWV,AETvg$G]?IH ACK<DP?

I : X → [0;∞]:KLgAEa]T+g$DPX<XLD,tK<?]vAxDvT^xAEKLuVACT^baIDPXwF

lim supε↓0

ε logµε(A) ≤ − infx∈A

I(x) 8]9L; g$DPBtTºPTBCkÂHUXLD0^nTF^CT A

A ⊆ X VP?IFlim inf

ε↓0ε logµε(O) ≥ − inf

x∈OI(x) 8]9 8M

g$DPBtTºPTBCkD/\(T?¢^CT AO ⊆ X 9

ijD/uvTUACK<uyT^tKLAfKw^fF:K !ÂH G]XLAACDlDPo:AWVK<?¢Vvg$GIXLXcemYVP^F]TÊH BEKLoTF¢K<?¢ACa]T¼F:TI?IKAEKLD/? Io]G]AfKAmK<^f\(D0^C^CK o]X<T+ACDyPTUAVyTV³ÂcemY9! .%$$'( 2)& & = g[ACaIT¼G]\I\(TBOoDPGI?IF¹K<?£F:TI?]KLACK<DP?*8:9Q8DP?IXLk¢a]D/X<F]^fg$DPBNVPXLXH D/uv\(VPHJA K<?I^nACTV/FDg[VPXLXH X<D/^CTFS ^nTUAE^ ]ACaITU?AEa]TdgVPuvK<X<k

(µε)ε>0ÊV,AEK<^ ITÂCa]T ¥À. 5 5 6..07- 5 ±,P±'. DPB^Ca]DPBCA jAEa]TNTV³ÂcefY 9

TV³¹cemY HUV?£o(T^nACBETU?]PACa]T?]TF¹ACD¢VÂg$GIXLX[cefY tKLACa£ACa]Tya]TXL\£Dg%AEa]Tvg$DPX<XLD,tK<?]c-TUuyuVta]KwHWaKw^VyF:K<BCTHJAH D/?I^CT0G]TU?(H T+DgXLTuyuyV;/9 8]9L; ÀK<? e&zP ©9

;P;

;8

! .%$$'( 2 &L^& gVPuvK<X<k(µε)ε>0

Dgt\]BEDPo(Vo]K<XLKLAxk¹uyTV/^nGIBCT^NDP?²V fVPGIÊF:DPBACD/\(D/XLD/PKwHUVPX^C\IVPHUT X Kw^ .0¾Z±%(.j 5 °M'6( KLgg$DPBtTUº/TUBEk

α > 0AEa]TUBETNT:Kw^xAW^tVvHUDPuy\IV/HJA^nTUA

K ⊆ X tKAEalim sup

ε↓0ε logµε(X \K) < −α.

3 .j 52 & 2&^*,+"-(µε)ε>0

3 + .0/ + V 5 / + / -76 .0LIL]> -76 A?- =<. N6 L]> 5 =V CE5 3`.M3 6 L 6I- > N+ .9U8 CE+ 9 Q = -I?S+8_VDV +"C 3 5 8a/H1 8:9<;_ ?S5 L 19;= 50C .0LIL 4`50N VS. 4"- 9 +"- 9: -I?S+ / 6I- .0L 9 5h?@5 L 19G= 5C .LIL 4 L 5 9 + 19 +"- 9 Q `ba]TBCTÂVPBCTyuV?jkAED0D/X<^ -ta]KwHWa²VBETyGI^nTUg$G]X[K<?*ACa]TÂVPBCTVDgbX<VPBC/TvF:TºjK<VACK<DP?I^ V?(F£taIK<HWa²HUVP?£oTg$DPBEuÀGIX<VACTF£tKLACa]D/G:A¼H D/?I^CK<F:TBCK<?]V¢^n\THUKHy\]BCD/o]X<TUu¢9¸D0^xAdDPgAEa]TBCT^nGIXAW^Na]TUBETa]TXL\ ACDG(^nTÂVT:\DP?]T?0ACKwVXZBWV,ACT+ta]KwHWaK<^VPXLBETV/F:k³j?]D,t? :ojklHVBEBCkjK<?]¼KLAD,ºPTB%AEDÂVyF:KTUBETU?0Af^CKAEGIV,AEKLD/?-9

`ba]T+g$D/XLX<D,tKL?Ijg$BET0G]TU?0AEXLkGI^nTFc-TuvuV^Ca]D,ÂCa(V,AtACaITdT:\DP?]T?/AEK<VPX-BWV,ACTNDPgV^CG]u K<^[xGI^xAfACa]TuV :K<uÀG]u DPgAEa]TNK<?IF:K<º0KwF:GIVPXZBEVACTÛ9

3 .j 52)&2)& 5C ./S> =<. N6 L]> 5 = / 6I-#+ L]> N ./S>O=8a/ 4"-76K5 /@9 f1, . . . , fn : +→ +

F + ? . f+

lim infε↓0

ε log(

n∑

k=1

fk(ε))

≥ maxk=1,...,n

(

lim infε↓0

ε log fk(ε))

./H1

lim supε↓0

ε log(

n∑

k=1

fk(ε))

= maxk=1,...,n

(

lim supε↓0

ε log fk(ε))

.

DPACTOACaITdVP^Ck0uyuyT AEBCklo(TUAxTTU?ACa]T ≥ V?IFACaIT=^nK<P?9:= AK<^tTVP^CkyACDÂHUDP?I^nACBEGIHJAtT:VPuy\]XLT^:taITUBETVÂ^nACBEK<H AfK<?]T0GIVPXLKLAxkaIDPXwF]^g$D/BAEa]T IBW^xA+HUV/^nT/9I`ba]TÀ?ITjAmX<TUuyuVK<XLX<GI^nACBWV,ACT^tAxDl^C\(TH KwVXHVP^CT(ta]TUBET

TOa(VºPT+T0GIVPXLKLAxkyg$DPBtoDAEaoDPGI?IF]^93 .j 5£2)& & *^+<-

f, g : + → +3 +-7F 5 =8a/ 4<-J6K5 /S9T./H1 .99U8 N+T-I? . - +"6I-%?@+"C5 / +P5 =-%?@+-JF 5 4`5 /21 6I-76K5 /@9 lim supε↓0 ε log g(ε) ≤ lim infε↓0 ε log f(ε)

50Clim supε↓0 ε log g(ε) <

lim infε↓0 ε log(

f(ε) + g(ε)) ?@5 L 19 Q R ?@+ / F + ? . f+lim inf

ε↓0ε log

(

f(ε) + g(ε))

= lim infε↓0

ε log f(ε)

./H1lim sup

ε↓0ε log

(

f(ε) + g(ε))

= lim supε↓0

ε log f(ε).

)+,-¨ VM 9-«K<BW^xAdVPÊ^CG]uyT lim supε↓0 ε log g(ε) ≤ lim infε↓0 ε log f(ε)9-«]BEDPu X<TUuyuV8:9Q8ÂT

³j?]D,lim infε↓0 ε log

(

f(ε) + g(ε))

≥ lim infε↓0 ε log f(ε)9µm?£ACa]TDPACa]TBNaIV?IFMg$DPBNTºPTBCk

c >lim infε↓0 ε log f(ε)

V?IFTUº/TUBEkE > 0

T I?IFVP?ε < E

tKLACaf(ε) < exp(c/ε)

V?(F ]ojklHWa]DjD/^CK<?]ε^CGa!ÂH K<TU?0ACX<kl^CuyVPXLX IVPX<^CD¼tKLACa

g(ε) < exp(c/ε)9Ii:DvTdHUVP?HUDP?IHUXLG(F:T

lim infε↓0

ε log(

f(ε) + g(ε))

≤ lim infε↓0

ε log(

2 exp(c/ε))

= c

g$DPBmVX<Xc > lim infε↓0 ε log f(ε)

9(`ba]Kw^\IBCD,º/TÂCaIT IBW^xAmHUX<VPKLu¢9`ba]TÀ^CTH D/?IF¢H XwVK<u Kw^fVF]KLBETH AmH D/?I^nT 0G]TU?(H T+DgXLTuyuyV8]9 8]9 o2 D,V/^C^CG]uyT lim supε↓0 ε log g(ε) < lim infε↓0 ε log

(

f(ε) + g(ε)) V?(FHWa]DjD0^nTNV

c ∈ tKAEalim sup

ε↓0ε log g(ε) < c < lim inf

ε↓0ε log

(

f(ε) + g(ε))

.

;_

`ba]T?¢ACa]TBCTdT:K<^nAE^mV?E > 0

^nGIHWaAEaIV,Amg$DPBTºPTUBEkε < E

TdaIVº/TNo(DPACaf(ε) + g(ε) > 2 exp(c/ε)V?(F

g(ε) < exp(c/ε)9·DP?(^nT/GITU?0ACX<k¹T I?IF

f(ε) > exp(c/ε)g$D/BdVX<X

ε < EV?IF£AEa0G(^+ACa]TX<D,TBNXLK<uyKA¼ÊV,AEK<^ IT^

lim infε↓0 ε log f(ε) ≥ cg$DPBdVX<X

c < lim infε↓0 ε log(

f(ε) + g(ε)) 9`ba]K<^

\]BED,ºPT^lim infε↓0 ε log

(

f(ε) + g(ε))

≤ lim infε↓0 ε log f(ε)V?IF¢VP/VK<?tKAEaAEa]T IBW^nAbT^xAEKLuVACT+g$BEDPu

X<TUuyuVy8]9 8¼T+PT AtAEa]TNT0GIVPXLKLAxkg$D/BbACa]T+X<D,TBoDPGI?IF]^9«]BEDPu¶ACa]Tv^nTH D/?IF\IVBCAfDg|X<TUuyuVl8:9Q8vTd³j?]D, AEaIV,AmACa]TÀBEVACT+g$DPBfACa]Tv^nGIu Kw^tAEa]TduV :K<uÀG]uÁDg

ACaITNKL?IF]KLºjKwF:GIVXBEVACTP9:THVGI^CT+Dglim sup

ε↓0ε log g(ε) < lim inf

ε↓0ε log

(

f(ε) + g(ε))

≤ lim supε↓0

ε log(

f(ε) + g(ε))

ACaITNuyV :KLu¼G]u uÀG(^xAtAEa]TU?¢oTdV,AnAWVK<?]TFlg$DPBlim supε↓0 ε log f(ε)

9]`baIK<^HUDPuy\]X<T AETÂCaITN\]BCDjDPgx9 0TF'

«]D/Bg$G:ACG]BETNBET g$TUBETU?(H T+TN^nAEVACT+ACaITmg$D/XLX<D,tK<?]yoIVP^CK<HOT^xAEKLuV,AETP93 .j 52)&<,& *^+<-

c1, . . . , cn ∈ iQ R ?@+ /c21α1

+ · · · + c2nαn

≥ (c1 + · · · + cn)2∑n

k=1 αk

= 5C .0LIL α1, . . . , αn > 0 ./H1 +m 8S.L 6I- > ?@5 L 19 6 = .0/21 5 /SL]> 6 = -I?S+<CE+h6 9h. λ ∈ FG6I-%?λαk = ck

= 50Ck = 1, . . . , n Q

)+,-¨ & c-TUAa =

∑nk=1 αk

pk = αk/a

IV?IFdk = ck/pk

g$DPBk = 1, . . . , n

9I`ba]T? ∑nk=1 pk = 1V?(F TU?I^CTU? ^bK<?]T0GIVPXLKLAxkÂ/KLº/T^

n∑

k=1

c2kpk

=

n∑

k=1

pkd2k ≥

(

n∑

k=1

pkdk

)2

=(

n∑

k=1

ck

)2 8]9 _d ta]TBCT+T0GIVPXLKLAxkÂa]D/X<FI^bDP?]X<kg$DPB

d1 = · · · = dn9IemKLºjKwF:KL?I 8]9 _d %ojk a \]BCD,º/TÂCaITdH XwVK<u9 0TFS

BET^CG]XLAm^CDPuyTUtaIVAO^CKLuyK<X<VPBAEDlX<TUuyuV8:9Q8yKwÂEa]Tdg$D/XLX<D,tK<?]I9(= AmPK<ºPT^fVP?¹G]\]\TUBOo(D/G]?IF¢g$D/BfAEa]TT:\DP?]T?0ACKwVXZBWV,ACT+DPgVy\]BED:F:GIHJA9

3 .j 52 &w¤ & *^+"-f1, . . . , fn : + → +

.0/21 lim supε↓0 ε log fk(ε) = −c2k = 5C k = 1, . . . , n Q*^+<-α1, . . . , αk > 0 QOR ?@+ / -I?S+ = 5 LIL 5FG6 / A+ 9 -J6IN . -#+ ?@5 L 19"c

lim supε↓0

ε log

n∏

k=1

fk(αkε) ≤ −(∑n

k=1 ck)2

∑nk=1 αk

.

)+,-¨ & `V³jKL?IÀAEa]TN\]BEDjF]GIHJAD/G:AtDgACa]TNX<DP0VBEKAEa]u T I?(F

lim supε↓0

ε log

n∏

k=1

fk(αkε) ≤n∑

k=1

lim supε↓0

ε log fk(αkε)

=

n∑

k=1

1

αklim sup

ε↓0ε log fk(ε)

= −n∑

k=1

1

αkc2k.

\I\]XLkjK<?]yACaITNT^nACK<uyVACTOg$BEDPu X<TUuyuVÂ8:9 r¼\IBCD,º/TÂCa]TdHUX<VPKLu¢9 0TF' `ba]Tvg$DPX<X<D,tKL?]¢X<TUuyuV¢tKLX<X|AEG]BC?*D/G:A+ACDoTºPTUBEk¹GI^CT g$GIX|g$DPBN\]BED,ºjKL?IG]\I\(TBdX<VPBC/TvF]TUºjK<VACK<DP?

oDPG]?IFIÛ9(= Ama]TXL\IÂCDÂ^C\]X<KA+V?G]\]\TUBmo(D/G]?IF¢K<?0ACDlV (?]KAETd?jG]uÀoTUBmDg[HUV/^nTÎta]K<HWaK<?¢ACGIBC?a]TXL\(^tACDV\I\]XLklXLTuvuVÂ8]9 8]9j`baIK<^bAEBCKwHW³ÂKw^bKLX<X<GI^xAEBEVACTFojkAEa]TN\]BEDjDgDPg\]BEDP\D/^CKLACK<DP?8:9 NoTUX<D,N9

;Ur

3 .j 52)&w® & 5Ce+ . 4`? δ > 0-%?@+"CE+Z6 9e. / 6I-#+ 9 +"- Dδ

n ⊆

α ∈ n≥0

∣

∣ α1 + · · · + αn = 1 + δ

FG6I-%?

(ε1, . . . , εn) ∈ n≥0

∣

∣ ε1 + · · · + εn ≤ ε

⊆⋃

α∈Dδn

(ε1, . . . , εn) ∈ n≥0

∣

∣ εj ≤ αjε= 50C j = 1, . . . , n

= 5C .0LIL ε > 0 Q)+,-¨ & c-TUA

δ > 09]THVGI^CTOACa]Td^CKLuy\]X<TK<^HUDPuy\IV/HJA jAEa]TdH D,º/TUBEKL?I

(x1, . . . , xn) ∈ n≥0

∣

∣ x1 + · · · + xn ≤ 1

⊆⋃

α∈ n≥0

‖α‖1=1+δ

(x1, . . . , xn) ∈ n≥0

∣

∣ xj < αjg$D/B

j = 1, . . . , n

HUVP?¹oT¼BETF:G(H TFACDV I?]KLACTvDP?]T/9i:DlT¼HUVP?MHWa]DjD/^CTÀV I?]KLACTy^nTUADδ

n

^nG(HWaACaIVAmAEa]TvKL?IHUXLG(^nK<DP?M^nACK<X<Xa]D/X<F]^fta]TU?¢AEa]TdG]?IKLD/?¹K<^fDP?]X<kÂAWV³/TU?D,ºPTB

α ∈ Dδn

9(¸¹G]XAEKL\IXLkjK<?]loDAEa¹^nKwF:T^ftKAEaε > 0

/KLº/T^bAEa]TH XwVK<u¢9 0TF' c-TuyuyVÂ8]9 ¼Kw^tGI^CT g$G]X IKLg|DP?]TdHV?T:\]X<DPKLAm^CDPuyT+³0K<?IF¢DPgK<?IF:T\(T?IF:TU?(H TÀ^nACBEGIHJAEG]BETP9I`ba]T+g$D/XLX<D,tK<?]

\]BEDP\D/^CKAEKLD/?¢g$TVACG]BET^mVoIVP^CKwHdV\]\IXLKwHUVACK<DP?ta]KwHWaM^Ca]D,^tAEa]TÀuVPHWaIKL?]TBCkV,AOD/BC³9]=@?MHWa(V\:AETUBOTtK<XLX-G(^nTOACaIK<^tKwF:TV¼K<?VvuvD/BCTNHUDPuy\]X<K<HV,ACTF^CKLACGIVACK<DP?-9

)+,I±(&$ 2 & 1 &,*,+"-X1, . . . , Xn

3 +Z6 /H1 + V + /21 + / - :'V 5 9 6I-76If+eC .0/21 5N f . CU6 .M3<L + 9 FG6I-I?lim sup

ε↓0ε logP

(

Xk ≤ ε)

= −c2k

./H1lim inf

ε↓0ε logP

(

Xk ≤ ε)

= −b2kF^?@+"CE+

bk, ck ≥ 0 = 50C k = 1, . . . , n QR ?@+ / F + ? . f+lim sup

ε↓0ε logP

(

X1 + · · · +Xn ≤ ε)

≤ −(c1 + · · · + cn)2.

./H1lim inf

ε↓0ε logP

(

X1 + · · · +Xn ≤ ε)

≥ −(b1 + · · · + bn)2.

)+,-¨ & c-TUAδ > 0

VP?IFDδ

n

V/^bKL?¢X<TUuyuVy8]9 qI9]`ba]Kw^tPK<ºPT^

P(

X1 + · · · +Xn ≤ ε)

≤∑

α∈Dδn

P(

X1 ≤ α1ε, . . . , Xk ≤ αkε)

.

«]D/BbACa]TÀKL?IF]KLºjKwF:GIVXACTBCu^tTdHUVP?G(^nTNX<TUuyuVÂ8:9Q:9IcT Alim supε↓0 ε logP

(

Xk ≤ ε)

= −c2ktKLACa

ck ≥ 09I`ba]TU?TNPT A

lim supε↓0

ε logP(

X1 ≤ α1ε, . . . , Xk ≤ αkε)

= lim supε↓0

ε log

n∏

k=1

P(

Xk ≤ αkε)

≤ − (c1 + · · · + cn)2

1 + δ

;

g$DPBtTºPTBCkα ∈ Dδ

n

9I¡m^nK<?]yXLTuyuyVÂ8]9 8ÀTdH DP?(H X<GIF:Tlim sup

ε↓0ε logP

(

X1 + · · · +Xn ≤ ε)

≤ maxα∈Dδ

n

lim supε↓0

ε logP(

X1 ≤ α1ε, . . . , Xk ≤ αkε)

≤ − (c1 + · · · + cn)2

1 + δg$DPBtTºPTBCk

δ > 0V?(FlAEajGI^

lim supε↓0

ε logP(

X1 + · · · +Xn ≤ ε)

≤ −(c1 + · · · + cn)2.

«]D/BtACaITdXLD,TUBtoDPGI?IF¢XLTUAlim infε↓0 ε logP

(

Xk ≤ ε)

= −b2ktKAEa

bk ≥ 09I«IBCD/u¶X<TUuyuVl8:9 rT

³j?]D,AEaIV,ATN^naIDPG]XwFHWaID0D0^nTαk\IBCD/\(D/BnAEKLD/?IVXACD

bkK<?D/BEF]TUBACD/T Ato(T^xA\D/Ê^CKLo]X<T+o(D/G]?IF

lim infε↓0

ε logP(

X1 + · · · +Xn ≤ ε)

≥ lim infε↓0

ε logP(

Xk ≤ bkb1 + · · · + bn

ε, k = 1, . . . , n)

= lim infε↓0

ε logn∏

k=1

P(

Xk ≤ bkb1 + · · · + bn

ε)

≥n∑

k=1

b1 + · · · + bnbk

lim infε↓0

ε logP(

Xk ≤ ε)

= −n∑

k=1

b1 + · · · + bnbk

b2k

= −(b1 + · · · + bn)2.

`ba]Kw^H D/uy\]XLTUACTÂCa]TN\]BEDjDgx9 0TF' `ba]TdBETUuVPKL?]K<?]y\IVPBnAfDPgACa]Kw^f^CTH ACK<DP?¹^CG]uyuyVPBCKw^CT^t^CDPuyTdK<uv\DPBCAEVP?0ABCT^nG]XLAE^g$BCD/u AEa]TÀX<KAETUBWV,AEG]BCT/9

`ba]TÂHUDP?0ACBWVPH ACK<DP? \]BCK<?IHUKL\]X<TÂVX<X<D,ÂED¢ACBWV?I^ng$DPBEu V?²cefY´D/? D/?]TÂ^C\IVPHUT¼AED¹V?]DPACa]TBd^n\(VPH TojkuyTVP?I^fDPg%VlHUDP?0ACK<?jG]DPGI^fuV\]\]K<?](9`ba]T¼g$DPX<X<D,tKL?]ÂACa]TDPBETUu AEa]TUD/BCTuÁr(9 8]9L;ÀKL? e&zP % m^xAWV,ACTÂEa]TBET^CG]XA9

d.jI,.j 2 & ¿ 45 / -JC . 4"-76K5 /TV CU6 / 4<6 VHL + Q *,+"- X .0/21 Y 3 + .089<1 5C -#5 V 5 L 5ÀD6K4 .0L 9JVS. 4+ 9h.0/21f : X → Y 3 + . 45 / -J6 /S8 5 89^=8a/ 4"-J6K5 / Q 5 /S9 6 1 +<C . Ad5 5 1 C . - + =8a/ 4<-J6K5 / I : X → [0;∞] Q . 5Ce+ . 4? y ∈ Y 1 + / +

I ′(y) = inf

I(x)∣

∣ x ∈ X , f(x) = y

.

R ?@+ / I ′ 6 9Z. Ad5 5 1 C . - + =8a/ 4<-J6K5 / 5 / Y Q 3 = I 45 / -7CE5 L 9 -%?@+ * X .99 5_4"6 . -#+ 1 FG6I-I? . =<. N\6 L > 5 = V CE5 3`.M3 6 L 6I- > N+ .9U8 CE+ 9 (µε)5 / X : -I?S+ /

I ′45 / -JCE5 L 9 -%?@+ *,X .99 5_4<6 . - + 1 FG6I-%?-%?@+ =<. N6 L]> 5 =GV CE5 3.D3 6 L 6I- > NT+ .98 C`+ 9 (µε f−1)

5 / Y Q= gfK<^f?]DAOKL?,xTHJAEKLº/TNACa]T?ACaITÀuV\]\IKL?]xX<DjD/^CT^bK<?:g$DPBEuV,ACK<DP?W9`ba0G(^tACaITÀH D/?/AEBEV/HJAEKLD/?\IBCK<?IH K<\]X<T

HUVP?¹AEBEVP?I^n\DPBCANV?*cemY g$BEDPu VXwVBEPTUB+^C\IVPHUTÀAED¢V^CuyVPXLX<TUB+D/?]TP9Z`ba]TeOV^nD/? R PBnAE?]TUBmACaITUDPBETUuF:DjTÂEa]TNDP\I\(D0^nKLACT/9:= Aa]TUX<\I^bAED¼AEBEVP?I^C\(D/BnAV?¢cemYg$BEDPu¶^CuVX<XLTBb^C\IV/H TÂEDV¼XwVBEPTN^C\IVPHUTP9=@?¢DPBWF:TUBAEDyg$DPBEuÀG]XwV,AET+ACa]T+AEa]TUD/BCTu¶TO?ITUTFAEa]TÀH D/?IH T\:ADg|V?\]BEDxTHJACK<ºPT+X<KLuyKLA9c-T A

(J,≤)oT

Vv\IVBCACKwVX<X<kÂDPBWF:TUBETF^nTUA I^CGIHWaACaIVAta]TU?ITUºPTBi, j ∈ J

ACaITUBETOT:K<^nAE^tVk ∈ J

tKAEai ≤ k

V?IFj ≤ k

9 ±,.j$7(.&°-&U.0 Kw^fVvgVuyK<XLk

(Yj)j∈JDg mVGIÊF:DPB¹AEDP\DPX<DPPKwHUVPX-^n\(VPH T^]ACDP/T AEa]TUBtKLACa¹VgVuyK<XLk

(pij)i,j∈JDPgHUDP?0ACK<?jG]DPGIûV\(^

pij : Yj → YiI^CGIHWaACaIVA

pik = pij pjkta]T?]TUº/TUB

i, j, k ∈ YtKLACai ≤ j ≤ k

9

;q

`ba]T ±,.j$7(.¢' DPgAEa]T¼^Ck:^xAETUu(Yj , pij)i,j∈J

Kw^tACaITÀ^nGIoI^nTUA X Dg[VPXLXTUX<TUuyTU?0AE^g$BCD/u AEa]T\]BED:F:GIHJA+^n\(VPH T Y =

∏

j∈J Yj(ta]KwHWaMVBETÀH D/?I^CK<^nACT?/AmtKAEaACa]T¼uV\I^

pi,j(K©9 T/9V

y = (yj)j∈J ∈ YKw^mK<? X ZKgbV?IFD/?]X<kKgyi = pij(yj)

ta]T?]TUº/TUBi < j

9-`ba]TyHUV?IDP?]KwHUVPX\]BEDxTHJACK<DP?(^pj : X → YjVBETyF:TI?ITF£o0k

pj(y) = yj9-`baITy^C\IV/H T X KwÔT0G]K<\]\(TF tKAEaMAEa]TvACDP\DPX<DP/k¢KL?IF]GIH TF£ojk Y 9-`baITHUVP?]DP?IK<HVX\]BCDPxTH ACK<DP?IÔVBET¼HUDP?0ACK<?jG]DPGIÔoTHVGI^CTdAEa]TUkMVPBCTfxGI^nAOAEa]TyBCT^xAEBCKwHJAEKLD/?¹DgACa]TyH D/?0ACK<?0GIDPGI^

H DjD/BEF:K<?IVACTOuyVP\I^ Y → YjACDyAEa]TN^n\(VPH T X 9`ba]TÀAxk0\IK<HVXT]Vuy\]X<TÀa]TBCT¼K<^

JoTUK<?]lAEa]Ty^nTUAmDg%VPXLX I?IKAETv^CG]oI^CT AW^fDPg%V?MK<?0ACTBCº,VX

[0; t] ⊆ tKLACaAEa]Tv^CT AOKL?(H X<GI^nK<DP?¹V/ÂEa]TÀ\(VBCACKwVXD/BEF]TUBEKL?](9(`ba]T¼\]BEDxTH ACK<ºPTd^Ck:^xAETUuK<^fACaITU?AEa]TÀgVPuyKLX<k¢Dg[VX<XI?]KLACTvF:K<uvT?I^CKLD/?IVX^n\IV/H T^ j g$DPB I?]KLACT¼^CT AW^ j ⊆ [0; t]

IK©9 T/9Ig$D/Bj ∈ J

9=@?¢ACaIK<ÔHUV/^nT+AEa]TÀ\]BEDxTHJAEKLº/TX<KLuyKLA X K<^NACa]TÂ^C\IVPHUTDgtVX<X|g$G]?IH ACK<DP?I^

[0; t] → T0G]K<\]\(TF*tKAEa ACa]TAEDP\DPX<DP/k¹Dgb\(D/KL?0AxtKw^nTH D/?jºPTUBEPT?IH T/9

`ba]TNg$D/XLX<D,tK<?]ACaITUDPBETUu AEa]TUD/BCTurI9 q]9<;NKL? e&zP 0I T:\]XwVK<?Iâ]D, AEDÂACBWV?(^n\DPBCAfV?McefY g$BCD/uACaITd^n\IV/H T^ Yj

AED¼AEa]TN\]BEDxTHJACK<ºPTOX<KLuyKLA X 9d.jI,.j 2 & Ã ( X . F 9 5 / CU- / +"C *^+"- (µε)ε>0

3 + . =<. N6 L]> 5 = V C`5 3.D3 6 L 6I- > NT+ .9U8 CE+ 9 5 / X Q 99U8 N+e-%? . - = 5C+ . 4`? j ∈ J-%?@+ V C`5 3.D3 6 L 6I- > N+ .9U8 CE+ 9 (µε p−1

j )ε>05 / Yj

9<. -76 9%=> -I?S+B*,XFG6I-%?iAd5 5 1 C . -#+ =8a/ 4"-76K5 / Ij Q R ?S+ / -%?@+ =<. N6 L]> (µε)9". -76 9 + 9 -%?@+e* X 5 / X FG6I-%? . Ad5 5 1 C . -#+=8a/ 4"-J6K5 / I :G1 + / + 13<>

I(x) = sup

Ij(pj(x))∣

∣ j ∈ J = 50C .0LIL x ∈ X Q

?IDACaITUB+KLuy\DPBCAEVP?/A+AED0D/X|K<ÔAEa]T[VBWVPF]aIV? c-TUuyuV `ba]TDPBETUu r(9 _I9L;vKL? ez/ % J9-cT AZε

o(TBWV?IF:D/u º,VBEK<VPo]X<T^ (AWV³jK<?]¢ºVPXLGIT^NK<? ACa]TBETU/G]X<VPBOACDP\DPX<DP/K<HVX|^C\IVPHUT X V?IF X<T A

µεo(TyAEa]TXwV

DgZε9

d.jI,.j 2)&' ;( . C .M1 ? .0/ *,+"N\N . W 8_VDV 5 9 +\-I? . - (µε)9". -J6 9 + 9 -I?S+Z*,XFG6I-I? . Ad5_5 1C . - + =8a/ 4<-J6K5 / I : X → [0;∞] : ./21L +<- ϕ : X → 3 + ./S> 45 / -76 /'8 5 89;=8a/ 4"-76K5 / Q 99U8 NT+ =8 CU-%?@+"C+"6I-I?S+<CB-I?S+Z- . 6 L 45 /H1 6I-J6K5 /

limM→∞

lim supε↓0

ε logE(

exp(ϕ(Zε)/ε)(ϕ(Zε) ≥M))

= −∞,

5CB-%?@+ = 5 LIL 5FG6 / APN5N+ / - 45 /21 6I-76K5 /h= 5C 9 5N+ γ > 1 :lim sup

ε↓0ε logE

(

exp(γϕ(Zε)/ε))

<∞.

R ?@+ /limε↓0

ε logE(

exp(ϕ(Zε)/ε))

= supx∈X

(

ϕ(x) − I(x))

.

& ! # # ! ! "#=@?¢ACaIK<^m^nTHJACK<DP?¢TNtKLX<X\IBCD,º/T+V^CK<uv\IXLTdXwVBEPTNF:TºjK<VACK<DP?\]BEKL?(H K<\]XLTNtaIK<HWa¹F:T^CHUBCK<oT^bAEa]TÀVP^Ckjuy\:ACDPACKwHoTUaIVºjK<DPG]BbDgACaITd^xAWV,ACK<DP?(VBEkÂF:K<^nACBEK<o]G:ACK<DP?Dg|VyF:K GI^CK<DP?¢\]BEDjHUTÊ^ta]TU?ACaITdF:BCKLgAoTH D/uyT^t^nACBEDP?]/TUB9

`ba]TN\]BEDjDgtKLX<XoT¼V^CK<uv\IXLTÀVP\]\]X<K<HV,AEKLD/?DPgACa]TÀcV\]XwVPHUT Y[BEKL?IHUKL\IXLT:ta]K<HWaKw^tg$D/BCu¼G]XwV,ACTFKL?¢AEa]Tg$DPX<X<D,tKL?]X<TUuyuV]9`ba]Kw^Kw^^nK<uyKLXwVBACDAEa]Th[VPBEV/F:aIV?X<TUuyuV AEa]TUD/BCTu8]9L;~M J9£Td/KLº/TdV?K<?IF:T\(T? F:T?/A\IBCDjDga]TUBET:oTHVGI^CTOACa]TN\IBCDjDgK<^t^CK<uv\IXLTdVP?IFl?ITUºPTBnAEa]TUX<TÊ^bPK<ºPT^b^nD/uvT+K<?I^CKL/a/A93 .j 52 & 2& * .V2L . 4+ V CU6 / 4"6 V2L + *,+"- A ⊆ d 3 +\NT+ .98 C .D3"L + ./H1 ϕ : d → 3 + .N+ .9U8 C .D3"L + =8a/ 4<-J6K5 / FG6I-%? ∫

A −ϕ(x) dx <∞ Q R ?S+ / F + ? . f+

limϑ→∞

1

ϑlog

∫

A −ϑϕ(x) dx = − ess inf

x∈Aϕ(x).

;

)+,-¨ & efT?]DAETÀAEa]Tc-TUoT^CPGITÀuyTV/^nG]BETvo0kλd 9Z«K<BW^xANHWa]DjD/^CT c > ess infx∈A ϕ(x)

9`ba]TU?£TaIVº/T

λdx ∈ A | ϕ(x) < c > 0V?IFACajGI^

lim infϑ→∞

1

ϑlog

∫

A −ϑϕ(x) dx ≥ lim inf

ϑ→∞

1

ϑlog(

−ϑcλdx ∈ A | ϕ(x) < c )

= −c.

`ba]Kw^ta]D/X<F]^g$D/BVX<Xc > ess infx∈A ϕ(x)

(ta]KwHWa/KLº/T^

lim infϑ→∞

1

ϑlog

∫

A −ϑϕ(x) dx ≥ − ess inf

x∈Aϕ(x).

«]D/Bess infx∈A ϕ(x) = −∞ TUº/TUBEk/AEa]K<?]lKw^+H X<TVB9ZfÊ^nGIuvT

ess infx∈A ϕ(x) > −∞ 9`baITU? o0kVPFIF:KL?IlVH D/?I^xAWV?0AAED

ϕTÀHV?¹VPÊ^CG]uyT

ess infx∈A ϕ(x) = 0tKAEa]DPG:AOXLD0^C^Dg|PT?]TUBWVX<KAxk/9ITHVGI^CT

−ϑϕ(x) g$DPB ϑ→ ∞ H D/?0º/TUBEPT^VI9 ^9:uyDP?]DPACDP?IK<HVX<XLkg$BEDPu VoD,ºPTmACDyACaITNKL?IF]K<HV,ACD/Bg$G]?IHJAEKLD/?DgACa]Td^CT Ax ∈ A | ϕ(x) = 0 V?IFKw^bo(D/G]?IF:TFo0kÂACa]TNK<?0ACTU/BEVPo]X<Tmg$GI?IHJAEKLD/? −ϕ(x) TNaIVº/T

limϑ→∞

∫

A −ϑϕ(x) dx = λdx ∈ A | ϕ(x) = 0

≤ λdx ∈ A | ϕ(x) ≤ 0 ≤∫

A −ϕ(x) dx <∞

V?(FÂAEa0G(^tT+/T AtACaITOGI\]\(TBoDPG]?IF

lim supϑ→∞

1

ϑlog

∫

A −ϑϕ(x) dx ≤ 0 = − ess inf

x∈Aϕ(x).

0TF' f^+VP? VPXLuyD/^nAfAEBCK<ºjK<VPX|H DP?(H X<GI^nK<DP?£DgACa]TcVP\]XwVPH T¼\]BEKL?IHUKL\IXLTyTvHV?£F]TUBEKLº/TvVX<VPBC/T¼F:TºjK<VACK<DP?

\]BEKL?(H K<\]XLTvg$D/Bd^xAWV?IF]VPBEFM?]DPBEuVX|F:Kw^xAEBCK<o]G:AETF BWV?IF:D/u º,VBEK<VPo]XLTÔKL? 9-`ba]Kw^+Kw^+KLX<XLG(^xAEBEVACTF£K<?£AEa]Tg$DPX<X<D,tKL?]yH D/BCD/XLXwVBEkP9»¼I, 5 ° 2 & ]2)& = X 6 9e.\9 - .0/21D. C 1/ 50CUN .0L C ./H1 50N f . CU6 .D3"L + : -I?S+ /

limε↓0

ε logP (√εX ∈ A) = − ess inf

x∈A

x2

2

= 5C +"f+"C > NT+ .9U8 C .M3<L + 9 +<- A ⊆ iQ)+,-¨ & `ba]TNF:Kw^xAEBCK<o]G:AEKLD/?¢Dg √

εXaIVP^tF:T?I^CKAxk

ψ0,ε(x) =1√2πε

exp(

−x2

2ε

)

.

`bajGI^btKLACaϑ = 1/ε

AEa]TNcVP\]X<V/H T+\]BEKL?(H K<\]XLT+/KLº/T^

limε↓0

ε logP (√εX ∈ A) = lim

ε↓0ε log

∫

A

exp(

−x2

2ε

)

dx− limε↓0

ε log√

2πε

= − ess infx∈A

x2

2+ 0.

0TF' ½m¾ 5 ²±. 2)&' '& c-T A

Bo(TyVDP?]T F:K<uyTU?I^CK<DP?IVPXBED,t?]K<VP?uyDACK<DP?9`baITU? Bt

Kw^NR+VGIÊ^nKwV?£F:K<^ ACBEK<o]G:ACTF tKAEa²T:\(THJAWV,ACK<DP?0V?(F£º,VPBCKwV?IHUT

t9¡m^nK<?]¢AEa]TÂH D/BCD/XLXwVBEk¢TÂHV? T]VuyK<?]TyACa]TXwVBEPT

F:Tº0KwV,AEKLD/?oTUa(Vº0K<DPGIBDPgAEa]TNTUº/TU?0A |Bt| > c g$DPB c→ ∞ 9 KAEaε = 1/c2

T+aIVº/T|Bt| > c ⇐⇒ √

ε · Bt/√t ∈

x ∈ ∣

∣ |x| > 1/√t

;

V?(FÂAEa0G(^lim

c→∞1

c2logP

(

|B1| > c)

= − 1

2t.

«]D/BmHUDPuy\IVBEKw^nD/?tKAEa¹T]Vuy\]X<Tv8]9 8Âo(TXLD, T¼VPX<^CDÂ?]DPACT¼ACaIVAfACaIT¼\]BEDPo(Vo]K<XLKLAxkDg |Bt| < εg$DPB

ε ↓ 0F:THUVk:^b^nX<D,TBACaIVP?Tj\DP?ITU?0ACKwVX<XLk/9 £TNa(VºPT

limε↓0

ε2 logP(

|B1| < ε)

= limε↓0

ε2 log

∫ ε

−ε

1√2π

−x2/2 dx

= limε↓0

ε2 log(

2ε1√2π

)

= 0.

`ba]T+uVK<?¢BCT^nG]XLAtDgACa]Kw^^CTHJAEKLD/?Kw^V/^g$DPX<XLD,^9d.jI,.j 2 & ([*^+"-

Φ: d → 3 + 1 6 +<C`+ / -76 .M3<L + ./21i9U8 4? -%? . - exp(−2Φ(x))6 9.

V C`5 3.D3 6 L 6I- >1 + /@9 6I- > 5 / d Q *,+"- Φ 3 + 3 5 8a/21 + 1 = CE5N 3 + L 5F[FG6I-%?Φ∗ = infΦ(x) | x ∈ d >

−∞ Q 6 /2.0LIL]>TL +<- b = − gradΦ 3 + *G6 V'9 4?6I-%bh45 / -76 /'8 5 89 QR ?@+ / = 5Ce+<f+"C > ϑ ≥ 1-%?@+ 9 -#5 4? .9 -J6K4 1 6 +<C`+ / -76 .0L +`m 8S. -J6K5 /

dXϑ = ϑ b(Xϑ) dt+ dW? .9Z.9 - . -J6K5 /H. C >1 6 9 -7CU6 3"8 -J6K5 / µϑ

.0/21 = 50Ce+"f+<C > NT+ .98 C .D3"L + 9 +"- A ⊆ d F + ? . f+lim

ϑ→∞

1

ϑlogµϑ(A) = − ess inf

x∈A2(

Φ(x) − Φ∗)

.

)+,-¨ & 0VK<? :X<T Aλd F:TU?IDACT+AEa]Tdc-To(T^n/G]T+uyTVP^CG]BETmD/? d V?(FF:TI?]T

Zϑ =

∫

d

exp(

−2ϑΦ(x))

dx.

`ba]T?T+aIVºPTZϑ =

∫

Φ>0exp(

−2ϑΦ(x))

dx+

∫

Φ≤0exp(

−2ϑΦ(x))

dx

≤∫

Φ>0exp(

−2Φ(x))

dx+ λdΦ ≤ 0 · exp(−2ϑΦ∗)

<∞.

¡f^CK<?]Zϑ

TNHUVP?F:TI?]TΦϑ = ϑ · Φ + ln

√Zϑ9]`ba]Kw^t?]T \DACT?0ACKwVX-aIV/^

∫

exp(−2Φϑ) dx =

∫

exp(−2ϑΦ) exp(− lnZϑ) dx = 1

V?(F− gradΦϑ = −ϑ gradΦ = ϑ · b.kmDPX<uvD/PD/BCD,º ^bACa]TDPBETUu ACa]TDPBETUu ;/9 r AEa]Ty\]BEDjHUTÊ^

Xϑ Kw^+BCTºPTUBW^CKLo]X<T¼VP?IF£aIVP^+V¢^xAWV,AEKLD/?IVBEkF:Kw^xAEBCK<o]G:AEKLD/?µϑtKLACaF:TU?(^nKLAxk

exp(−2ϑΦ)/Zϑ9

¡f^CK<?]vACa]Kw^F:T?I^nKLAxkÂT+PT A

limϑ→∞

1

ϑlogµϑ(A) = lim

ϑ→∞

1

ϑlog

∫

A

exp(−2ϑΦ(x))1

Zϑdx

= limϑ→∞

1

ϑlog

∫

A

exp(

−2ϑΦ(x))

dx− limϑ→∞

1

ϑlogZϑ

= − ess infx∈A

2Φ(x) + ess infx∈ d

2Φ(x),

ta]TBCTNAEa]TÀXwVP^nAfX<KL?ITdK<ÔVÂHUDP?I^CT0G]T?IH TNDPg[XLTuvuVl8]9L;/;P9I«K<XLX<K<?]lK<?¢ACa]TvF:T(?]KAEKLD/?¹DgΦ∗

I?]Kw^naIT^tAEa]T\]BEDjDgx9 0TF'

;z

DAEa jACa]TNX<TUuyuVyV?IFAEa]T+ACa]TDPBETUu a(VºPTOBCT^nG]XLAE^bDPgAEa]TOg$DPBEu

limϑ→∞

1


x∈AI(x) 8]9 r

g$DPB+VX<XuyTVP^CG]BWVo]X<TÀ^CT AWÂ ⊆ d VP?IFg$D/BO^CDPuyTdg$GI?IHJAEKLD/? I 9ZfÊ^nG]uyT¼ACaIVA+VÂgVuyK<XLkDPg%uyTVP^CG]BETâIV/^tACaITÀ\]BEDP\TUBCAxk 8:9 rd 9`ba]T?T¼HUV?MGI^nT infx∈A I(x) ≤ ess infx∈A I(x)

AEDlPTUAACaITÀGI^CGIVXG]\]\TUBoDPG]?IFg$BEDPu AEa]TNXwVBEPT+F:TUºjKwV,ACK<DP?\]BEKL?(H K<\]XLT:g$DPBHUXLD0^nTF^CT AW^

A ⊆ d T+aIVº/T

lim supϑ→∞

1


x∈AI(x) ≤ − inf

x∈AI(x).

µm?¹AEa]TyDAEa]TUB+aIVP?IFMTyF:DP?- A+PT A+AEa]TyXLD,TUBOo(D/G]?IF£KL?MACa]Ty/TU?]TBEVPXHVP^CTP9µm?]X<kKL?*HUV/^nT¼ACaIVANTaIVº/T

infx∈O I(x) = ess infx∈O I(x)g$DPBOTUºPTBCkDP\TU?M^CT A

O ⊆ d TP9 I9(Kg I Kw^mHUDP?0ACK<?jG]DPGI^ ]T¼HV?H D/?IH X<GIF:TOAEa]TNXLD,TUBboDPG]?(Flim infϑ→∞

1

ϑlogµϑ(O) ≥ − inf

x∈OI(x)

g$DPBtTºPTBCkÂD/\(T?^nTUAO VPH ACGIVPXLX<klK<?ACa]Kw^HUV/^nT+TNaIVºPT+TºPTU?T0GIVPXLKLAxklaITUBET_ J9ijDV,AX<TVP^nAtg$DPBHUDP?0ACK<?jG DPG(^bBEVACTOg$G]?IH ACK<DP?IÎAEa]TdH D/?IF:KLACK<DP? 8]9 r KLuy\]X<KLT^bACaITdcemYhtKLACa¢BWV,AETmg$GI?IHJAEKLD/? I 9`ba]T+Kw^t?]DAbAEBCG]TNK<?PTU?ITUBWVX©9:= gT+aIVºPT+V?¢cemY KL?AEa]T+g$DPBEu DgF:T(?]KAEKLD/?¹8:9Q8 jACaITU?ACa]TNX<K<uvKLA

limϑ→∞

1

ϑlogµϑ(A)

F:DjT^b?IDA?]TH TÊÊVBEKLX<kT:Kw^xA9]´^nTUAAta]TUBET+T+aIVº/T

infx∈A

I(x) = infx∈A

I(x),

K©9 T/9]ta]TUBETOACa]TNX<K<uvKLAfF:DjT^bT:K<^nA ]Kw^tHUVPXLX<TFV P'ZE°²&.0 DPgAEa]TNBWV,ACTOg$G]?(HJACK<DP?9

& " ! "#`ba]TyTUuy\]K<BEK<HVX|F:Kw^xAEBCK<o]G:AEKLD/?£DPgVBCTºPTBE^CKLoIXLTvF:KGI^CKLD/? HUDP?jºPTBC/TÂCDACa]T^nAEVACK<DP?IVPBCk¹F:Kw^nACBEKLo]G]ACK<DP?ta]T?

t→ ∞ 9]=@?lAEa]Kw^^nTHJAEKLD/?TNtKLX<XZPK<ºPTNVvXwVBEPT+F:TUºjKwV,ACK<DP?BET^CG]XAbg$D/BbACa]Kw^HUV/^nT/9`ba]T .j²±$,' 5 |&U,$

Lωt ∈ Prob( d)

DPg[V\]BEDjHUTÊ^XtKLACa¹º,VX<G]T^K<? d K<ÔF:T(?]TFojk

Lωt (A) =

1

tλd

s ∈ [0, t]∣

∣ ωs ∈ A g$DPBVPXLX

ω ∈ C(

[0;∞), d) A ∈ B( d),

ta]TBCTλd F:T?]DAET^tAEa]T d F:KLuyT?I^nK<DP?(VXc-TUoT^CPGITduyTVP^CG]BETP9ijKL?(H T Lt

K<^mVÂuV\]\IKL?]lg$BCD/u¶AEa]TÀ\(V,ACa^C\IVPHUT

C(

[0;∞), d) K<?0ACD

Prob( d)TvHV?¹GI?IF:TUBW^nAEV?(F

LtVPÔVlBWV?IF]DPu\]BEDPoIVPo]KLX<KLAxkuvTVP^CG]BET

tKLACaH D/BCBET^C\(D/?IF:K<?]v\]BCD/oIVoIKLX<KAxkÂ^C\IV/H TC(

[0;∞), d) 9

c-TUAXoTdVyBETUºPTBE^CK<o]XLT+F:K G(^nK<DP?¢\]BED:H TÊ^btKLACa^xAWV,AEKLD/?IVBEklF:Kw^xAEBCK<o]G:AEKLD/?

µ9]=@?ACa]Kw^^CKAEGIV,AEKLD/?AEa]T DPBETUuÁq]9Q8:9Q8:;+Dg|emTUGIÊHWa]TUXV?IFi0AEBCDjD:HW³ eOi] PzV\]\IXLK<T^9I`ba]Td?IDAEVACK<DP?ACaITUBETNK<^VÂXLKLAnAEXLTdoIKAmF]KTUBETU?0A

g$BEDPuÁDPG]BWÎACa]TKLBVKw^D/G]B

2vVP?IF

UACaITUBET¼H D/BCBET^C\DP?IF]ÂED

2Φa]TBCT/9`baITNACa]TDPBETUuT:\]BETÊ^CT^bAEa]T

BWV,ACTNtKLACa¢AEa]Tda]TXL\DPgACa]TÀemK<BCKwHWa]X<T Ag$D/BCu E HJgx9(o(TXLD, Ita]KwHWaK<^fVPÊ^nD:H KwV,AETFtKAEaACaITd\]BEDjHUTÊ^ X F:TI?]T

JE : Prob( d) → [0;∞]o0k

JE(ν) :=

E(f, f),KLgν µ

tKAEadν = f2dµ

g$DPBVf ∈ D(E)

(VP?IF+∞ TX<^CTP9

d.jI,.j 2 & j;(G*^+<-Pν

3 +-%?@+ 1 6 9 -7CU6 3"8 -76K5 / 5 =T. C`+<f+<C 9 6 3"L + 9 5 L 8 -76K5 / 5 = -%?@+ 9 -#5 4? .9 -76K41 6 +<CE+ / -J6 .L +m 8S. -J6K5 /

dXt = b(Xt) dt+ dBt 8]9 M FG6I-%?j6 / 6I-J6 .L1 6 9 -7CU6 3"8 -J6K5 / L(X0) = ν ./H1T9 - . -J6K5 /2. C >i1 6 9 -JCU6 3<8 -J6K5 / µ Q X+ / + v = (b2 + div b)/2./H1.99U8 NT+e-I? . - x ∈ d | v(x) ≤ c 6 9 4`50N VS. 4"- = 50Ce+ . 4`? c ≥ 0 Q 5 /@9 6 1 +"C Prob( d)+m 8 6 VDV + 1

8~

FG6I-%? -%?@+\F + . -#5 V 5 L 5À >j.0/21 -I?S+ 50CE+ L σ .L Ad+ 3 C . QBR ?@+ / JE 6 9\. Ad5 5 1 C . -#+ =8a/ 4<-J6K5 / ./H1 -I?S+= 5 LIL 5FG6 / AT* Xl?@5 L 19"c

− infΓJE ≤ inf

ν∈Prob(

d)lim inft→∞

1

tlogPν

(

ω | Lωt ∈ Γ

)

≤ supν∈Prob(

d)

lim supt→∞

1

tlogPν

(

ω | Lωt ∈ Γ

)

≤ − infΓJE

= 5C .0LIL N+ .9U8 C .M3<L + 9 +"- 9 Γ ⊆ Prob( d) Q`ba]TNemKLBEK<HWaIXLTUA g$DPBEu E g$D/BbACa]TN\IBCD:H T^C^ X HUVP?oTdH D/?I^xAEBCG(HJACTFV/^g$DPX<XLD,^9afKwV

Ptf(x) = Ex

(

f(Xt))

g$DPBOVX<Xx ∈ d V?IFVPXLX t ≥ 0

Td/T A+VÂ^xAEBCD/?]PX<k¢H DP?0AEKL?jG]D/GI^tDP\TUBWV,AEDPBf^CTUuyK<PBEDPG]\(Pt)t≥0

DP?¢AEa]T^C\IVPHUT L2( d,B( d), µ)

9I`ba]TN/TU?]TBEVACDPBLDgACa]Kw^t^CTUuyKL/BCD/G]\¢^CVACKw^ IT^

Lf = b · ∇f +1

2∆f

g$DPBdVPXLXf ∈ C2

c ( d)9THUVPGI^CT

XKw^+BCTºPTUBW^CKLo]X<TAEa]TPT?]TUBWV,AEDPB

LKw^N^nTXg VPF,xDPK<?/A9 D,F:TI?ITÂV0GIVPF]BEVACKwHg$D/BCu E0

ojkE0(f, g) :=

1

2

∫

d

∇f · ∇g dµg$DPBVPXLX

f, g ∈ C∞c ( d, )

93 .j 52 & ]¤)& *,+"-

X 3 + . C`+<f+<C 9 6 3"L + 9 5 L]8 -J6K5 / 5 = -%?@+ZWXZY 8:9QD FG6I-%?P*G6 V'9 4`?a6I-Ib 45 / -76 /'8 5 891 CU6 = - b : d → d ./H1\9 - . -76K5 /2. C >1 6 9 -JCU6 3<8 -J6K5 / µ QOR ?@+ / -%?@+ m 8S.D1 C . -J6K4 = 50CUN E09<. -76 9 O+ 9

E0(f, g) = (−Lf, g)µ

= 5C .0LIL f, g ∈ C∞c ( d) : F^?S+<CE+ ( · , · )µ

6 9 -%?@+ 9 4 .L . C V CE5 18 4"-5 / L2( d,B( d), µ) QtHUHUDPBWF:K<?]lAEDACa]TDPBETUu¼9 8P_DPgbªTUTF V?(F*i:KLuyDP? ªfiI8%BCT^n\9ACa]TDPBETUu qI9 8]9 zDgtefTGIÊHWa]TUX

V?(F¹i0ACBEDjDjHW³ IACa]Tv/G(VPF:BWV,AEK<HOg$D/BCu E0aIVP^mVÂH X<D/^CG]BET (E ,D(E)

) 9£TÀaIVºPTf ∈ D(E)

KLg%V?(F¢DP?]X<kKLgACaITUBETvVPBCT

fn ∈ C∞c ( d)

V?IFg1, . . . , gd ∈ L2( d,B( d), µ)

tKLACafn → f

V?(F∂jfn → gj

KL?L2( d,B( d), µ)

g$DPBn→ ∞ VP?IF

j = 1, . . . , d9]=@?ACa]Kw^HVP^CTOT+PTUA

E(f, f) = limn→∞

E0(fn, fn) =1

2

∫

d

d∑

j=1

g2j dµ.

a]K<XLT+AEa]TN\]BETUºjK<DPGI^t^CTH ACK<DP?IÂEBCTV,ACTFÂAEa]TNXwVBEPT+F:TUºjKwV,ACK<DP?oTUaIVºjK<DPG]BtDPg^CDPuyTNF:TUBEK<ºPTFl\]BEDP\TUBCACK<T^bDgVy\]BED:H TÊ^ ]KAOK<^fVXw^nD\D/Ê^nK<o]X<TNACDÂHUDP?I^CKwF:TUBtAEa]TdXwVBEPTdF:TºjK<VACK<DP?¢o(TaIVºjKLD/G]BDPgAEa]Td\]BED:H T^C^bKLAE^CTUXLg (K©9 TP9g$DPBbAEa]TdF:Kw^xAEBCK<o]G:AEKLD/?¢DgAEa]TN\]BED:H TÊ^CT^\IV,AEaI^bDP?AEa]Td^n\(VPH TODg|VX<X-ÊVuy\]X<TO\(V,ACa(Û9

`ba]TNoIV/^nKwHOX<VPBC/TNF:TUºjKwV,ACK<DP?BET^CG]XAaITUBETNK<^fi]HWa]KLXwF:TB ÂCa]TDPBETUuÁVoDPG:AXwVBEPTNF:Tº0KwV,AEKLD/?I^g$DPBÊHUVPXLTFF:D,t?BCD,t?IK<VP?uyDAEKLD/? ^CTUT+ACaITUDPBETUu:9Q8:9<;+KL? eZzP % J9k

C0([0; t], d)TdF:TU?]DPACT+AEa]TÀ^C\IVPHUTNDg

VX<X-H D/?0ACK<?0GIDPGI^g$G]?(HJACK<DP?(^ω : [0; t] → d ^xAWVBCACK<?]yKL?¢~IT0G]K<\]\(TFtKAEaAEa]Td^CG]\]BETUuÀGIu ?]DPBEu¢9

d.jI,.j 2 & :® W)4`?a6 L 1 +"C Q *,+"- B 3 + .9 - ./21M. C 1 CE50F / 6 ./ NT50-76K5 / Q 50C ε > 0 L +"- ε

3 +-%?@+ L . F 5 = -I?S+ 9 4 .L + 1 1 50F / V CE5 4+ 99 √εB Q R ?@+ / -%?@+iNT+ .98 C`+ 9 ε9". -J6 9%=> 5 /

(

C0([0; t], d), ‖ · ‖∞) .0/ * X FG6I-%?TAd5 5 1 C . - + =8a/ 4<-J6K5 /

I(ω) =

12

∫ t

0|ωs|2 ds,

6 = ω 6 9e.D3<9 5 L]8 -#+ L]> 45 / -76 /'8 5 89:G.0/21+∞ + L 9 + Q

8:;

½m¾ 5 ²±. 2)&2)& ·DP?I^CKwF:TUBNVDP?]T F:K<uyTU?I^CK<DP?IVPXBED,t?]K<VP?¹uyDAEKLD/?£D/?¹ACaIT¼AEKLuyTyK<?/AETUBEº,VX [0; t]9

TÂHUV?*GI^CTli:HWa]K<XwF:TUB ÔAEa]TUD/BCTu ACD¹HVXwH G]XwV,AETvACa]TÂT:\(D/?]TU?0ACKwVX[F]THUVkMBEVACT^+DPgAEa]T\]BEDPoIVPo]K<XLKLAxkP(

‖B‖∞ > c) g$D/B

c→ ∞ 9 KLACa

ε = 1/c2T+aIVºPT

sup0≤s≤t

|Bs| > c ⇐⇒ √εB ∈

ω∣

∣ |ωs| > 1g$D/B^nD/uyT

s ∈ [0; t]

=: A

V?(FloTHVGI^CTAK<^VyHUDP?0ACK<?jG]KAxk^nTUADgAEa]TNBWV,ACTOg$G]?(HJACK<DP?

IT I?IF

limc→∞

1

c2logP

(

‖B‖∞ > c)

= limε↓0

ε logP(√εB ∈ A

)

= − inf1

2

∫ t

0

ω2s ds

∣

∣

∣ ω ∈ A

= −1

2

∫ t

0

1/t2 dt = − 1

2t,

ta]TBCTNTNGI^CTFACaITNgVPH AACaIVAACaITdKL?IuÀG]uÁKw^fV,ACAEVPKL?]TFg$DPBωs = s/t

9·DPuy\IVPBCK<?]vACaIK<^ftKAEaT]Vu \]X<TÀ8:9<;+T+?]DPACKwH T+ACa(V,AtACaITNT:\(D/?]TU?0AEK<VPXZBEVACTOg$DPBsup0≤s≤t |Bs| > c

Kw^bAEa]Td^CVPuyT+V/ÂCa]TNT:\(D/?]TU?0ACKwVXBWV,ACTOg$D/BbACa]TNTºPTU?0A |Bt| > c

9µm?¢ACaITÀDAEa]TUBfaIV?IF¹KAOK<^f?]DAf\D/Ê^CKLo]X<TdAEDACBETVA

P(

‖B‖∞ < ε) g$DPB

ε ↓ 0AEa]TÀÊVuyTdbVkP9i:KL?IHUT

TOa(VºPTsup

0≤s≤t|Bs| < ε ⇐⇒ 1

εB ∈

ω∣

∣ sup0≤s≤t

|ωs| < 1

a]TBCTITND/G]X<F¢?ITUTFAEa]TdXwVBEPTdF:TºjK<VACK<DP?¢o(TaIVºjKLD/G]Btg$D/BtACa]T¼o]X<D,t? GI\¢BED,t?]KwV?uvDPACK<DP?K<?I^nACTV/FDg|g$D/BfAEa]TdAEa]Ty^CHVX<TF F:D,t?BCD,t?]KwV?uyDPACK<DP?-9£TvHUVP?MF:TUBEK<ºPTNAEa]T¼XwVBEPTdF]TUºjK<VACK<DP?¹oTUa(Vº0K<DPGIBfDgACaIK<^N^C\(TH KwVXTUº/TU?0A+?]TUº/TUBCACa]TXLT^C^ (oTHVGI^CTyV? T:\]X<K<HUKAOg$DPBEuÀG]XwVlg$DPBOACa]Ty\]BEDPo(Vo]K<XLKLAxk¢K<^+³j?]D,t?9(=@?^CTHJAEKLD/? ¼9 \-9I_Pr08D %Dg «]TUX$j;<KAKw^t^Ca]D,t?lAEaIV,A

P(

|Bs| ≤ εg$DPBVPXLX

s ∈ [0; t])

=4

π

∞∑

n=0

1

2n+ 1exp(

− (2n+ 1)2π2

8ε2t)

sin((2n+ 1)π

2

)

.

`ba]TdF:D/uyKL?IVACK<?]vACTBCu K<?ACa]Kw^^CG]u¶Kw^4

π

1

2 · 0 + 1exp(

− (2 · 0 + 1)2π2

8ε2t)

sin((2 · 0 + 1)π

2

)

=4

πexp(

− π2

8ε2t)

ta]KwHWaHUDPBEBCT^n\DP?(F]ÂCDn = 0

9]«IDPBbACaIT+AEVK<X-DPgACaITd^nG]u T I?(FlAEa]TNT^nACK<uV,AET∞∑

n=1

1

2n+ 1exp(

− (2n+ 1)2π2

8ε2t)

sin((2n+ 1)π

2

)

≤∞∑

n=1

exp(

− (2n+ 1)2π2

8ε2t)

≤∞∑

n=1

exp(

−π2t

2ε2n)

=exp(

−π2t2ε2

)

1 − exp(

−π2t2ε2

)

V?(FÂAEa0G(^lim sup

ε↓0ε2 log

4

π

∞∑

n=1

1

2n+ 1exp(

− (2n+ 1)2π2

8ε2t)

sin( (2n+ 1)π

2

)

≤ −π2t

2< −π

2t

8.

¡f^CK<?]yXLTuvuVÂ8]9 _¼TN/T Alimε↓0

ε2 logP(

‖B‖∞ < ε)

= limε↓0

ε2 log4

πexp(

− π2

8ε2t)

= −π2t

8.

`ba]Kw^NH D/KL?IHUK<F]T^mtKAEaMAEa]TH DPBEBET^C\(D/?IF:K<?]lBET^CG]XLAmg$BEDPu X<TUuyuV8:9 _lK<? c-Kw~];U 9-`ba]TyoTUaIVºjK<DPG]BOa]TUBETKwÔF]KTUBETU?0A+ACaIVP?MK<?£T]Vuy\]X<Ty8]9L;/9`ba]Ty\]BEDPo(Vo]K<XLKLAxk¢Dg

sup0≤s≤t |Bs| < εF:THVk:^fTj\DP?ITU?0ACKwVX<XLk

g$DPBε ↓ 0

taIKLX<TOACa]TN\]BEDPo(Vo]K<XLKLAxkg$D/B |Bt| < εF:DjT^b?]DPA9

8P8

´/TU?]TBEVPXLKw^CVACK<DP?DgACaITUDPBETUu8:9<;qyKw^tAEa]TÀ^CD HUVX<X<TF¢«]BETUKwF:X<KL? TU?0ATUX<X`ba]TDPBEk ^CTUT¼HWaIV\:AETUBm]9 qDg ez/ -g$D/BfVF:TUAEVK<X<TFT:\]XwV?IVACK<DP?' 9I`ba]TBCT]DP?]TdHUDP?I^CKwF:TUBW^tV^xAED:HWaIVP^nACKwHNF:K TBCT?/AEK<VPX-T0GIVACK<DP?¢DgACaIT+g$DPBEudXε

t = b(Xεt ) dt+

√ε dWt

g$DPBt ∈ [0; 1]

]V?IFXε

0 = 0,

ta]TBCTb : → K<^G]?]KLg$DPBEuvX<kcKL\(^CHWa]KLA/9afKwVV?¢V\]\IXLKwHUVACK<DP?DgACa]TdHUDP?0ACBWVPH ACK<DP?l\IBCK<?IH K<\]X<TNDP?]TNHV?

H D/?IH X<GIF:TOg$BEDPu AEa]TUD/BCTu 8:9<;q¼AEa]T+g$DPX<XLD,tK<?]vBCT^nGIXA9d.jI,.j 2)&' 1 CE+"6 1L 6 / T+ / -Ib_+ LIL Q R ?@+ =<. N6 L]> (Xε

t ) 9". -76 9 + 9 -I?S+B*,X 6 / C0[0; t]FG6I-I?

-%?@+ Ad5 5 1 C . - + =8a/ 4<-J6K5 /

I(ω) =

12

∫ t

0|ωs − b(ωs)|2 ds,

6 = ω ∈ H1:G.0/21

+∞ + L 9 + Q«]D/BtACaITÀ\]BEDjDg ]AEa]TÀHUDP?0ACBWVPH ACK<DP?¢\]BEKL?IHUKL\IXLTdKw^fVP\]\]X<KLTF¢ACDÂACa]TduVP\

F : C0[0; t] → C0[0; t]ta]TUBET

f = F (g)K<^tF:TI?]TFlAEDo(T+AEa]TNG]?]Kw0G]TN^nD/XLG]ACK<DP?DgACa]TND/BEF:K<?IVPBCkÂF:K TUBETU?0ACKwVXZT/G(V,ACK<DP?

f(t) =

∫ t

0

b(f(s)) ds+ g(t)g$D/BVX<X

t ∈ [0; 1]9

«]GIBnAEa]TUBt/TU?]TBEVPXLKw^CVACK<DP?I^VBET+\(D0^C^CK<o]XLT:TP9 I9jAED¼AEa]TdHUV/^nT+DPgAEa]Tdi:efsdXε

t = b(Xεt ) dt+ √

εσ(Xεt ) dWt

g$DPBt ∈ [0; 1]

]V?IFXε

0 = x,

ta]TBCTx ∈ d b : d → d K<^fG]?]KLg$DPBEuyXLk¢c-K<\IÊHWa]KAIVP?IF¢ta]TBCTÀVPXLXH D/uv\DP?ITU?0AE^tDPgACa]TduVACBEK σVBET+o(D/G]?IF:TF ]G]?IKg$D/BCuyX<klc-K<\IÊHWa]KANH DP?0AEKL?jG]D/GI^g$G]?IH ACK<DP?I^9

>§ t Á

=@?¹AEa]KwÔHWa(V\:AETUBfTvtKLX<XG(^nT¼ACa]T¼ACaITµmBE?I^nACTUK<? ¡a]X<TU?joTHW³\]BEDjHUTÊÂEDlTj\IX<VPKL?M^CDPuyT¼Dg[AEa]Tv0G]T^ ACK<DP?(^fta]KwHWaMTÀtK<X<XVP?I^nTUBmuyDPBETÀPT?]TUBWVX<X<kK<?¹XwV,ACTBOHWa(V\:AETUBWÛ9= AmtK<XLXAEG]BC?£DPG:AmACaIVAmAEa]T¼^CK<uv\IXLT^nACBEGIHJAEG]BCTÀDg|ACaITµmBE?I^nACTUK<? ¡a]X<TU?joTHW³\]BEDjHUTÊ^tK<XLXTV/^nTÀuyVP?jk¢HVXwH G]XwV,AEKLD/?I^ ]o]G:A+Tv^xAEKLX<XHUV?£^nTTtaIVAb³jK<?IFDgBCT^nGIXAW^bTNHUVP?T:\(THJAtg$DPBbAEa]TNPT?]TUBWVXZHVP^CTP9

! "#c-TUA

BoT¼V

d F:KLuyT?I^nK<DP?(VXBED,t?]KwV?¢uyDACK<DP?¹VP?IF α > 0oT¼VBETVPXº,VX<G]TF¢\(VBWVuyT AETUB9I`ba]TU?¹ACa]T

^CDPX<G:ACK<DP?DgACa]TN^nACD:HWaIV/^xAEK<H+F:K TBCT?0ACKwVXZT0GIVACK<DP?dXt = −αXt dt+ dBt, _I9L; X0 = x0 ∈ d

Kw^HUVPXLX<TFACaIT y,&U.0'-W.j%.j ±,-/.0&& tKLACa\IVPBEVPuyT ACTBαVP?IF¢^xAWVBCAKL?

x0 ^CTUT:g$DPBT]Vu \]X<T]º9 TUK ^PHW³/TUBtV?IF K<?]³jXLTB z/~ ]\-9-;~M J9]¸¹D/^nACX<kÂT+tK<XLXH DP?(^nKwF:TUBbAEa]TdHUV/^nTx0 = 0

9¡f^CK<?]ÂAEa]TÀBET^CG]XLAE^fg$BCD/uHWa(V\:AETUBv;ÀKLAOKw^mHUXLTVBfACaIVAmAEa]K<ÔT/G(V,ACK<DP?a(VPÔVÂG]?]Kw0G]Ty^nD/XLG:AEKLD/?-9G]A

GI^CKL?I¼AEa]TNº,VBEK<VACK<DP?DgH DP?(^xAWV?0AE^buyT AEa]D:FTNHUVP?TºPT?Tj\IXLKwH KLACX<k^CDPX<ºPTmACa]Kw^tT/G(V,ACK<DP?9]c-T A

Xt = −αtX0 +Bt − α

∫ t

0 −α(t−s)Bs ds

DPBtT0G]KLº,VPXLT?/AEXLkXt = −αtX0 +

∫ t

0 −α(t−s) dBs _I9 8M

g$DPBOVX<Xt ≥ 0

9(`baITU?¹KLAmKw^TVP^CklAEDlHWa]THW³ACa(V,AfAEa]TÀ\IBCD:H T^C^XF:T(?]TF¢ojkACa]Kw^ (Kw^fV^nD/XLG:AEKLD/?DgACaIT

i:ems _I9L; J9«K<PGIBCTd_I9L;d^Ca]D,^ Iº/TN\IV,AEaI^DPg[V?£µmBE?I^xAETUK<? ¡a]X<TU?joTHW³Â\]BED:H T^C^ttKLACa\(VBWVuyT AETUB α =1V?IF¢^nAEVPBnAtK<?~I9= gTdF:TI?]TOACa]TN\DAETU?0ACKwVX

Φ: d → VP^

Φ(x) =α

2|x|2 − d

4log

α

π

g$D/BVX<Xx ∈ d

ACaITU?ACa]TdF]BCKLgAb(x) = −αx HUV?¢oTNT:\]BETÊ^nTFlV/^ b = − gradΦ

V?IFT I?(F

exp(

−2Φ(x))

=(α

π

)d/2

exp(

−α|x|2)

=1

(

2π 12α

)d/2exp(

−|x|22 1

2α

)

.

KLACa¹AEa]TUD/BCTu;P9 rlT¼HUVP?MH D/?IH X<GIF:TÀACaIVAmAEa]Tv\]BCD:HUTÊ^XKw^fBETUº/TUBW^nK<o]X<TÀV?(FaIVPÔV

d F]KLuyTU?(^nK<DP?IVPX?]D/BCuVX-F]K<^nACBEKLoIG:ACK<DP?¢tKLACa¹H D,º,VPBCKwV?IHUTmuVACBEK 12αId

V/^bKAW^^nAEV,AEKLD/?IVBEklF:Kw^nACBEKLo]G]ACK<DP?-9I«IBCD/u HWaIVP\:ACTBN;T+VPXLBETV/F:k³j?]D,AEa]TdF:T?I^nKLAxkÂDPgAEa]TdF:Kw^xAEBCK<o]G:AEKLD/?D/?AEa]TN\IVACa¢^n\IV/H T

ϕt(ω) = exp(

−α2

(ω2t − ω2

0 − t · d) − α2

2

∫ t

0

ω2s ds

)

. _I9 _d

8_

8,r

0 2 4 6 8 10

−0.4

−0.2

0

0.2

0.4

t

Xt

«KL/G]BET¼_]9<; R ?6 9 A 8 C`+ 9 ?S5F 9 f+ V@. -I? 9 5 =./ C /@9 - +"6 / ? L + /23 +4 BV CE5 4+ 99 5 / -I?S+h6 / - +"CUf .L [0; 10]FG6I-%? V@. C . N+<-#+"C α = 1 Q ! # =@?MHWaIVP\:ACTBOÂTdtK<X<XVP?I^CTBtACa]TÀg$DPX<XLD,tK<?]l0G]T^xAEKLD/? taIV,AOK<^fACa]T¼X<VPBC/TÀF:Tº0KwV,AEKLD/?IôTUa(Vº0K<DPGIBfDgACaITvT?IF:\DPK<?0A

XtDPgVF:KGI^CKLD/?£\IBCD:H T^C^ KgTvK<?IH BETV/^nTÀACa]TF:BEKgA«]D/BNV?µmBC?I^nACTKL? ¡fa]XLT?jo(THW³\]BED:H TÊÂCa]Kw^tuyTVP?IÂCa(V,ATOBETU\IX<V/H TOACa]TdHUDP?I^nAEVP?0A

αtKAEa

ϑαg$D/B

ϑ > 0VP?IFÂAWV³/T

ϑ→ ∞ AEa]TU?-9c-TUA

Xϑ o(TvACaITy^CDPX<G:AEKLD/?MDg _I9L; mtKAEa£\IVPBEVPuyT ACTB ϑα VP?IFM^nAEVPBnA+KL?*~]9Z`ba]Ty\]BED:H T^C^maIVP^mAEa]TF:T?I^nKLAxk

ϕϑt (ω) = exp

(

−ϑα2

(ω2t − ω2

0 − t · d) − ϑ2α2

2

∫ t

0

ω2s ds

)

= exp(

ϑF (ω) − ϑ2G(ω))

tKLACaF (ω) =

α

2(ω2

0 − ω2t + d)

V?(F

G(ω) =α2

2

∫ t

0

ω2s ds.

`ba]TÀHVP^CTNDg[V?£µmBE?I^nACTKL? ¡aIXLT?0oTHW³\]BEDjHUTÊ^tKw^T^C\TH KwVX<X<k^CKLuy\]X<Tda]TBCTIoTHUVPGI^CTdTÀ³0?ID, AEa]TT:\]X<KwH KLAmF:Kw^xAEBCK<o]G:AEKLD/?DPgXϑ

t

9I= AfK<^fVd F:K<uvT?I^CKLD/?IVXR+VPGIÊ^nKwV?¢F:Kw^xAEBCK<o]G:AEKLD/?tKLACaT:\(THJAEVACK<DP?~V?IFH D,º,VPBCKwV?IHUTfuV,AEBCK

Σ =1

2αϑ

(

1 − exp(−2αϑ t))

· Id

^CTUT¼g$DPB+T:VPuy\]XLTy^CTHJAEKLD/?£ I9 _Dg BE?,rI 9-¡f^CKL?IAEa]TycV\IX<V/H TvY[BCK<?IHUKL\]X<T X<TUuyuV8]9L;/;_ fDP?]TyHUVP?TV/^nK<X<kl^nTTOACaIVAlim

ϑ→∞

1

ϑlogP (Xϑ

t ∈ A) = − ess infx∈A

αx2 _I9 r g$DPBVPXLX

t > 0a]D/X<FIÛ9 DACT+AEaIV,AtAEa]Kw^tBEVACTNF:DjT^b?IDAF:TU\TU?(FD/? t 9

TBCTÀT¼VP?/AmACDPK<ºPTÀVP?]DAEa]TUBm\]BCDjDPgg$D/BfAEa]TvDP?]T F:K<uyTU?I^CK<DP?IVPXHUV/^nT(ta]KwHWaMF:DjT^?]DPAOGI^CTdAEa]TT:\]X<KwH KLA+F:Kw^nACBEKLo]G]ACK<DP?MDPgXϑ

t

o]G:AdHUVP?¹oT¼/TU?]TBEVPXLKw^nTFACDuvD/BCT¼PT?]TUBWVXF:BCKLgA ITUXwF]^b9Z`DTV/^nTÀACa]T

?]DPAEV,AEKLD/?T+DP?]X<klHUDP?I^CK<F]TUBbACaITdHUVP^CTt = 1

9]«]D/BCu¼G]XwV ;;P9 z]9 0 [g$BCD/u ti:z/q^nAEVACT^

Ex

(

exp(

−γ2

2

∫ t

0

B2s ds

)

;Bt ∈ dz)

=

√γ

√

2π sinh(tγ)exp(

− (x2 + z2)γ cosh(tγ) − 2xzγ

2 sinh(tγ)

)

,

8P

ta]KwHWa/KLº/T^∫

1A(ω1) exp(

−ϑ2α2

2

∫ 1

0

W 2s ds

)

d (ω)

=

∫

A

√ϑα

√

2π sinh(ϑα)exp(

−z2ϑα cosh(ϑα)

2 sinh(ϑα)

)

dz

K<?D/G]BHUV/^nT/9:`ba]K<^K<^tVvPT?]TUBWVX<Kw^CVACK<DP?lDPgAEa]T¼·VuyTUBEDP? ¸¹VPBnAEKL? «IDPBEuÀG]XwVE(

e−λR

10

B2t dt)

=(

cosh√

2λ)−1/2

^CTUT ªOz/z HWaIV\]ACTUB m= J9 £TlVPBCTyK<?0ACTUBET^nACTF KL? ACaITÂT:\DP?]T?/AEK<VPX|AEVPKLXw^NDPgAEa]K<^dT:\]BETÊ^nK<DP?£g$DPBϑ→ ∞ 9

ªtTHUVX<X<KL?]¼ACaITdF:TI?IKAEKLD/?I^

sinh(x) = x − −x

2

VP?IFcosh(x) = x + −x

2TOD/oI^CTUBEºPTmACaIVAtACa]TBCTNVBETNH D/?I^xAWV?0AE^0 < c1 < c2

tKAEa

c1 −αϑ/2 ≤√α

√

2π sinh(ϑα)≤ c2 −αϑ/2 g$DPBVX<X

ϑ > 19 _I9 M

`baITNºVPXLGITv;+Kw^VBEo]KLACBWVBEk:V?jk\D/^CKAEKLº/T+?0GIuÀoTUBDPG]XwFF:DI9 Xw^nDT I?IFcosh(ϑα)

sinh(ϑα)= ϑα + −ϑα

ϑα − −ϑα−→ 1

g$DPBϑ→ ∞ 9 _]9 qM

THVGI^CT _]9QD bVP?IF _]9 qM VBET+KL?(F:TU\TU?IF]TU?0ADg|V?IFlACajGI^G]?]KLg$DPBEu KL?zTNHUV?¢H D/?IH X<GIF:T

limϑ→∞

1

ϑlog

∫

1A(ω1) exp(

−ϑ2α2

2

∫ 1

0

W 2s ds

)

d (ω)

= limϑ→∞

1

ϑlog

√ϑ

∫

A

√α

√

2π sinh(ϑα)exp(

−z2α

2· cosh(ϑα)

sinh(ϑα)· ϑ)

dz

= limϑ→∞

1

ϑlog

∫

A

exp(

−α2ϑ− z2α

2ϑ)

dz

= − ess infz∈A

(α

2+z2α

2

)

= −α2

(1 + ess infz∈A

z2). _I9QD `ba]Kw^tKwÂCa]TdV/^nkjuy\:ACDPACKwHOo(TaIVºjKLD/G]Bg$DPBbACaIT

exp(ϑ2G)ACTUBEu¢9

D, :TdH D/?I^CK<F:TBbACa]Tdg$G]XLXF:TU?(^nKLAxklojk ]ACACK<?]ÂK<?¢ACa]T+gV/HJAEDPBexp(ϑF )

9(`ba]Kw^tKw^tTV/^nko(THUVG(^nTOACaITg$G]?IH ACK<DP?

FDP?]X<klF:T\(T?IF]^tD/?lAEa]TNTU?IF]\(D/KL?0A

zDgACa]TN\IVACa-9£TN/T A

limϑ→∞

1

ϑlogP (Xϑ

1 ∈ A)

= limϑ→∞

1

ϑlog

∫

1A(ω1) exp(

ϑF (ω) − ϑ2G(ω))

d (ω)

= limϑ→∞

1

ϑlog

√ϑ

∫

A

√α

√

2π sinh(ϑα)

· exp(α

2(1 − z2) · ϑ− z2α

2· cosh(ϑα)

sinh(ϑα)· ϑ)

dz

= limϑ→∞

1

ϑlog

∫

A

exp(

−α2ϑ+ (

α

2− z2α

2) · ϑ− z2α

2ϑ)

dz

= − ess infz∈A

αz2.

8q

`ba]Kw^tKwÂCa]TNTj\TH ACTFBET^CG]XLA9f^duvT?0ACK<DP?]TF VPo(D,º/TvACa]TÂ^nAEVACK<DP?IVPBCkMF]K<^nACBEKLoIG:ACK<DP?

µϑDgACa]TµmBE?I^nACTKL? ¡aIXLT?0oTHW³M\]BCD:HUTÊ^tKLACa¹\IVPBEVPuyT ACTB

ϑαKw^fV

d F:KLuyT?I^nK<DP?(VXR+VPGI^C^CKwV?¹F:Kw^xAEBCK<o]G:AEKLD/?tKAEa¹uyTVP?¹~ÂV?IFHUD,º,VBEK<VP?IH T+uV ACBEK 12ϑαId

9`bajGI^tAEa]TdBET^CG]XLAg$BCD/u¶AEa]TUD/BCTu8]9L;_H D/KL?IHUK<F]T^ttKLACaAEa]TÀXwVBEPTdF]TUºjK<VACK<DP?¢BET^CG]XAOVoDPG:AR+VG(^C^CK<VP?F]K<^nACBEKLoIG:ACK<DP?I^g$BEDPu H DPBEDPX<XwVBEk8]9L;8dK<?ACa]Kw^HVP^CTP9]¡f^CK<?]yTUKLACa]TBtBCT^nG]XLAT (?IF

limϑ→∞

µϑ(A) = − infz∈A

αz2

g$DPBmTUº/TUBEkluyTV/^nG]BWVoIXLTN^CT AA ⊆ 9ijK<?IH TÀACa]TdBEK<Pa0AfaIVP?IF^nKwF:TÀHUDPK<?IH KwF:T^ttKAEaACaITÀBWV,ACT+g$BEDPu _I9 r g$DPBVP?jk

t > 0ACa]TNXwVBEPT+F:TUºjKwV,AEKLD/?oTUaIVºjK<DPG]BbDPgAEa]TN\(D/KL?0A

Xϑt

g$DPBϑ→ ∞ K<^tACa]TdÊVuyTNVPÂCaITNX<VPBC/T

F:Tº0KwV,AEKLD/?oTUa(Vº0K<DPGIBDPgAEa]Td^nAEV,AEKLD/?IVBEkÂF:Kw^xAEBCK<o]G:AEKLD/?-9

Á Á

=@?AEa]K<ÔHWaIVP\:ACTBfT¼^nACGIF:kg$D/Bf\D/^CKAEKLº/TÀBWV?IF:D/u º,VPBCKwVo]X<T^tACa]TvBCTX<VACK<DP?oT AxTUT?¢ACa]Tvo(TaIVºjKLD/G]BDPgACaIT¼cVP\]XwVPH T+AEBEVP?I^ng$DPBEu ?]TVBK<? I?IKAxkV?(FACa]TvF:K<^nACBEK<o]G:ACK<DP?M?]TVPB UTBCD(9(`ba]TDPBETUu^tDg|AEa]Kw^f³jK<?IFVBETHUVPXLX<TF`VG]oTUBEK<VP?lAEa]TUD/BCTuÛ9

BCT^nGIXADPgF:TdBEG]K x?¢^Ca]D,ÂCaIVA

E( −λX) ∼ r√

λ g$DPBλ → ∞ V?IF

P (X ≤ ε) ∼ s/ε g$DPBε ↓ 0

VBETyKL?²^CDPuyT^nT?I^CTT0G]K<ºVPXLT?0ANVP?IF PK<ºPTÔV¢BETUXwV,AEKLD/?£oT AxTUT? ACa]TÂHUDP?I^nAEVP?/AW^rVP?IF

s9 TPK<ºPT

^CaIVBE\¹oDPG]?IFI^fg$D/BfAEa]TvG]\]\TUBNV?IFMXLD,TUBmXLK<uyKAW^mg$DPBOACa]KwÔBCTX<VACK<DP?-9`ba]T^nT¼BCT^nG]XLAE^mtK<XLXACGIBC?£DPG:AOACDoT¼V\D,TBng$GIX-ACDjD/XAEDF:TUACTUBEuyKL?ITNACa]T¼X<VPBC/TÀF:Tº0KwV,AEKLD/?o(TaIVºjKLD/G]BDPg[BEVP?IF:DPuÁº,VBEK<VPo]X<T^tta]TBCTdACa]TcVP\]XwVPH TmACBWV?(^xg$D/BCu Kw^b³j?]D,t?-9

# «]BEDPu F:TdBCG]K x?- ^t`VPG]oTUBEK<VP?lAEa]TUD/BCTu TNHV?TVP^CK<XLklHUDP?IHUXLG(F:TOACa]T+g$D/XLX<D,tKL?IvBET^CG]XLA9

d.jI,.j ^& ( *^+<-X ≥ 0 3 + . C ./H1 5N f . CU6 .D3"L + .0/21 A .0/ +<f+ / - FG6I-%? P (A) > 0 QOR ?@+ / -%?@+L 6IN6I-

r = limλ→∞

1√λ

logE( −λX · 1A)

+ D6 9 - 9 6 =e./H1 5 /SL > 6 =s = lim

ε→0ε logP (X ≤ ε, A)

+ D6 9 - 9B./H1 6 / -I?a6 9 4 .9 +ZF + ? . f+ s = −r2/4 Q)+,-¨ & =@? `baITUDPBETUu rI9<;8:9 zg$BEDPu tR+`t 0_[XLTUA

α = −1φ(x) = 1/x

ψ(x) = 1/x2 -VP?IF

B = |s| 9`ba]Kw^f/KLº/T^tAEa]T¼HVP^CTdDg A = Ω9«]D/Bm/TU?]TBEVPX^CT AE^

AT¼^CtKLAEHWaAEDÂACaITÀuyTV/^nG]BET

Q( · ) =P ( · ∩ A)/P (A)

V?IFAEa]T¼HUDPBEBCT^n\DP?IF]KL?]yTj\TH AEVACK<DP?-9`ba]K<^fBCTF:GIHUT^bAEa]TÀHUV/^nTÀDg|PT?]TUBWVXAACDyACaIT

IBW^xAHVP^CTP9 0TF' a]K<XLTÂF]TVX<K<?]¢tKAEa T:\]BCT^C^CK<DP?IÔX<KL³/T¼AEa]T

P (X ≤ ε, A)VoD,ºPTvTytK<XLX[g$BCT/GITU?0ACX<k¹GI^CTyACa]Tg$DPX<X<D,tKL?]¼ACBEK<º0KwVX-ÊHUVPXLK<?]y\]BEDP\TUBCAxkP9

3 .j 5Â,&2)& . 99U8 N+B-I? . - s = limε↓0 ε logP (X ≤ ε)+M6 9 - 9 Q R ?@+ /h= 5C +"f+"C > c > 0 : F +? . f+

limε↓0

ε logP (cX ≤ ε) = sc

8/

8

./H1limε↓0

ε logP (X ≤ cε) = s/c.

3 R ?@+ 9". N+ZCE+ L . -76K5 /@9 ?@5 L 1 = 5CB-I?S+ lim sup .0/21 -%?@+ lim inf Q # #

L2 #

=@?¢ACaIK<^^CTH ACK<DP?¢TNPK<ºPTdV IBW^xAfVP\]\]X<K<HV,AEKLD/?DPgAEa]TÀ`VG]oTUBACa]TDPBETUu g$BEDPuÁ^nTHJAEKLD/?r(9L;/9]kV\I\]XLkjK<?]ACaIT+ACa]TDPBETUu AED¼AEa]TNBWV?IF:D/u º,VBEK<VPo]X<T

X =

∫ t

0

B2s ds

ta]TBCTBKw^mVP?MD/?]T F:K<uvT?I^CKLD/?IVXBED,t?]KwV?uyDAEKLD/?¹T¼HUVP?MF:TBCK<ºPTvV?McefY g$DPB+BED,t?]KwV?\IVACaI^H D/?IF:KLACK<DP?]TFÂAEDaIVºPT+^CuyVPXLX

L2 ?]DPBEu¢9`ba]TIBW^xA+^xAETU\MDPg[ACaIK<^+\]BEDP/BEVPuyuvTÀK<^fACD¢HUVPX<HUG]X<VACT¼ACa]T¼AEVPKLXwÔDg[AEa]TycV\IX<V/H TdAEBEVP?I^ng$DPBEuÁDPgX9

«]D/BCu¼G]XwV ;P9 z]9 0 %g$BCD/u$ ti:zPq_^nAEVACT^

Ex

(

exp(

−γ2

2

∫ t

0

B2s ds

)

;Bt ∈ dz)

= ϕ(x; t, z)

ta]TBCTϕ(x; t, z) =

√γ

√

2π sinh(tγ)exp(

− (x2 + z2)γ cosh(tγ) − 2xzγ

2 sinh(tγ)

)

.

«]D/BfV^nAEVBCACK<?]y\DPK<?/AxIuyTVP^CG]BEVPo]X<T+^CT AW^

A1, . . . , An ⊆ IVP?IF :TFACK<uyT^0 < t1 < · · · < tn = tACaITd¸¹VBE³PD,ºy\]BEDP\TUBCAxkÂDgBCD,t?]KwV?uyDACK<DP?PK<ºPTÂCa]T?

Ex

(

exp(

−γ2

2

∫ t

0

B2s ds

)

1A1(Bt1) · · · 1An(Btn

))

=

∫

A1

· · ·∫

An

ϕ(x; t1, z1)ϕ(z1; t2 − t1, z2)

· · ·ϕ(zn−1; tn − tn−1, zn) dzn · · · dz1. TÀVBET+DP?]X<kK<?0ACTBCT^xAETFKL?¢AEa]TdTj\DP?ITU?0ACKwVXAEVK<Xw^tDgACaIK<^fT:\]BETÊ^nK<DP?g$DPB

γ → ∞ 9«K<BW^xAfD/oI^CTUBEºPTACa(V,AtACaITUBET+VPBCTNHUDP?I^nAEVP?/AW^0 < c1 < c2

V?(FG > 0

tKLACa

c1 −γt/2 ≤ 1√

2π sinh(γt)≤ c2 −γt/2 g$DPBVPXLX

γ > G9

`ba]T?TNHUVP?GI^CTOACa]TNBETUXwV,AEKLD/? |2xy| ≤ x2 + y2 AEDPT A(x2 + z2)

2· cosh(γt) − 1

sinh(γt)≤ (x2 + z2) cosh(γt) − 2xz

2 sinh(γt)≤ (x2 + z2)

2· cosh(γt) + 1

sinh(γt)

g$DPBVPXLXx, z ∈ 9c-TUAε > 0

9]THUVPGI^CTmDPgcosh(γt) ± 1

sinh(γt)= γt + −γt ± 1

γt − −γt−→ 1

g$D/Bγ → ∞ 9

T+HV?ACa]T? I?IF¢Vγ0 > 0

]^CGIHWaACaIVAta]TU?]TºPTBγ > γ0

ACa]TNT^xAEKLuVACT(x2 + z2)

2· (1 − ε) ≤ (x2 + z2) cosh(γt) − 2xz

2 sinh(γt)≤ (x2 + z2)

2· (1 + ε)

L2 8z

a]D/X<F]^g$D/BVX<Xx, z ∈ 9 £TdHUDP?IHUXLGIF]T

lim supγ→∞

1

γlogEx

(

exp(

−γ2

2

∫ t

0

B2s ds

)

1A1(Bt1) · · · 1An(Btn

))

≤ limγ→∞

1

γlog γn/2cn2

∫

A1

· · ·∫

An

−γt1/2 exp(

−γ x2 + z2

1

2(1 − ε)

)

· −γ(t2−t1)/2 exp(

−γ z21 + z2

2

2(1 − ε)

)

· · · ·

· −γ(tn−tn−1)/2 exp(

−γ z2n−1 + z2

n

2(1 − ε)

)

dzn · · · dz1

= limγ→∞

1

γlog

∫

A1

· · ·∫

An

exp(

−γtn/2− γ(x2/2 + z21 + · · ·

· · · + z2n−1 + z2

n/2)(1− ε))

dzn · · · dz1. DPACTyAEa]TÂ^n\THUK<VPX[B PX<TvDPgAEa]TTU?(F:\(D/KL?0A tn = t

9 KAEa*ACa]TaITUX<\²Dg%AEa]TlcV\]XwVPHUTv\IBCK<?IH K<\]X<T ^CTUTX<TUuyuVy8]9L;/;_ T+HV?¢HUVPX<HUG]X<VACTOACaITNXLK<uyKAD/?AEa]TNBEKL/a/AbaIVP?IF¢^nKwF:TOACD/T A

lim supγ→∞

1

γlogEx

(

exp(

−γ2

2

∫ t

0

B2s ds

)

1A1(Bt1) · · · 1An(Btn

))

≤ − ess infz1∈A1,...,zn∈An

(

t/2 + (x2/2 + z21 + · · · + z2

n−1 + z2n/2)(1 − ε)

)

.

g$DPBVPXLXε > 0

V?(FÂAEa0G(^

lim supγ→∞

1

γlogEx

(

exp(

−γ2

2

∫ t

0

B2s ds

)

1A1(Bt1) · · · 1An(Btn

))

≤ − ess infz1∈A1,...,zn∈An

(t/2 + x2/2 + z21 + · · · + z2

n−1 + z2n/2).

H D/uv\IXLTUACTUX<kVP?IVX<DP/DPGI^HUVPX<HUG]XwV,ACK<DP? G(^nK<?]vACa]TNX<D,TUBbo(D/G]?IF]^tg$BCD/u¶VoD,ºPT [/KLº/T^

lim infγ→∞

1

γlogEx

(

exp(

−γ2

2

∫ t

0

B2s ds

)

1A1(Bt1) · · · 1An(Btn

))

≥ − ess infz1∈A1,...,zn∈An

(t/2 + x2/2 + z21 + · · · + z2

n−1 + z2n/2).

V?(FÂAEDPPTUACa]TBbACa]Kw^^Ca]D,^

limγ→∞

1

γlogEx

(

exp(

−γ2

2

∫ t

0

B2s ds

)

1A1(Bt1) · · · 1An(Btn

))

= − ess infz1∈A1,...,zn∈An

(t/2 + x2/2 + z21 + · · · + z2

n−1 + z2n/2).

r(9L;

»¼I, 5 ° ^& &)*^+<-B 3 + . 5 / + 1 6IN+ /S9 6K5 /2.0L C`5F / 6 .0/ 5-J6K5 / Q R ?S+ /

limε↓0

ε · logPx

(∫ t

0

B2s ds ≤ ε,Bt ∈ A

)

= − (t+ x2 + ess infz∈A z2)2

8

= 5C +"f+"C > x ∈ ./H1 +"f+"C >h9 +"- A FG6I-I?P (Bt ∈ A) > 0 .0/21 6 /VS. CU-J6K4 8aL . C

limε↓0

ε · logP

(∫ t

0

B2s ds ≤ ε

)

= − t2

8.

)+,-¨ & ijTUAnACK<?]λ = γ2/2

KL?¢T/G(V,ACK<DP? r(9L; PK<ºPT^

r = limλ→∞

1√λ

logEx( −λR

t

0B2

s ds · 1A(Bt)) = − 1√2

(t+ x2 + ess infz∈A

z2).

D, THUVP?£G(^nT¼ACa]TÂ`VGIo(TBmAEa]TUD/BCTu ACDPTUA+ACa]T IBW^nAOT/G(VX<KAxk/9-`ba]T^nTH DP?(FMH XwVK<u g$D/XLX<D,Ôo0kAEVP³jKL?]

x = 0VP?IF

A = 9 0TFS

_P~

ANV IBW^nAO/X<VP?IH T¼KANuVk¹^nTTUu ^nACBWV?]/TdAEaIV,AOACa]TyBWV,AETÀKw^N/G(VPF:BWV,AEK<HÀKL?MACa]TyK<?/AETUBEº,VXXLT?]AEat9

G:ANACaITg$DPX<XLD,tK<?]¢a]TG]BCKw^nACKwHyBCTºPTVPX<^mAEaIV,AdAEa]Kw^dVPH ACGIVPXLX<kMuV³/T^N^CTU?I^CTZtBEKLACTyACaITACK<uyTÂKL?0AETUBEºVPX[0; t]

VP^fACa]TyF:Kw^$xD/KL?0AOG]?]K<DP?MDPg[KL?0AETUBEºVPX<Î1, . . . , In

VP?IFMV/^C^CG]uyTNg$DPBmACaIT¼uyDPuyT?/AOACa(V,AfAEa]TvTUºPT?0AE^∫

IkB2

s ds < εVBETyV/^nkjuy\:AEDACKwHUVPXLX<kK<?IF:TU\TU?(F:TU?0A9«]G]BCACaITUBNKLAduyVP³PT^N^nT?I^nTvAED¢V/^C^CG]uyTvACaIVANACa]T

H D/?0ACBEKLo]G]ACK<DP? DgtV?*KL?0AETUBEºVPXIkACDAEa]TÂK<?/AETU/BEVPX ∫ t

0 B2s ds

K<^N\IBCD/\(D/BnAEKLD/?IVXACDKLAE^dXLT?]AEa-9ijDTH D/?I^CK<F:TB

limε↓0

ε · logP

(

n⋂

k=1

∫

Ik

B2s ds ≤

|Ik|t

· ε

)

= −1

8

n∑

k=1

|Ik|2 t/|Ik| = −1

8t2,

ta]TBCTyAEa]TBEVACT^+TBCTÂHVXwH G]XwV,AETF£GI^nK<?]¢AEa]TlÊHUVX<K<?]¢\]BCD/\(TBnAxk¹g$BCD/u X<TUuyuV¢rI9Q8:9-`ba]TÂBET^CG]XLAdK<ÂCaITvÊVuyT¼V/ÂEa]TvBEVACTÀtaIK<HWa¹TvPDAmg$DPBmACa]T¼g$G]X<XKL?0ACTBCº,VPX'9ZijDACa]Ty0GIVPF:BWV,AEK<HÀF:TU\TU?IF]TU?IHUkDP?

tKw^

H D/uy\IV,AEKLo]X<TÀtKLACa¹ACa]TyVPÊ^nGIuv\]ACK<DP?¢ACa(V,AfAEa]TyH DP?0AEBCK<o]G:AEKLD/?I^DPg|ACa]T¼KL?0ACTBCº,VPX<Î1, . . . , In

VBETÀVP^Ckjuv\ ACDPACKwHUVPXLX<kÂK<?IF:TU\TU?(F:TU?0AfV?(F\IBCD/\(D/BnAEKLD/?IVXACDvAEa]TNKL?0AETUBEºVPXZXLT?]AEa-9½m¾ 5 ²±. ^& 2& KAEaACaITda]TUX<\¹DPg|HUDPBEDPX<X<VPBCkr(9 _ÂTdHV?BETU\]BED:F:GIH T+AEa]TÀBET^CG]XLAE^tDPg|T]Vuy\]X<T¼8:9Q8

g$DPBOAEa]TL2 ?]D/BCu K<?I^nACTV/F¹DPg%ACaITy^CG]\]BETUu¼G]u ?]D/BCu¢9Z`ba]TyT:\DP?]T?/AEK<VPXBEVACTvDg

P(

‖B‖2 > c) g$DPB

c→ ∞ Kw^tV/VPKL?¢HUVPX<HUG]XwV,ACTFltKLACa¹i:HWaIKLXwF:TUB ^%AEa]TUD/BCTu¢9 KAEaε = 1/c2

T+aIVºPT∫ t

0

B2s ds > c2 ⇐⇒ √

εB ∈

ω∣

∣

∫ t

0

ω2s ds > 1

=: A.

`ba]TNBWV,AETfg$G]?(HJACK<DP?I(ω)

G]?IF]TUBtAEa]TdH D/?I^xAEBEVPKL?0A ∫ t

0ω2

t dt = βKw^tuyKL?IKLuVXg$D/BbACa]T+g$GI?IHJAEKLD/?

ωtKAEa

ωs =√

2β/t sin(sπ/2t)

g$DPBVPXLXs ∈ [0; t]

V?IFTO/T AtAEa]TNuyKL?]K<uVX-º,VPXLG]T

I(ω) =1

2

∫ t

0

2β

t· π

2

4t2cos2

(sπ

2t

)

ds =βπ2

8t2.

`bajGIÂCaITd^nTUAAK<^tVyH D/?0ACK<?0GIKAxkl^CT ADPgAEa]TNBEVACTOg$G]?IH ACK<DP?V?(FT I?IF

limc→∞

1

c2logP

(

‖B‖2 > c)

= limε↓0

ε logP(√εB ∈ A

)

= − inf1

2

∫ t

0

ω2s ds

∣

∣

∣ ω ∈ A

= − infβ>1

βπ2

8t2= − π2

8t2.

`ba]Kw^NKw^ÀH D/?I^CK<^nACT?/AdtKLACaACa]TÂBET^CG]XLAE^NDPgtT]Vuy\]X<Tl8:9Q8:9i:KL?IHUTACaITÂTUº/TU?0A ‖B‖2 > cK<uy\]XLK<T^

‖B‖∞ > c/√tTNT:\(THJA

limc→∞

1

c2logP

(

‖B‖2 > c)

≤ limc→∞

1

c2logP

(

‖B‖∞ > c/√t)

=1

tlim

c→∞1

c2logP

(

‖B‖∞ > c)

V?(FK<?IF:TTFACaITÀH D/BCBET^C\(D:F:K<?]vBEVACT^tVBET −π2/8t2g$DPBtAEa]TNXLTUgAfaIVP?IF¢^nKwF:TdV?(F −4/8t2

g$D/BbACa]TÀBCK<Pa0A aIVP?IF¢^nKwF:TP9`ba]T+XwVBEPT+F:TUºjKwV,ACK<DP?oTUaIVºjK<DPG]BbDg ‖B‖2 < ε

g$DPBε ↓ 0

Kw^F:TÊH BEK<o(TFlojklHUDPBEDPX<X<VPBCkyrI9 _]9 TNPTUA

limε↓0

ε2 logP(

‖B‖2 < ε)

= limε↓0

ε2 logP(

∫ t

0

B2s ds < ε2

)

= − t2

8,

L2 _];

ta]KwHWa¹BETU\]BED:F:GIH T^tAEa]T¼HUDPBEBCT^n\DP?IF]KL?]BET^CG]XLAfg$BEDPuX<TUuyuV8:9 _DPg cK<~I;U©9Zf/VK<? (TvHUV?£H DPuy\IVPBCTACaIK<^ttKLACa¢AEa]TNBET^CG]XAW^bg$BEDPu¶T]Vuy\]X<TN8]9 8]9 a]T?]TUº/TUBtTNaIVºPT ‖B‖∞ < ε/

√tTdVXw^CDva(VºPT ‖B‖2 <

εVP?IFACajGI^bTd^na]D/G]XwFlaIVº/T

limε↓0

ε2 logP(

‖B‖2 < ε)

≥ limε↓0

ε2 logP(

‖B‖∞ < ε/√t)

= t limε↓0

ε2 logP(

‖B‖∞ < ε)

.

`ba]TNBET^CG]XLAE^g$BEDPu VPo(D,º/TOVP?IFg$BCD/u¶T:VPuy\]XLTd8]9 8vVBET −t2/8 g$D/BtACa]TNX<T gAmaIV?IF^CK<F:TdVP?IF −π2t2/8g$DPB

ACaITNBCK<Pa0AtaIV?(F^CK<F]TI^CDyTUºPTBCk0AEa]KL?I ]AW^bACD/PTUACa]TBTUX<X©9cVACTBOTtK<XLX[?ITUTFMACaITBCT^nG]XLAE^dDgt·D/BCD/XLXwVBEk¢rI9 _G]?]KLg$DPBEuvX<kMK<? ACa]TK<?]KLACKwVX%HUDP?IF]KAEKLD/?

x9`D

VPHWaIKLº/T+ACa]Kw^fGI?]Kg$D/BCuyKLAxkTdGI^CTÀVº/TUBW^nK<DP?Dg%f?IF:TBE^CDP?- ^tK<?]T0GIVPXLKLAxk ACaIK<^mK<^mH D/BCD/XLXwVBEkyK<?¹?IF]TUB ^CDP?- ^bDPBEKL/KL?(VX\IV\TUBe ?(FIP_% 3 .j 5^&L^&l*^+"-

(Xs)0≤s≤t./H1 (Ys)0≤s≤t

3 +\-JF 5 9 + V@. C .D3"L + .899 6 ./V CE5 4+ 99 + 9\./H1k ∈ [0; 1]

FG6I-%?E(Xs) = kE(Ys)

./21 Cov(Xr, Xs) = Cov(Yr, Ys) = C(r, s) = 5C .0LIL 0 ≤ r, s ≤ t Q 99U8 N+B-I? . - C 6 9 45 / -76 /'8 5 89 QOR ?S+ /P(

∫ t

0

X2s ds ≤ ε

)

≥ P(

∫ t

0

Y 2s ds ≤ ε

)

./H1P(

sup0≤s≤t

|Xs| ≤ ε)

≥ P(

sup0≤s≤t

|Ys| ≤ ε)

= 5C .0LIL ε > 0 Q3 .j 5Â,&¤)& *^+<-

B 3 + . 5 / + 1 6IN+ /S9 6K5 /2.0L CE50F / 6 .0/ 5-J6K5 /./H1 A ⊆ 4 L 5 9 + 1 Q R ?@+ /limε↓0

ε · log supx∈A

Px

(

∫ t

0

B2s ds ≤ ε

)

= − infx∈A

(t+ x2)2

8.

)+,-¨ & c-TUAx, y ∈ A

tKLACa0 < |x| < |y| 9I`ba]T?X<TUuyuVyrI9 rV\]\IXLK<TFlACD X = B+|x| Y = B+|y|V?(F

k = |x/y| VP?IFACa]Td^CkjuyuvTUACBEkÂDgBCD,t?]KwV?uyDACK<DP?PK<ºPT^

Px

(

∫ t

0

B2s ds ≤ ε

)

≥ Py

(

∫ t

0

B2s ds ≤ ε

)

. r(9 8M D, HWa]DjD/^CT x ∈ A

tKLACa |x| = inf |y| | y ∈ A 9]`ba]TU?AEa]TNT^nACK<uV,ACT rI9Q8D PK<ºPT^

Px

(

∫ t

0

B2s ds ≤ ε

)

= supy∈A

Py

(

∫ t

0

B2s ds ≤ ε

)

V?(FÂAEa]TdH XwVK<u g$D/XLX<D,^tKAEa¢HUDPBEDPX<X<VPBCkyr(9 _I9 0TF'

c-TUA X o(TdACa]TÀ^C\IV/H TNDg|VX<X-uV\(^ω : [0; t] → ^nGIHWaAEaIV,A

ω0 = 0T0G]K<\]\TF¢tKLACa¢AEa]TNACD/\(D/XLD/Pk

Dg\DPK<?/AxtKw^CT+HUDP?jºPTBC/TU?IHUTP9]µm? X F:TI?]T+AEa]T+gVuyK<XLk(Pε)ε>0

DguvTVP^CG]BET^bojk

Pε(A) = (

A∣

∣

∣

∫ t

0

W 2s ds ≤ ε

)

g$DPBVPXLXZuyTVP^CG]BEVPo]X<TA ⊆ X 9

d.jI,.j ^&<®;( O/ -I?S+ 9JVS. 4+ X -I?S+ =<. N6 L]> (Pε)ε>09". -J6 9 + 9 -%?@+Z* X FG6I-%? -%?@+ AM5_5 1 C . - +

=8a/ 4"-J6K5 /I(ω) = sup

(t+ 2ω2t1 + · · · + 2ω2

tn+ ω2

t )2/8− t2∣

∣ n ∈ g , 0 < t1 < · · · < tn < t= 5C .0LIL ω ∈ X Q

_/8

)+,-¨ & «IDPBOuyTVP^CG]BWVo]X<TÀ^CT AWÂ1, . . . , An ⊆ VP?IF jTF¢ACK<uyT^

0 < t1 < · · · < tn = tACa]T

`VG]oTUBbAEa]TUD/BCTu rI9<;NV\]\]X<K<TFlACDT0GIVACK<DP? r(9L; %/KLº/T^

limε↓0

ε · logP(

(Bt1 , Bt2 , . . . , Btn) ∈ A1 ×A2 × · · · ×An

∣

∣

∣

∫ t

0

B2s ds ≤ ε

)

= limε↓0

ε · logP(

Bt1 ∈ A1, Bt2 ∈ A2, . . . , Btn∈ An,

∫ t

0

B2s ds ≤ ε

)

− limε↓0

ε · logP(

∫ t

0

B2s ds ≤ ε

)

= −(

t+ ess infz∈A1×A2×···×An

(2z21 + · · · + 2z2

n−1 + z2n))2/8 + t2/8. r(9 _d

¡f^CK<?]An = TvHUV?¹F]BCD/\¢ACa]TyVPÊ^nGIuv\]ACK<DP?

tn = tV?IFVPBCBEK<ºPTdV,AmACa]TÀg$DPX<XLD,tK<?]ÂBET^CG]XLA9«]D/BmVX<X

uyTV/^nG]BWVoIXLT+^nTUAEÂ1, . . . , An ⊆ VP?IF :TFACK<uyT^

0 < t1 < · · · < tn ≤ tT+aIVºPT

limε↓0

ε · logP(

(Bt1 , Bt2 , . . . , Btn) ∈ A1 ×A2 × · · · ×An

∣

∣

∣

∫ t

0

B2s ds ≤ ε

)

= − ess infz∈A1×A2×···×An

It1,...,tn(z)

ta]TBCTIt1,...,tn

: n → +

K<^tF]TI?]TFo0k

It1,...,tn(z) =

1

8

(

t+ 2z21 + · · · + 2z2

n

)2−t2, KLgtn < t

(V?IF(

t+ 2z21 + · · · + 2z2

n−1 + z2n

)2−t2 g$D/Btn = t

9 KLACa£ACa]TvT]H T\:ACK<DP?MDg[AEa]T¼T?IF:\DPK<?0A

tAEa]TyVPHJAEGIVX\D/^CKAEKLD/?I^fDPg[ACaIT

tiaIVºPT¼?]DlK<?IGITU?IHUT¼D/?¹ACaIT

BWV,ACT/9-`ba]TTU?(F:\(D/KL?0AtKw^N^n\THUK<VPX -oTHVGI^CTyACaIT\]BCD:HUTÊ^NF:DjT^+?]DPAd?]TUTFMAEDBCTUACG]BE? ACDACaITÂDPBEKL/KL?

0G]KwHW³0X<klVgACTUBmVyº0Kw^CKAK<?An

V,AtAEKLuyTtI^CDÂV,AtAEa]TNTU?(FDPgACa]TNK<?0ACTUBEº,VX-KLAKw^nHWaITV\TUB mAEDo(T+gVPBfVbVk

g$BEDPu AEa]TNDPBEKL/KL?9THVGI^CTOACa]TdBWV,AET+g$G]?IH ACK<DP?

It1,...,tn

Kw^fHUDP?0ACK<?jG]DPGI^bTd/T AfV?cefYD/? n VP^tK<?¢ACa]TÀBCTuyVPBC³lDP?\IVPPTy;zI9j«IBCD/u ACaIK<^tTdHUVP?/T AbACaITdcemYhD/?ACa]TN\IVACa¢^n\(VPH T+tKLACa¢BWV,ACTOg$G]?(HJACK<DP?I(ω) = sup

It1,...,tn(ωt1 , . . . , ωtn

)∣

∣ n ∈ g , 0 < t1 < · · · < tn ≤ tojkMV\I\]XLkjK<?]ACa]TÂeOV^CDP? R BCAC?]TBfAEa]TUD/BCTu VoDPG]AdX<VPBC/TvF:TºjK<VACK<DP?IÔg$D/BN\]BCDPxTH ACK<ºPTvXLK<uyKAW^ ^CTUTACaITUDPBETUuÁ8:9 zM J9 0TF'

DPACTvACa(V,A+ACa]TBWV,AETvg$G]?IHJAEKLD/? I KL?£ACa]TyAEa]TUD/BCTu tK<XLXAxkj\]KwHUVPXLX<kAEVP³PTvKAW^NK<? IuÀGIu g$DPBdV?]D/? H D/?0ACK<?0GIDPGIÔ\IVACaω9-fÊ^nG]uyT

ωKw^+H D/?/AEKL?jG]D/GI^+V?IF£?]D/? TUBEDI9c-TUA ε = ‖ω‖∞/2

9`baITU? T I?IFK<? I?]KLACTXLkuV?jkF:Kw^xAEKL?(HJAfAEKLuyT^

ttKLACa

ω2t > ε2

V?(F¢ACajGIÎ(ω) = +∞ 9ZijKL?(H T

IKw^

+∞ g$DPBm?]DP? UTBCD (HUDP?0ACK<?jG]DPGI^tg$G]?IHJAEKLD/?I^mKAOtKLX<X?IDAfoTÀ\D/Ê^nK<o]X<TdAEDÂ\]BED,ºPT+ACaIT¼ÊVuyTNACaITUDPBETUuÁtKLACa X BETU\IX<V/H TFojk (

C( +), ‖ · ‖) 9

# `ba]TyBETUuVK<?]K<?]l\IVPBnA+Dg%ACa]Kw^NHWaIV\]ACTUBNHUDP?0AEVPKL?I^fACaITv\IBCDjDgDgVlACa]TDPBETUu VPo(D/G:ANG]\]\TUBNV?IF£X<D,TBX<KLuyKLAE^mKL?MACa]Tv`VPG]o(TBCKwV?¢AEa]TUD/BCTu¢9(=@?MHUDP?0ACBWVP^nAAEDÂACaITUDPBETUuÁrI9<;NAEa]T¼BET^CG]XLAfDg|AEa]KwÔ^CTH ACK<DP?¹VP\]\]X<KLT^tKLACa]D/G:AfV?jklV/^C^CG]uy\:ACK<DP?DP?ACaITdF:K<^nACBEK<o]G:ACK<DP?¢DPg

X9

d.jI,.j ^& 1 (*,+"-X ≥ 0 3 + . C ./H1 50N f . CU6 .D3"L + .0/21 A .0/ +"f+ / -FG6I-I? P (A) > 0 Q X+ / +-%?@+ 8_VMV +"C .0/21L 50F +<C L 6IN6I- 9

r = lim supλ→∞

1√λ

logE( −λX · 1A) ./H1 C = lim inf

λ→∞

1√λ

logE( −λX · 1A)

.9 F + LIL,.9s = lim sup

ε→0ε logP (X ≤ ε, A) ./H1 9 = lim inf

ε→0ε logP (X ≤ ε, A).

R ?@+ / −r2/4 = s ./H1 = 5CZ-I?S+ L 50F +<C L 6IN6I- 9 F + ? . f+e-I?S+ 9 ? . C V + 9 -J6IN . - + 9 − C 2 ≤ 9 ≤ − C 2/4 Q

_P_

)+,-¨ & m^K<?¢ACa]Td\IBCDjDgDgAEa]TUD/BCTu¶rI9<;NKLAfK<^fTU?]D/G]PaAEDlH D/?I^nKwF:TBbACa]TvHUVP^CTA = 9«KLBW^xAm?]DAETACa(V,A ZoTHVGI^CT

XKw^m\D/^CKLACK<ºPTAEa]TyT:\THJAWV,AEKLD/?

E( −λX)T:K<^nAEÔg$D/BNVX<X

λ ≥ 0V?(F¹Kw^NVl?jG]u¼o(TBoT AxTUTU?¢~V?(FM;/9IijDÂVX<XACa]TNº,VPXLG]T^

r C

s(V?IF 9 tKLX<X-oTN?]TU0V,ACK<ºPT/9`ba]T+T^nACK<uV,AET

s ≤ −r2/4 g$DPX<XLD,^g$BEDPu AEa]TNT:\DP?]T?0ACKwVX-¸¹VPBC³/D,ºvK<?]T/G(VX<KAxk ]c-TUAε > 0

9]«]BEDPuE( −λX ) ≥ −λεP

(

−λX ≥ −λε)

= −λεP(

X ≤ ε)

TO/T AP (X ≤ ε) ≤ λεE( −λX )

VP?IFlACajGI^ε logP (X ≤ ε) ≤ ε

(

λε+ logE( −λX)) g$D/BVX<X

λ ≥ 09

«]D/Bλ = r2/4ε2

AEa]TNoDPG]?IFoTHUDPuyT^ε logP (X ≤ ε) ≤ r2/4 + ε logE( −Xr2/4ε2

).

`V³jKL?IvGI\]\(TBtXLK<uyKAW^tTNPT A

s = lim supε↓0

ε · logP (X ≤ ε) ≤ r2

4+ lim sup

ε↓0ε · logE( −Xr2/4ε2

)

=r2

4− r

2lim sup

ε↓0

2ε

|r| logE( −X(r/2ε)2)

=r2

4− r

2· r = − r

2

4.

ªtT\]X<V/H K<?]yVX<X-G]\]\TUBtX<KLuyKLAE^tK<?ACa]TN\IBCTº0K<DPG(^tVBEPG]uyTU?0AbtKAEa¢XLD,TUBbX<KLuyKLAE^t/KLº/T^ 9 ≤ − C 2/49

uyD/BCTÀHVBET g$G]XV?(VX<kj^CKw^Kw^f?]TH T^CÊVBEkÂAEDl\]BED,ºPTs ≥ −r2/4 9 £TyHUVP?T:\]BETÊ^ r ºjK<VÂACa]TvX<D,TBAEVPKLXw^bDg

X

r = lim supλ→∞

1√λ

logE( −λX)

= lim supλ→∞

1√λ

log

∫ 1

0

P ( −λX ≥ t) dt

t =−u

= lim supλ→∞

1√λ

log

∫ ∞

0

P (X ≤ u/λ) −u du

= lim supε↓0

ε log

∫ ∞

0

P (X ≤ uε2) −u du.

`ba]TÀF]TI?]KLACK<DP?¢DPgsPK<ºPT^tACaIVAtg$DPBTºPTUBEk

δtKLACa

0 < δ < |s| ACa]TBCTNT:K<^nAE^VP? E > 0I^CGIHWaACaIVAg$DPBTUº/TUBEk

η < ET+aIVºPT

P (X ≤ η) ≤ η−3/2 (s+δ)/η .=VP?/AbAEDvG(^nTOACaITNBCTX<VACK<DP?∫ ∞

0

zu−3/2 exp(

−z2

u− u)

du =√π −2z .

=@?ACa]TdHUDP?0ACTjAbDPgAEa]TdVoD,ºPTOT^nACK<uV,ACT+AEa]Kw^tPK<ºPT^∫ ∞

0

P (X ≤ uε2) −u du ≤∫ E/ε2

0

(ε2u)−3/2 exp(

−(

√

|s+ δ|ε

)2 · 1

u− u)

du

+

∫ ∞

E/ε2

1 · −u du

≤ ε−3 ε√

|s+ δ|√π −2

√|s+δ|/ε + −E/ε2

`ba]Td^CG]u Kw^F:D/uvK<?IVACTFojkACaIT IBE^nAtACTBCu ]^CDyT+PTUAr ≤ −2

√

|s+ δ| taITU?]TºPTUB0 < δ < |s|

V?(FÂAEa0G(^r ≤ −2

√

|s| 9]THUVPGI^nT+oDAEa r VP?IF s ]VPBCT+?]T/V,AEKLº/TfAEa]Kw^^naID,^ s ≥ −r2/4 9

_r

«KL?(VX<XLkITdHUV?\IBCD,º/T − C 2 ≤ 9P9¡f^CKL?IACa]T¼T^nACK<uV,ACT −λx ≤ 1[0;ε](x) + −λε1(ε;∞)(x)g$DPBfVPXLX

x ≥ 0tKLACa

λ = | 9 |/ε2 PK<ºPTÊ(

−|s|X/ε2) ≤ P (X ≤ ε) + −| 9 |ε/ε2

P (X > ε) ≤ P (X ≤ ε) + −| 9 |/ε.

THVGI^CTÀg$D/BOACa]TÂ^CTHUDP?IF¹ACTUBEu K<?MAEa]T^nGIu ACa]TX<K<uvKLAlimε↓0 ε log −| 9 |/ε = −| 9 | T:K<^nAE^ ZTHV?H D/?IH X<GIF:T

−| C |√| 9 | = lim infε↓0

ε logE(

−|s|X/ε2)

≤ max(

lim infε↓0

ε logP (X ≤ ε) , limε↓0

ε log −| 9 |/ε)

= max(

−| 9 | , −| 9 |) = −| 9 |.`V³jKL?IyÊ0GIVBETÂEa]TNT^nACK<uV,ACT+oTHUDPuyT^ C 2 ≥ | 9 | V?IFuÀG]XLACK<\]X<K<HV,AEKLD/?tKLACa −1

PK<ºPTÂEa]TNBET^CG]XA9`ba]TNG]\I\(TBfoDPG]?IF¢D/? 9 Kw^f^CaIVPBC\ :oTHVGI^CTNK<?¢ACa]TÀHVP^CTNDgACa]TDPBETUu rI9<;NTNaIVº/T+T0GIVX<KLAxkÂACaITUBETP9

`ba]TÀgVPHJAOACa(V,AfAEa]TvXLD,TUBmo(D/G]?IFg$D/BAEa]T¼X<D,TUBfX<K<uvKLA 9 K<Ô^CaIVBE\¹Kw^m^Ca]D,t?ojkACa]TvT]Vuy\]X<T¼VAfAEa]TTU?(FDPgAEa]K<^f^nTHJACK<DP?9 0TF' »¼I, 5 ° ^& ¿ & /H1 +"CB-%?@+ .99U8 N V -76K5 /@9 5 = -%?@+5C`+<N Q F + ? . f+ r = −2

√

|s| ./H1 = 5Ce-%?@+L 50F +<C L 6IN6I- 9 F + ? . f+ -I?S+ 9 ? . C V + 9 -J6IN . - + 9 −2√

| 9 | ≤ C ≤ −

√

| 9 | Q)+,-¨ & µm?

(−∞; 0]AEa]TyuV\

x 7→ −√

|x| Kw^N^nACBEK<H ACX<kuvD/?]DAEDP?]KwHUVPXLX<k¢KL?(H BETVP^CK<?]I9`bajGIÔg$DPBr, s ≤ 0

TNaIVºPTs ≤ −r2 Kg[VP?IFDP?]X<klKLg −√|s| ≤ r

9(f\]\]X<kjKL?]yAEa]Kw^tACDyAEa]TdBET^CG]XLAE^tDPgAEa]TUD/BCTu¶rI9 \]BED,ºPT^%AEa]TdH D/BCD/XLXwVBEkP9 0TF' DPACT+AEaIV,AtAEa]TUD/BCTu¶rI9 vF:DjT^?]DAfF:K<BETHJAEXLkKLuy\]X<kACa]TDPBETUu r(9L;/9]= gACa]TNX<KLuyKLA

sg$BCD/u ACaITUDPBETUu¶rI9<;

T:Kw^xAW^jAEa]TU?¢T+PT As ≤ − C 2/4 ≤ −r2/4 = s,K©9 T/9IAEa]TÀX<KLuyKLA

rVXw^CDÂT:Kw^xAW^fV?(F^CVACKw^ IT^

s = −r2/4 9G]AmKLg[TÀV/^C^CG]uyTNAEaIV,A r T:K<^nAE^ IAEa]TU?¹ACa]TD BETUu r(9QÀDP?IXLkÂ/KLº/T^−r2 ≤ 9 ≤ s = −r2/4V?(FlTdHUVP?]?]DAF:K<BETHJAEXLkH DP?(H X<GIF:TOACaIVAtACa]TNX<K<uvKLA

sg$BEDPu ACa]TDPBETUu r(9L;+T:Kw^xAWÛ9

`ba]T+g$D/XLX<D,tKL?IT]Vuy\]X<Td^naID,^bACaIVAg$DPBmPT?]TUBWVXZBWV?IF]DPu¶º,VPBCKwVo]X<T^XAEa]TÀX<D,TBto(D/G]?IF − C 2 ≤9 DP?ACa]TNX<D,TUBbXLK<uyKA 9 Kw^^na(VBE\-9

½m¾ 5 ²±. ^&w2 & c-T As < 0

V?IF(εn)n∈ 0

oTÀV^xAEBCKwHJAEXLkF:THUBCTVP^CKL?Iy^CT0G]T?IH TNtKLACaε0 = ∞ VP?IF

limn→∞ εn = 09(`ba]TU?T+aIVº/T∑

n∈

(

−|s|/εn−1 − −|s|/εn

)

= −|s|/ε0 − limn→∞

−|s|/εn = 1 − 0 = 1

V?(FlTdHUVP?F:TI?]TdVvBWV?IF:D/u º,VBEK<VPo]XLTXtKAEa¢ºVPXLGIT^bK<?AEa]Td^CT A εn | n ∈ g o0k

P (X = εn) = −|s|/εn−1 − −|s|/εn

g$DPBVPXLXn ∈ g 9]`ba]Kw^tBWV?IF]DPu º,VBEK<VPo]X<TOa(VP^

P (X ≤ ε) =

∞∑

n=n(ε)

(

−|s|/εn−1 − −|s|/εn

)

= −|s|/εn(ε)−1

tKLACan(ε) = minn ∈ g | εn ≤ ε V?IF¢HUDP?I^CT0G]T?/AEXLk

ε logP (X ≤ ε) = −|s| ε

εn(ε)−1.

_/

kF]TI?]KLACK<DP?DPgn(ε)

TdaIVºPTεn(ε) ≤ ε < εn(ε)−1

9`ba]Kw^fVPXLX<D,^tGIÂCDlHVXwH G]XwV,AET+ACa]T¼T:\(D/?]TU?0AEK<VPXAEVPKLXBEVACT^ 9 = s

V?IF Io(THUVG(^nTsKw^t?]T/V,AEKLº/T

s = s · lim infn→∞ εn/εn−19

·a]DjD/^CK<?]F:KTUBETU?0Am^CT0G]T?IH T^(εn)

X<TV/F]^bAEDlF:K TBCT?/AmºVPXLGIT^bg$D/Bsr(V?(F C 9(«]D/BDPGIBT]Vuy\]X<T

X<T Aq < 1

V?IFMF:TI?ITεn = qn g$D/BmVX<X n ∈ g 9`ba]TyVoD,ºPTdHUVPX<HUG]XwV,ACK<DP?¹^Ca]D,^ 9 = s

V?IFs = qs

9`ba]T+AEa]TUD/BCTu¶PK<ºPT^

r = −2√

q|s| VP?IF C ∈ [−2√

|s|;−√

|s|] 9£TÀVP?/AbAEDyg$G]BnAEa]TUBfT]VuyK<?]T Cj9I`ba]TcVP\]XwVPH TmACBWV?(^xg$D/BCu DPgXHUVPX<HUG]X<VACT^tVP^

E( −λX) =∑

n∈ −λqn(

−|s|/qn−1 − −|s|/qn)

=∑

n∈ −λqn−|s|/qn−1(

1 − −|s|(1−q)/qn)

.

THVGI^CT+Dgexp(−|s|(1 − q)/qn) → 0

g$DPBn → ∞ TNaIVºPT

1/2 < 1 − exp(−|s|(1 − q)/qn) < 1g$DPB

^CGa!ÂH K<TU?0ACX<klXwVBEPTn9(emT(?]T

n(λ)ojk

qn(λ) ∈ [q√

|s|/λ;√

|s|/λ) 9 KLACa f(x) = exp(−λx − q|s|/x)TOa(VºPTE( −λX ) > exp

(

−λqn(λ) − |s|/qn(λ)−1)1

2=

1

2f(qn(λ))

g$DPBO^nG@!HUKLT?0ACX<kXwVBEPTλ9THUVG(^nT

fKw^K<?IH BETV/^nK<?]DP?¢AEa]TÀK<?0ACTBCº,VX

(0;√

q|s|/λ] V?(FF:THUBCTVP^CKL?]D/?[√

q|s|/λ;∞)TyHUV?MT^nACK<uyVACT

fDP?

[q√

|s|/λ;√

|s|/λ) o0kKLAEÔº,VX<G]T^fD/?¹ACaIT¼oDPG]?(F]VBEKLTÛ9`ba]Kw^X<TVPFIÂCDE( −λX) >

1

2min

(

f(q√

|s|/λ), f(√

|s|/λ))

=1

2exp(

−(1 + q)√

λ|s|)

g$DPB^CGa!ÂH K<TU?0AEXLkX<VPBC/Tλ9I`V³jKL?I¼X<D,TBbXLK<uyKAW^tTN/T A

−√

|s| ≥ C ≥ −(1 + q)√

|s|ta]TBCTOACaIT IBE^nAtKL?IT0GIVX<KLAxklH D/uvT^g$BCD/u ACaIT+ACa]TDPBETUu D/BtT0G]K<ºVPXLT?0ACX<k

− C 2 ≤ 9 ≤ − C 2/(1 + q)2.

`ba]Kw^+^na]D,^fACa(V,ANojk¹HWa]DjD/^CKL?]^nuVX<Xº,VX<G]TÔDgqT¼HUVP?¹g$D/BEHUT 9 ACDo(TVBEo]KLACBWVBEKLX<k¢H X<D/^CTdACD − C 2tKLACa]D/G:A − C 2 oTUK<?]ÂHUXLD0^nTOACD~I9IijDyACa]TNX<D,TUBbo(D/G]?IFDP? 9 g$BCD/u ACa]T+AEa]TUD/BCTu Kw^^na(VBE\-9

_Pq

> b> % O

=@?¹AEa]Kw^NHWaIV\:AETUBOTyF:TBCK<ºPTvV?£cefYg$DPBOAEa]T¼oTUa(Vº0K<DPGIBmDPg[ACa]TyT?IF:\DPK<?/AXt

DgbVlF:K G(^nK<DP? ta]TU?ACaITÀF:BEKgAOK<^^nACBEDP?II9I`ba]Kw^Kw^fVy/TU?]TBEVPXLKwÊV,ACK<DP?DgAEa]TdBET^CG]XAfg$DPBtAEa]TvµmBE?I^xAETUK<? ¡a]X<TU?joTHW³l\IBCD:H T^C^KL?HWaIVP\:ACTB_]9Q8:9

TÀbV?0AfAEDF:TUACTBCuyK<?]TdAEa]T¼XwVBEPT¼F:TUºjKwV,ACK<DP?(^foTUaIVºjK<DPG]Bfg$DPBmACa]TvTU?(F:\(D/KL?0AXt

DPg^CDPX<G:ACK<DP?I^fDgACaIT º,VPXLG]TF^nACD:HWaIV/^xAEK<H+F:K TUBETU?0ACKwVXZT/G(V,ACK<DP?dXϑ

s = ϑb(Xϑs ) ds+ dBs

DP?[0; t]

Xϑ0 = z ∈ ]9L;

g$DPBtXwVBEPTOº,VX<G]T^ϑ9

`ba]TÀ^CKLACGIVACK<DP?¹a]TBCTÀK<^mF:KTUBETU?0Amg$BCD/u AEa]T¼^CKLACGIVACK<DP?K<?ACa]Tv«]BETUKwF:X<KL? TU?0ATUX<XAEa]TUD/BCTu9(=@?¹DPGIBHUV/^nT+AEa]TÀX<TU?IACatDPgACa]T¼KL?0ACTBCº,VPXKw^:TF VP?IF¢KL?(^xAETVPF

ϑ/D0T^bACDlKL? (?]KAxk/9Zµm?ITÀHUVP?BCT^CHVX<TNT0GIV ACK<DP? :9<;_ VP^g$D/XLX<D,Û9]emT(?]T Y ϑ

s = Xϑs/ϑ

g$DPBVPXLXs ∈ [0;ϑt]

9]`baITU?¢ojklX<TUuyuV;/9 _¼AEa]TN\]BEDjHUTÊ^Y ϑ K<^Vy^nD/XLG]ACK<DP?DgACa]TÀi:ems

dY ϑs = b(Ys) ds+

1√ϑdBs

D/?[0;ϑt]

Y ϑ0 = z

V?(FlTNaIVºPTP (Xϑ

t ∈ A) = P (Y ϑϑt ∈ A).`ba]TvBCT^CHVX<TF\IBCD/o]XLTuXLDjDP³:^muyDPBET¼^CKLuyK<X<VPBfAEDlAEa]Tv^CKLACGIVACK<DP?¹g$BEDPuÁAEa]T«]BETUKwF:XLK<? £T?0AUTXLXAEa]TUD/BCkoTHUVPGI^CTv?ID, ACaITv?IDPKw^nTF:TH BETVP^CT^9ZG:ANACa]TyX<TU?]PACa*DgACa]TyAEBEVP?I^ng$DPBEuvTFAEKLuyTK<?/AETUBEº,VX[F:T\(T?IF]^

DP?ϑI^CD¼AEa]Td«]BETUKwF:X<KL? TU?0ATUX<XZACaITUDPBETUu ^xAEKLX<X-HUVP?]?]DAoTNV\]\IXLK<TFTV/^nK<X<kP9«]BEDPu cTUuyuVM;/9 ÂT¼³0?ID, ACa]TF:T?I^nKLAxkDg|AEa]TF:Kw^xAEBCK<o]G:AEKLD/?MDg

Xϑt

ZV/^C^CG]uyKL?IXϑ

0 = 0V?IF

b = −∇ΦT+PT A

P (Xϑt ∈ A) =

∫

1A(ωt) exp(

ϑF (ω) − ϑ2G(ω))

d (ω)

ta]TBCT

F (ω) = Φ(ω0) − Φ(ωt) +1

2

∫ t

0

∆Φ(ωs) dsVP?IF

G(ω) =1

2

∫ t

0

b2(ωs) ds.

= A+tKLX<XACG]BE?MD/G:A ACa(V,AOACa]TvDP?]X<k\IVACaI^mta]KwHWa£H D/?0ACBEKLo]G]ACTdg$D/BOACa]TvX<VPBC/TÀF:Tº0KwV,AEKLD/?IÔo(TaIVºjKLD/G]BDgXϑ

t

VBETdACaID/^CTta]KwHWa£H D/BCBET^C\DP?IFAEDº/TUBEk^CuVX<Xº,VPXLG]T^fDPgG9-`ba]Kw^+HWaIV\:AETUB+H DP?(^nKw^xAW^fDPg|ACa]BETUT

_0

_P

\IVPBnAWÛ9«K<BW^xAOT¼T]VuyK<?]TNACaITv^CKAEGIV,AEKLD/?¢ACaIVAOF:GIBCK<?]Vl^na]D/BnAmACK<uyTÀK<?/AETUBEº,VX-AEa]Tv\]BCD:HUTÊ^BEG]?I^D,º/TUBVÂX<DP?IlF:Kw^xAWV?IHUTÀta]K<XLTv^xAEKLX<X³PTTU\]K<?] ∫

b2(ωs) ds^CuyVPXLX©9`ba]K<^mtK<XLXoTÀGI^CTF¹g$DPBAEa]TvKL?]KLACKwVXV?(F¢ACa]T

I?IVPX-\]K<THUTNDgACa]TN\IVACa-9I=@?AEa]Td^CTH D/?IF¢^nTHJAEKLD/?TNT]VuyK<?]TOACa]Td^CKLACGIVACK<DP?ACaIVA ∫b2(ωs) ds

Kw^^CuyVPXLXD,ºPTBtVyXLD/?]KL?0AETUBEºVPX-DgACK<uyTP9`ba]K<^tKLX<X-oTdGI^CTFACDAEBCTV,AbACaITduVK<?\IKLTH TNDgAEa]TN\IV,AEa-9I=@?AEa]TNACaIKLBWF^CTHJAEKLD/?T ]AtACaIT^CT+AxDyBET^CG]XAWÂCD/PT AEa]TUBbK<?¢DPBWF:TUBACD (?IFACa]TNTj\DP?ITU?0ACKwVXZBWV,AETmg$D/BbACa]TdcefY9

" " # "`ba]TNg$D/XLX<D,tK<?]\IVBCAfa]TXL\(^tACDlT^nACK<uyVACTNAEa]TÀ\IBCD/oIVo]K<X<KAxklACaIVAfACaITd\IV,AEa¢ACBWVºPTX<^0G]KwHW³jXLko(TUAxTTU?¹VP?T0G]K<X<KLo]BEK<G]u\(D/KL?0AfDPg|ACa]TÀF]BCKLgAOVP?IF¢ACaIT I?IVPXBET^C\-9IK<?]KLACKwVX\(D/KL?0A9 TBCTvi:HWa]K<X<F:TB ^tAEa]TUD/BCTuÁHV?oTV\I\]XLK<TF¢VP?IFlT+tKLX<X-BETF:G(H TOACa]TNTº,VX<GIV,AEKLD/?DPgACaITNBEVACTOg$G]?IH ACK<DP?ACDÂVvº,VBEK<VACK<DP?IVPX\]BCD/o]X<TUu¢9

`ba]T¼³PTUkg$DPBOTUº,VPXLGIVACK<?]AEa]T¼BWV,AETNg$G]?IH ACK<DP?£KL?¹\IBCD/\(D0^nKLACK<DP?M]9 _oTUX<D,´KwÂEa]TÀg$D/XLX<D,tK<?]ÂX<TUuyuV]9 TdF:T g$TBbACa]TN\]BEDjDgDgACa]TNX<TUuyuVvG]?0ACK<XZACa]TNT?IFDgACa]Td^CTH ACK<DP?-9

3 .j 5¤ & 2&;*^+"-v : → [0;∞) 3 + . V 5 9 6I-76If+ : -JF 5-76IN+ 9 45 / -J6 /S8 5 89UL]>1 6 +"CE+ / -J6 .D3"L +=8a/ 4"-J6K5 / FG6I-%? lim inf |x|→∞ v(x) > 0 ./H1 m ∈ FG6I-I?

v(x) = 06 =./21 5 /'L]> 6 = x = m ./H1

v′′(m) > 0 Q 5C a, z ∈ .0/21 β ≥ 0 1 + / +

Ma,z,βt =

ω ∈ C[0; t]∣

∣

∣ω0 = 0, ωt = a− z,

1

2

∫ t

0

v(ωs + z) ds = β

./H1J(a, z) =

1

4

(

∣

∣

∫ m

z

√

v(x) dx∣

∣+∣

∣

∫ a

m

√

v(x) dx∣

∣

)2

.

5 /S9 6 1 +<CZ-%?@+ZC . -#+ =8a/ 4"-76K5 /It(ω) =

12

∫ t

0 |ω|2 ds, 6 = ω 6 9e.D3<9 5 L]8 -#+ L]> 45 / -76 /'8 5 89:;.0/21+∞ + L 9 + .

*^+<-K1,K2 ⊆ 3 +h4`50N VS. 4"- 9 +"- 9 FG6I-I? m /∈ K1 ∩K2

.0/21 B ⊆ +3 + 3 5 8a/H1 + 1 FG6I-%? 0 ∈ B QOR ?@+ /F + ? . f+

inf

It(ω)∣

∣

∣ ω ∈⋃

β∈B

Ma,z,βt

−→ 1

supBJ(a, z) = 5C t → ∞,

8a/ 6 = 50CUN L > 50f+"C a ∈ K2./H1 z ∈ K1 Q

3 .j 5¤)&2)&,*^+<-Ma,z,β

t3 + .9 6 / L +"N\N . QMQ R ?S+ /\= 5C +<f+"C > VS. 6IC K1,K2 ⊆ 5 = 45N VS. 4"-

9 +<- 9 -%?@+ 9 +"-M =

⋃

z∈K1

⋃

a∈K2

⋃

0≤β≤1

Ma,z,βt

6 9 4 L 5 9 + 1 6 / C0([0; t], ) Q)+,-¨ & klF:TI?]KLACK<DP?¢DgACa]Td^CT AW^

Ma,z,βt

TOa(VºPT

M =⋃

z∈K1

ω ∈ C[0; t]∣

∣

∣ ω0 = 0, ωt + z ∈ K2,1

2

∫ t

0

v(ωr + z) dr ≤ 1

.

fÊ^CG]uyT+ACaIVAω ∈ C0([0; t], ) \M 9:`ba]TNTUKLACaITUB

ωt + z /∈ K2g$DPBVPXLX

z ∈ K1:K'9 TP9

ωtX<KLT^bDPG:AW^nKwF:TOACaIT

H D/uy\IVPH At^CT AK2 −K1

:DPB1

2

∫ t

0

v(ωr + z) dr > 1

g$DPBtTºPTBCkz ∈ K2

]K'9 TP9inf

z∈K2

1

2

∫ t

0

v(ωr + z) dr > 1

oTHUVPGI^CTK2

K<^+HUDPuy\IVPH AOVP?IFvV?(FACaIT¼K<?0ACTU/BEVPXVPBCT¼H DP?0AEKL?jG]D/GIÛ9=@?MoDAEa HVP^CT^fTyHUVP? I?IF V?

ε > 0]^nGIHWaACa(V,AAEa]TNoIVX<X

B(ω, ε)VXw^nDyX<KLT^tKL?

C0([0; t], ) \M 9I`bajGI^MK<^bAEa]TdH D/uy\]XLTuyTU?0ADgV?

DP\TU?^CT A9 0TF'

_Pz

`ba]T+uVK<?¢BCT^nG]XLAtDgACa]Kw^^CTHJAEKLD/?Kw^bAEa]T+g$DPX<XLD,tK<?]v\]BEDP\D/^CKAEKLD/?-9)+,I±(&$ ¤ & &,*,+"-

B 3 + . 5 / + 1 6IN+ /S9 6K5 /2.0L CE5F / 6 ./ N5-J6K5 / FG6I-%? 9 - . CU- 6 / z ∈ ./21b : → 3 + . -JF 5-76IN+ 9 45 / -J6 /S8 5 89UL]>1 6 +<C`+ / -76 .M3<L + =8a/ 4"-J6K5 / FG6I-%? lim inf |x|→∞ |b(x)| > 0 ./H1m ∈ FG6I-I?

b(x) = 06 =B.0/21 5 /SL]> 6 = x = m ./21 b′(m) 6= 0 QOR ?S+ / = 50Ce+"f+"C > V@. 6ICe5 = 4`50N VS. 4"- 9 +"- 9

K1,K2 ⊆ F + ? . f+

lim supt→∞

lim supε↓0

supz∈K1

ε logPz

(1

2

∫ tε

0

b2(Bs) ds ≤ ε,Btε ∈ K2

)

≤ −1

4inf

z∈K1

infa∈K2

(

∣

∣

∫ m

z

|b(x)| dx∣

∣+∣

∣

∫ a

m

|b(x)| dx∣

∣

)2

./H1 = 5C +"f+"C > z ∈ ./21 +<f+<C > 5 V + /j9 +"- O ⊆ F + ? . f+

lim inft→∞

lim infε↓0

ε logPz

(1

2

∫ tε

0

b2(Bs) ds ≤ ε,Btε ∈ O)

≥ −1

4infa∈O

(

∣

∣

∫ m

z

|b(x)| dx∣

∣ +∣

∣

∫ a

m

|b(x)| dx∣

∣

)2

.

`ba]T+uyD:F:G]X<GI^tDgACa]TNK<?0ACTPBWVXw^Kw^bAWV³PT?lAEDy\]BCD/\(TBCX<kÂaIVP?IF:X<TOACa]TdHVP^CT^m < z

VP?IFa < m

9)+,-¨ & TÂbV?0ANACD¹VP\]\]X<k*i:HWaIKLXwF:TUB ^+ACa]TDPBETUu V?IF£ACDMTUº,VX<GIV,AETyACa]TlBEVACTyg$G]?IH ACK<DP?²GI^CK<?]X<TUuyuVl:9<;P9IcT A

K1,K2 ⊆ o(TÀHUDPuy\IV/HJA9efTI?]TÀACa]Td\IBCD:H T^C^Bojk^CT ACACK<?]

Br = (Brε − z)/√εg$DPBtTºPTBCk

r > 09]`ba]TU?

BKw^tVyBED,t?]KwV?uvDPACK<DP?tKAEa^xAWVBCAbK<?~yV?IFTO/T A

Pz

(

Btε ∈ K2,1

2

∫ tε

0

b2(Bs) ds ≤ ε)

s = rε= Pz

(

Btε ∈ K2,1

2

∫ t

0

b2(Brε) dr ≤ 1)

= P(√

εBt + z ∈ K2,1

2

∫ t

0

b2(√εBr + z) dr ≤ 1

)

= P(√

εB ∈⋃

a∈K2

⋃

β≤1

Ma,z,βt

)

V?(FÂAEa0G(^

supz∈K1

Pz

(

Btε ∈ K2,1

2

∫ tε

0

b2(Bs) ds ≤ ε)

≤ P(√

εB ∈⋃

z∈K1

⋃

a∈K2

⋃

β≤1

Ma,z,βt

)

. ]9 8M

«]BEDPu X<TUuyuV¹:9Q8lTy³j?]D,ÁEaIV,ANAEa]TÂ^nTUA ⋃z∈K1

⋃

a∈K2

⋃

β≤1Ma,z,βt

Kw^dHUXLD0^nTF¹K<?£AEa]Ty\IV,AEa^C\IVPHUT (

C0[0; t], ‖ · ‖∞) I^CDyTNHUVP?VP\]\]X<ki:HWa]K<XwF:TUB ÂEa]TUD/BCTu AEa]TUD/BCTu 8:9<;qd %ACDyPTUA

lim supε↓0

ε log supz∈K1

P(√

εB ∈⋃

z∈K1

⋃

a∈K2

⋃

β≤1

Ma,z,βt

)

≤ − inf

It(ω)∣

∣

∣ ω ∈⋃

z∈K1

⋃

a∈K2

⋃

β≤1

Ma,z,βt

= − infz∈K1

infa∈K2

infβ≤1

inf

It(ω)∣

∣ ω ∈Ma,z,βt

.

«KLBW^nAfVPÊ^nG]uyTm ∈ K1 ∩K2

9(emTI?]TdACa]T¼\IV,AEaωojk

ωs = 0g$D/BfVX<X

s ∈ [0; t]9(`baITU?¹H X<TVPBCX<klT

aIVº/Tω ∈Mm,m,0

t

g$DPBtTºPTUBEktV?IF¢^CKL?IHUT+T I?IF

It(ω) = 0T+aIVº/T

inf

It(ω)∣

∣

∣ ω ∈⋃

β∈B

Ma,z,βt

= 0

r/~

g$DPBVPXLXt ≥ 0

9(µm?ACa]T+DAEa]TUBta(V?IFTOa(VºPTJ(m,m) = 0

9µfACaITUBEtK<^CT+ACa]T¼TUº,VX<GIVACK<DP?¢Dg|AEa]TdK<? IuÀGIuÁKwÔF:DP?]TÀKL?X<TUuyuV]9L;/9I¡f^CKL?I

v(x) = b2(x)Td/T A

v′′(m) = 2(b′(m))2 > 0VP?IFg$DPBtTºPTUBEk

η > 0TNHUVP? I?IF¢V

t0 > 0]^nG(HWalAEaIV,A

infβ≤1

inf

It(ω)∣

∣ ω ∈Ma,z,βt

≥ J(a, z) − η

g$DPBVPXLXz ∈ K1

a ∈ K2

V?(Ft ≥ t0

9]`ba]Kw^tPK<ºPT^lim sup

ε↓0ε log sup

z∈K1

P(√

εB ∈⋃

z∈K1

⋃

a∈K2

⋃

β≤1

Ma,z,βt

)

≤ − infz∈K1

infa∈K2

infm∈N

J(a, z) + η

= −1

4inf

z∈K1

infa∈K2

(

∣

∣

∫ m

z

|b(x)| dx∣

∣+∣

∣

∫ a

m

|b(x)| dx∣

∣

)2

+ η

g$DPBtTºPTBCkη > 0

9]`DP/T AEa]TUBttKLACaACaITNBCTX<VACK<DP? :9Q8D AEa]K<^b\IBCD,º/TÂCa]TNGI\]\(TBoDPG]?IF-9«]D/B+ACa]TX<D,TUBNoDPG]?IF Tyg$DPX<XLD, ACa]T^CVPuvTy\]BED:H TF:G]BETP9 KLACa]D/G:AÀX<D/Ê^+DgbPTU?ITUBWVX<KAxk¹THUVP?VPÊ^CG]uyTmAEaIV,A

OKw^bo(D/G]?IF:TFZ9TBCTNT+PTUA

Pz

(

Btε ∈ O,1

2

∫ tε

0

b2(Bs) ds ≤ ε)

≥ Pz

(

Btε ∈ O,1

2

∫ tε

0

b2(Bs) ds < ε)

= P(√

εB ∈⋃

a∈O

⋃

β<1

Ma,z,βt

)

ta]TBCTOACaITd^nTUA⋃

a∈O

⋃

β<1

Ma,z,βt =

ω ∈ C[0; t]∣

∣

∣ ω0 = 0, ωt ∈ O − z,1

2

∫ t

0

b2(ωr + z) dr < 1

Kw^dDP\TU?§KL? (C0[0; t], ‖ · ‖∞

) 9i:D¹THUV?²G(^nTAEa]TlX<D,TUBNoDPG]?IF²g$BCD/ui:HWaIKLXwF:TUB ^+ACa]TDPBETUuV?IFX<TUuyuVy]9L;OAED I?]Kw^naACa]TN\IBCDjDgx9 0TF'

»¼I, 5 ° ¤ &L^& /21 +"CB-I?S+ .99U8 N V -76K5 /@9 5 = V CE5 V 5 9 6I-J6K5 / Q F + ? . f+

limη↓0

lim inft→∞

lim infε↓0

ε log infm−η≤z≤m+η

Pz

(1

2

∫ tε

0


≥ −1

4infa∈O

(

∫ a

m

|b(x)| dx)2

= 5C +"f+"C > 5 V + /j9 +"- O ⊆ iQ)+,-¨ & «]D/B

z ∈ F:TI?]T

Mzt =

ω ∈ C[0; t]∣

∣

∣ ω0 = 0, ωt + z ∈ O,1

2

∫ t

0

b2(ωs + z) ds < 1

.

c-TUAδ > 0

9Z·a]DjD0^nTvV?ω ∈ Mm

t

tKLACaIt(ω) < inf It(ω) | ω ∈ Mm

t + δ9THVGI^CT

OKw^fD/\(T?

V?(FbV?(FlAEa]TNK<?/AETU/BEVPXVPBCTNHUDP?0ACK<?jG]DPGI^bTNHUVP? I?(FV?

E > 0I^CGIHWaACaIVAtg$DPBtTºPTUBEk

η < EACaITNoIVX<X

Bη(ω) ⊆ C0([0; t], )K<^HUDP?0AEVPKL?ITFKL?¢VX<X-DgACa]Td^CT AW^

Mzt

g$D/Bm− η < z < m+ η

9I`baIK<^b/KLº/T^

lim infε↓0


Pz

(1

2

∫ tε

0


= lim infε↓0


Pz

(√εB ∈Mz

t

)

≥ lim infε↓0


Pz

(√εB ∈ Bη(ω)

)

.

rI;

V?(FlGI^CK<?]li:HWa]K<X<F]TUB ÂCaITUDPBETUu V?IFACaITNBCTX<VACK<DP?− inf

It(ω)∣

∣ ω ∈ Bη(ω)

≥ −It(ω) > − inf

It(ω)∣

∣ ω ∈ Mmt

− δ

T (?IF

lim infε↓0


Pz

(1

2

∫ tε

0


≥ − inf

It(ω)∣

∣ ω ∈ Bη(ω)

> − inf

It(ω)∣

∣ ω ∈ Mmt

− δ.

D, TÀHUVP?TUº,VX<GIV,AETNACaITdKL?IuÀG]uD/?¢ACa]TdBEK<Pa0AfaIVP?IF^nKwF:TÀV/^tTÀF:KwFKL?\IBCD/\(D0^nKLACK<DP?¹:9 _]9 £TPTUA

lim inft→∞

lim infε↓0


Pz

(1

2

∫ tε

0


≥ −1

4infa∈O

(

∫ a

m

|b(x)| dx)2

− δ

g$DPBtTºPTBCkη < E

9I`V³jK<?]vACa]TNX<K<uvKLAδ ↓ 0

I?IK<^Ca]TÂEa]TN\]BED0DPgx9 /TFS

T g$D/BCT+TdHUV?¢\]BED,ºPT+X<TUuyuVÂ:9<;]TN?]TTF¢^nD/uyTN\]BCT\IVBWV,AEKLD/?I^9]«]DPBACa]TNBETUuVK<?]K<?]\IVPBnADPgAEa]K<^^CTHJAEKLD/?¢TdV/^C^CG]uyTOACa]BEDPG]/a]DPG]AAEaIV,A

vK<^?]DP? ?]T/V,AEKLº/TNV?IFAxD¼AEKLuyT^fH D/?0ACK<?0GIDPGI^CXLkÂF]KTUBETU?0ACKwVoIXLTV?(FÂAEaIV,A

a, z ∈ VBET jTFZ9( 5 «]D/B

x, y ∈ TÀtKLX<XtBEKAET[x; y]

g$D/BAEa]TÀH X<D/^CTFK<?0ACTUBEº,VX-oT AxTUT?xVP?IF

y KL?¢AEa]T

HUV/^nTx < y

ACa]Kw^tKwÂCDoTNBCTVPFVP^[y;x]

K<?I^nACTVPFZ9f^tVIBW^nAf^xAETU\ACD,bVBWF]ÂCa]TN\]BEDjDgDgX<TUuyuVÂ:9<;+T+/T AtBCKwFDgACa]TN\IVPBEVPuyT ACTB

β9

3 .j 5¤)&¤)& *^+<- 0 ⊂ B ⊆ +3 + 3 5 8a/H1 + 1 Q 99U8 N+B-I? . -

limt→∞

inf

It(ω)∣

∣ ω ∈ Ma,z,1t

= J(a, z)

L 5_4 .LIL]>T8a/ 6 = 50CUN 6 / a, z ∈ iQ R ?S+ / L +<NN . Q ?@5 L 19 Q)+,-¨ & c-TUA

β > 09I«IDPB

ω ∈ C0[0; t]F:TI?]T

ω ∈ C0[0; t/β]o0k

ωr = ωrβg$DPBVPXLX

r ∈ [0; t/β]9

`ba]T?T+aIVºPTω0 = 0

ωt/β = ωt

IV?(F

1

2

∫ t/β

0

v(ωr + z) drs = rβ

=1

β· 1

2

∫ t

0

v(ωs + z) ds.

`bajGI^ω 7→ ω

Kw^VvDP?]T AED DP?]TOuV\]\]K<?]vg$BEDPu Ma,z,βt

DP?0ACDMa,z,1

t/β

9THVGI^CTmDPg

It/β(ω) =1

2

∫ t/β

0

˙ω2r dr =

β2

2

∫ t/β

0

ω2rβ dr

s = rβ=

β

2

∫ t

0

ω2s ds = βIt(ω)

T (?IFinf

It(ω)∣

∣ ω ∈Ma,z,βt

=1

βinf

It/β(ω)∣

∣ ω ∈Ma,z,1t/β

.

r08

D,XLTUA z ∈ K1V?(F

a ∈ K29ijKL?(H T

m /∈ K1 ∩K2TUºPTBCkH DP?0AEKL?jG]D/GI^b\IV,AEa

ωtKLACa

ω0 = 0VP?IF

ωt = a− zaIV/^

1

2

∫ t

0

v(ωs + z) ds > 0,

ACaITd^nTUAMa,z,0

t

Kw^bTUuy\:AxkKL?ACaIK<^HVP^CTNV?IFT (?IF

inf

It(ω)∣

∣

∣ ω ∈⋃

β∈B

Ma,z,βt

= infβ∈B\0

inf

It(ω)∣

∣ ω ∈Ma,z,βt

= infβ∈B\0

1

βinf

It/β(ω)∣

∣ ω ∈ Ma,z,1t/β

.

D,XLTUA K1,K2 ⊆ oTdH DPuy\IV/HJA9]c-T Aη > 0

V?IFHWa]DjD/^CT+Vt0 > 0

tKLACa∣

∣

∣inf

It(ω)∣

∣ ω ∈Ma,z,1t

− J(a, z)∣

∣

∣ ≤ η supB

g$DPBVPXLXt > t0

z ∈ K1

(V?IFa ∈ K2

9]`ba]TU?g$DPBtTºPTBCkt > t0 supB

VP?IFlTºPTBCkβ > 0

T+aIVºPT∣

∣

∣inf

It(ω)∣

∣

∣ ω ∈⋃

β′∈B

Ma,z,β′

t

− 1

βJ(a, z)

∣

∣

∣

=∣

∣

∣

1

βinf

It(ω)∣

∣ ω ∈ Ma,z,1t/β

− 1

βJ(a, z)

∣

∣

∣ ≤ η · supB

β·a]DjD/^CK<?]

β = supB/KLº/T^∣

∣

∣inf

It(ω)∣

∣

∣ ω ∈⋃

β∈B

Ma,z,βt

− 1

supBJ(a, z)

∣

∣

∣ ≤ η

g$DPBVPXLXt > t0 supB

z ∈ K1

(V?IFa ∈ K2

9(THUVPGI^nTηV/^VPBCoIKAEBEVPBCkyAEa]K<^ (?]K<^Ca]T^bACa]TN\IBCDjDgx9

0TF' THVGI^CT

It(ω + z) = It(ω)TNHUVP?^Ca]KLgAfTUº/TUBEk\IV,AEalg$BEDPu

Ma,z,1t

ojkzV?IFPTUA

inf

It(ω)∣

∣ ω ∈Ma,z,1t

= inf

It(ω)∣

∣

∣ ω0 = z, ωt = a,1

2

∫ t

0

v(ωs) ds = 1

.

«]D/BtACaITNuvD/uyTU?0AfVPÊ^nGIuvT+AEaIV,AtAEa]TUBETNKw^Vy\IV,AEaωtKLACa

It(ω) = inf

It(ω)∣

∣ ω ∈ Ma,z,1t

9(cV,ACTBtTtK<XLX^naID, ]ACa(V,AO^CGIHWa¹VP?

ωK<?gV/HJAOF]D0T^TjKw^nA9=@?¹DPBWF:TUBACDlTº,VX<GIV,AETNACa]T¼BEVACTNg$GI?IHJAEKLD/?

Itg$D/BACaIK<^

\IVACaω(T¼^CDPX<ºPT+AEa]TÀs%G]XLTB cVPPBWV?]/T+T0GIV,AEKLD/?I^ ^nTTÀ^CTHJAEKLD/?;8vDPg R+«q/_I g$DPBmTjACBETUuVPXº,VX<G]T^Dg

ItG]?IF:TBbACa]TdHUDP?I^nACBWVK<?0A

K(ω) =1

2

∫ t

0

v(ωs) ds!= 1

V?(FltKLACaACaITNo(D/G]?IF]VPBCklHUDP?IF]KAEKLD/?I^ω0 = z

V?IFωt = a.

THVGI^CTv ∈ C2( )

T¼HUVP?GI^CTdACaITUDPBETUu ;dg$BEDPu^CTHJAEKLD/?;8]9L;NDPg R+«q/_AED I?(F IAEaIV,Amg$DPBOTUº/TUBEkTjAEBCTuyVPX-\DPK<?/A

ωDgIGI?IF:TUBACa]Td/KLº/TU?¢H DP?(^xAEBEVPKL?0AEÂEa]TUBETNKw^VH D/?I^xAWV?0A

λ(^CGIHWalAEaIV,A

ω^CDPX<ºPTÂCa]T

T0GIVACK<DP?I^ωs = λv′(ωs)

g$D/BVX<Xs ∈ (0; t]

]VP?IFω0 = z :9 _PVd

1

2

∫ t

0

v(ωs) ds = 1 :9 _o' ωt = a. :9 _PH_

s :Kw^xAETU?IHUTdDg%^CDPX<G:ACK<DP?(^:AEa]T¼VPG:ACD/?]DPuyD/GI^^CTH D/?IF¢DPBWF:TBT0GIVACK<DP? ]9 _/VM tF]TÊH BEKLoTÂCa]TduyDPACK<DP?DgVlH XwVPÊ^CK<HVX\(VBCACKwH X<TÀDP?MACa]TvBETVXX<KL?]TvK<?¹ACaIT¼\DAETU?0ACKwVX −λv 9Z`ba]TyF:KTUBETU?0AEK<VPXT/G(V,ACK<DP?£HUV?Mo(T

r/_

BETF:GIHUTF£ACDVP?*VPG:ACD/?]DPuyD/GI^ IBW^xANDPBWF:TBNT0GIV,AEKLD/?£K<? ACa]T\IX<VP?]TtKAEa*ACa]TG(^nGIVPXACBEK<HW³ F]TI?]K<?]x(s) = (ωs, ωs)

V?(FF (x1, x2) =

(

x2, λv′(x1)

) ACa]T+T0GIVACK<DP?o(TH D/uvT^x(s) = F (x(s))

g$DPBVPXLXs ∈ [0; t]

9ijTTNTP9 I9]^CTHJAEKLD/?¢]9 _vDPgO bª Pzg$DPBF:TUAEVK<XwÛ9]THVGI^CT

v′V?IFACajGI^

FKw^tXLD:HVX<XLklcKL\(^CHWa]KLANHUDP?0ACK<?jG]DPGI^

g$DPBfTUºPTBCkl\IVK<Bω0 = z

ω0 = v0

DPgK<?]KLACKwVXHUDP?IF]KAEKLD/?I^V?(FTºPTBCkÂoDPGI?IF:TFBETU/KLD/?T I?IFVyG]?IK<0G]T^CDPX<G:ACK<DP?DPg|ACa]TµOefs VAmX<TV/^xAfGI\ACDlACa]T¼o(D/G]?IF]VPBCkDg|AEaIV,AOBCTPK<DP? ^CTUTÀACa]TDPBETUu KL?£^nTHJACK<DP?¹qI9 zDg ªf Pz_% J9

`ba]TBCTdVPBCTOAxDÂF:TU/BCTT^tDPgg$BETUTF:DPuÁKL? :9 _PVd o(THUVG(^nTNTdHUVP?¹HWa]DjD/^CT ω0V?(F

λ9(=@?¢ACa]TNg$D/XLX<D, K<?]ÂTNtKLX<X^Ca]D, :AEaIV,AAEa]TdAxDÂVPFIF:KAEKLD/?IVXH DP?(F:KAEKLD/?I^ :9 _o' tVP?IF ]9 _/H /GIVBWV?0ACTTOACa]T¼T:K<^nACT?IH TDg|VvG]?]Kw0G]TN^nD/XLG:AEKLD/?lAEDyACa]Td^Ck:^xAETUu :9 _M 9«]D/B

λ = 0AEa]TNDP?]X<kl^CDPX<G:ACK<DP?Dg :9 _PVM VP?IF :9 _PH_ %Kw^t/KLº/TU?lojk ωs = z+(a− z)s/t

g$DPB0 ≤ s ≤ tV?(FHUDP?I^CT0G]T?/AEXLklKL?ACaIK<^HVP^CTmT+aIVºPT

1

2

∫ t

0

v(ωs) ds = t · h(z, a)

tKLACah(z, a) =

12(a−z)

∫ a

zv(x) dx,

KLga 6= z

]VP?IF12v(z)

TUXw^CTP9ijK<?IH T

m 6= K1 ∩ K2z ∈ K1

V?(Fa ∈ K2

TdaIVº/Th(z, a) > 0

g$DPBmTUº/TUBEkz ∈ K1, a ∈ K2

V?(FoTHUVPGI^CT

K1 × K2K<ÔH D/uv\(VPHJAmT I?IF

c = inf(z,a)∈K1×K2h(z, a) > 0

9I=@?AEa]Tdg$D/XLX<D,tK<?]lVPÊ^CG]uyTt > 1/c

9:`ba]TU?¢TO³j?]D, g$BCD/u ]9 _Po' ACa(V,AtTUº/TUBEkl^nD/XLG]ACK<DP?Dg ]9 _d %a(VP^ λ 6= 09

`ba]TNK<?0ACTUBE\]BET AWV,ACK<DP?VP^bAEa]TNuyDAEKLD/?DPgVHUX<V/^C^CKwHUVX\IVPBnAEK<HUXLTNaITUX<\I^tGI^bAEDF:T AETUBEuvK<?]T+AEa]TdoTUaIVºjK<DPG]BDgACaIT+^CDPX<G:AEKLD/?IÛ9 TdHUVP?G(^nTdHUDP?I^CTUBEº,V,ACK<DP?DgTU?]TBC/k]THUVG(^nTODg

∂s

(1

2ω2

s − λv(ωs))

= ωsωs − λv′(ωs)ωs = ωs

(

ωs − λv′(ωs))

= 0

TOa(VºPT1

2ω2

s − λv(ωs) =1

2ω2

0 − λv(ω0) =: Eg$D/BVX<X

s ∈ [0; t]9 :9 rd

`ba]Kw^fHUDP?I^CTUBEº,V,ACK<DP?¢XwV´F:TÊH BEKLoTÂCa]Tv^n\TUTFg$DPBOV?jk\DPK<?0AfDg|AEa]TÀ\(V,ACa ]ACaIT¼^C\(TTFDPgAEa]TÀ\(V,ACa£V,A\DPK<?/A

ωsKw^

|ωs| =√

2(E + λv(ωs)). ]9 M `bajGIÂCaITNBEVACTOg$G]?IH ACK<DP?ItHUV?oTNT:\]BETÊ^nTFlVP^tV¼g$G]?(HJACK<DP?¢DPg

EV?IF

λV/^g$DPX<XLD,^9

It(ω) =1

2

∫ t

0

ω2s ds =

∫ t

0

E + λv(ωs) ds

= tE + 2λ, :9 qM ta]TBCT

λVP?IF

EVPBCTNF:TUACTUBEuyKL?ITFo0kT0GIVACK<DP?I^ ]9 _Po' bV?(F :9 _PH 9THVGI^CTÀDPgBETUXwV,ACK<DP? ]9 r fTI?IF AEaIV,Adta]T?]TUº/TUB

ωKw^NV^CDPX<G:AEKLD/?£DPg :9 _PVM fT¼aIVº/T E ≥

−λv(ωs)g$DPBmVPXLX

s ∈ [0; t]VP?IFACaITÀ\IVACaMHV?DP?IXLk¢^xAEDP\¹VP?IF¢ACGIBC?¹VAm\DPK<?/AW^

xtKAEa −λv(x) = E

9c-TUA

x ∈ o(Ty^CGIHWaMV\(D/KL?0ANV?IF£VPÊ^nG]uyTv′(x) = 0

9`baITU?ηtKLACa

ηs = xg$D/BOVPXLX

s ≥ 0K<^mACaIT

G]?]Kw0G]T¼^CDPX<G:AEKLD/?Dg :9 _PVM ttKAEa η0 = xVP?IF

η0 = 09 D,´VPÊ^nG]uyTNAEaIV,A ωs = x

g$D/BO^CDPuyTs > 0

9`ba]T?

(ωs−r)r∈[0;s]K<ÔVXw^CDlVl^CDPX<G:ACK<DP?MDg :9 _PVd ttKLACa£^nAEVPBnAOKL? x VP?IF¹K<?]KLACKwVX^n\TUTFM~^CDlTÀaIVº/T

ωs−r = ηr = xg$DPBVX<X

r ∈ [0; s]9I`ba]Kw^^Ca]D,ÂCaIVAfVv\(D/KL?0A

x 6= ztKLACa

E = −λv(x) VP?IF v′(x) = 0HUVP?]?]DPAmoTÀBETVPHWaITF¢ojkVl^CDPX<G:ACK<DP?ωDPg :9 _PVM J9`bajGI^mta]TU?ITUºPTBOVÂ?]DP? H D/?I^nAEV?0Af\(V,ACa£BCTVPHWa]T^V?

x ∈ tKLACaE = −λv(x) AEa]TU?T+aIVºPT ωs = λv′(ωs) 6= 0

VP?IFACa]TN\IVACaVX<bVkj^bHWaIVP?]PT^bF:KLBETH ACK<DP?ACaITUBETP9(«K<PG]BETÀ:9<;+KLX<X<GI^xAEBEVACT^bAxDÂF]KTUBETU?0Af³jK<?IF]^DPg^CDPX<G:AEKLD/? ]DP?ITdta]TUBETωsuyD,ºPT^buyDP?]DPACDP?IK<HVX<XLk

V?(FlD/?]TNta]TUBETOACa]TN\(V,ACaBETVPHWaIT^bVv\(D/KL?0AbtKAEa −λv(b) = E

V?(FlAEG]BE?IÂCa]TBCT/9ijK<?IH TdACa]TyF:K TBCT?/AEK<VPXT/G(V,ACK<DP? :9 _PVd bK<ÔVG]ACDP?IDPuyDPGI^fV?IF¹^CK<?IH TvVÂ^nD/XLG]ACK<DP? ω HWaIVP?]PT^F:K<BCTH ACK<DP?¢TºPTUBEkAEKLuyTNKw^tBCTVPHWa]T^bVy\(D/KL?0A

xtKLACa −λv(x) = E

:AEa]Td\IVACaHUVP?BETV/HWaVAuyD/^nAAxDÂF:Kw^xAEKL?(HJA

rPr

xz

−λv(x)E

m a xz

−λv(x)

E

ma b

«KL/G]BET:9<; R ?a6 9 Â 8 CE+6 LIL]89 -JC . -#+ 9 -JF 5j- >`V + 9 5 =9 5 L 8 -76K5 /= 5Ci+m 8S. -J6K5 / ]9 _/VM Q \+<C`+F +5 /SL > 4`5 / 9 6 1 +"C\-%?@+T4 .9 + λ > 0 Q R ?@+4 8 CUf+ 1L 6 / +6 9 -I?S+eA0C .`V ? 5 = -%?@+ =8a/ 4<-J6K5 / x 7→ −λv(x) QBR ?@+ 3 5 L 1 V@. CU-5 = -I?S+ L 6 / + 9 45CUC`+ 9JV 5 /219 -#5-%?@+ V 56 / - 9 f_6 9 6I- + 1 3"> -%?@+ VS. -%? ω QR ?@+-%?6K4 1 5- 9. CE+ (ω0,−λv(ω0))

./H1 (ωt,−λv(ωt)) Q 50-I? 9 5 L]8 -J6K5 /@9 9 - . CU- . - z ∈ K1

: ?S+ .D1 -#5F . C 19 .T/ +"6 A? 3 5 8 CE?S5 5 1 5 = -%?@+Bb_+<CE5 m :./H1 /H.LIL]> CE+ . 4`? .V 506 / - a ∈ K2 QR ?S+ L + = -Z? ./H1 6IN . AM+ 9 ?@50F 9 .= C`+`+ 9 5 L]8 -J6K5 /S: 6 Q + Q 5 / + FG6I-%?E > 0 : -%?@+iCU6 A_?a- ? .0/21 6IN . AM+ 9 ?@50F 9P. 3 5 8a/21 9 5 L]8 -J6K5 /S: 6 Q + Q 5 / +iFG6I-I? E ≤ 0

F^?S+<C`+P-%?@+ V@. -I? ω- 8 C /@9 . - -I?S+ V 506 / - b FG6I-%? −λv(b) = E Q\DPK<?/AWÔDg[AEa]T^CT¼?(V,ACGIBCT/9=@?MACa]Kw^NHUV/^nTÀACa]Ty^CDPX<G:ACK<DP?£D/ÊH K<XLXwV,AET^mo(TUAxTTU?¹AEa]T^CTv\(D/KL?0AEÔ\TUBEKLD:F:KwHUVPXLX<kP9`bajGI^bTUº/TUBEkl^CDPX<G:ACK<DP?Dg ]9 _/VM HWaIV?IPT^bF:K<BETHJAEKLD/?D/?]X<kVI?]KLACTN?jG]u¼o(TBtDgACK<uvT^bo(TUg$DPBET+ACK<uyT

t9

=@?¢DPBWF:TUBbAED I?IFACaITd\IV,AEa¢ta]K<HWa¢uyK<?]KLuyKw^nT^tAEa]TNBEVACT+g$G]?(HJACK<DP?ItTN?]TUTFACD³PTTU\ACBWVPHW³ÂDPgAEa]T

F:K TBCT?0A\(D0^C^CKLoIXLTOACBWVPHUT^DgACaITN\IV,AEa-9I«]D/BbACa]TNBETUuVPKL?]K<?]v\IVBCAtDgACa]Kw^^CTH ACK<DP?T+GI^CT+ACa]T+g$D/XLX<D,tK<?]?]DPAEV,AEKLD/?-9(`baITd\IV,AEa

(ωs)0≤s≤tK<^m^CVPK<FAEDÂaIVº/T U 5 /.

T = (x0, x1, . . . , xn)ta]T?

ω0 = x0ωt =

xnIV?IFAEa]TN\IVACa

ωuvD,º/T^buyDP?IDACD/?]KwHUVX<X<kÂKL?TKAEa]TUBmF:KLBETH ACK<DP?g$BEDPu

xi−1AEDxig$D/B

i = 1, . . . , nK<?

DPBWF:TBbVP?IF¢HWaIV?IPT^bF:K<BETHJAEKLD/?D/?]X<klV,AbACaITN\(D/KL?0AE^x1, . . . , xn−1

9 TNGI^nT+AEa]TdVoIo]BCTºjK<VACK<DP?

|T | =

n∑

i=1

|xi − xi−1|

g$DPBOAEa]TXLT?]AEaMDPg%ACaIT¼AEBEV/H TyV?IF ^CDPuyT AEKLuyTÔKwF:TU?0AEKg$kTtKLACa£ACa]T^CT A ⋃n

i=1[xi−1, xi]DgbH D,º/TUBETF

\DPK<?/AW^bACDtBEKLACTminT

maxT

v|T

(DPBinfx∈T v(x)

9«]D/Bt\(D0^nKLACK<ºPT+g$GI?IHJAEKLD/?I^f : → TNGI^CT+ACa]T

?]DPAEV,AEKLD/?∫

T

f(x) dx :=

n∑

i=1

∣

∣

∫ xi

xi−1

f(x) dx∣

∣.

`ba]TyVoI^CDPX<G:AETdº,VX<G]T^+VBETNAWV³PT?ACDuV³/TNACa]TvK<?/AETU/BEVPX\D/^CKAEKLº/T¼TºPTU?Mta]TU?xi < xi−1

9= gV^nD/XLG ACK<DP?ωDg :9 _PVM aIV/^bACBWVPHUT T = (x0, x1, . . . , xn)

IACa]Kw^tAEa]TU?¢K<uy\]XLK<TÂCaIVAv(x1) = · · · = v(xn−1) =

−E/λ V?IF¢TVPHWaDPgAEa]T x1, . . . , xn−1K<^fTUKLACa]TB

minTDPB

maxT9IT AxTUT?¢ACa]TÀ\(D/KL?0AE^

xiACaITÀ\IVACa

KwÔ^nACBEKwHJACX<kuyDP?]DPACD/?]K<H(K©9 TP9ZV,gAETUBOACa]Ty^nAEVBCAmK<?zKLAOD0^CHUKLX<XwV,ACT^ TUBEDlD/BmuyDPBETdAEKLuyT^moT AxTUT?

minTV?(FmaxT

o(TUg$DPBETÀKLAOBETV/HWa]TâVAfAEKLuyT

t9¡m^nK<?]ACa]KwÔ?]DAWV,AEKLD/?¹TvHUVP?g$DPBEuÀG]XwV,AETdAEa]TÀg$D/XLX<D,tK<?]c-TuyuyVI9

3 .j 5¤)&w® &,*^+<-λ,E ∈ .0/21. -JC . 4`+ T = (x0, . . . , xn) 3 + A06If+ / Q R ?S+ / -I?S+ = 5 LIL 50FG6 / A-JF 545 /H1 6I-J6K5 /@9 . C`+ +m 8 6If .L + / - Q R ?@+ 8a/ 6Km 8 + 9 5 L]8 -76K5 / ω : [0; t] → 5 =

ωs = λv′(ωs)= 5C .LIL s ∈ [0; t]

FG6I-I? 6 / 6I-J6 .L 45 /H1 6I-J6K5 /S9 ω0 = z .0/21 ω0 = sgn(x1 − x0)√

2(E + λv(0))? .9 -JC . 4+ T ./H1

9 5 L f+ 9 :9 _o' .0/21 ]9 _/H Q 6 +B? . f+ x0 = z : xn = a : E = −λv(xi)= 5C i = 1, . . . , n − 1 : .9 F + LIL^.9 E > −λv(x) = 5C.LIL minT < x < maxT :;.0/21 -%?@+ VS. 6IC (λ,E) 9 5 L f+ 9

∫

T

v(x)√

E + λv(x)dx =

√8 ]9Q,Vd

r0

./H1∫

T

1√

E + λv(x)dx =

√2t. :9 o2

)+,-¨ & fÊ^nG]uyTvACaITyHUDP?IF]KAEKLD/?I^mg$BCD/u(j)9-`ba]T?

ωKw^+V^CDPX<G:AEKLD/?MDg ]9 _/VM AEa]TUBETVBETdACK<uyT^

t0, t1, . . . , tntKLACa

ωti= xi

g$DPBi = 0, . . . , n

V?IF¢oT AxTUT?¢ACa]TNAEKLuyT^tiACa]T¼\]BCD:HUTÊ^tuyD,ºPT^buyD/?]D ACD/?]KwHUVX<X<kP9]«]D/BV?jkÂK<?0ACTU/BEVPo]X<Tj\(D0^nKLACK<ºPTOg$G]?(HJACK<DP?

g : → ^CG]oI^nACKLACG:AEKLD/?¢GI^nK<?] :9QD kjKLTX<FI^∫ t

0

g(ωs) ds =

n∑

i=1

∫ ti

ti−1

g(ωs) ds

=n∑

i=1

∫ xi

xi−1

g(x)dx

sgn(xi − xi−1)√

2(E + λv(x))

=

∫

T

g(x)√

2(E + λv(x))dx. ]9 d

\I\]XLkjK<?] :9 M %ACDyAEa]TOg$G]?IHJAEKLD/? g = v/KLº/T^

1 =

1

2

∫ t

0

v(ωs) ds =

1√8

∫

T

v(x)√

E + λv(x)dx.

`ba]Kw^tKw^bT0GIV,AEKLD/? :9 ,Vd W9I\I\]XLkjK<?] :9 M %ACDvACaITdH DP?(^xAWV?0Abg$G]?IHJAEKLD/? g = 1PK<ºPT^

t =

∫ t

0

1 ds =

1√2

∫ a

0

1√

E + λv(x)dx,

ta]KwHWaKw^tT/G(V,ACK<DP? ]9Q,o' J9 D, VPÊ^nGIuvT+H D/?IF:KLACK<DP? K E J9(«]DPB i = 1, . . . , nF:T(?]T+ACa]T+g$GI?IHJAEKLD/?

Fiojk

Fi(x) =1√2

∣

∣

∫ x

xi−1

1√

E + λv(x)dx∣

∣

g$DPBmVX<XxoT AxTUT?

xi−1VP?IF

xi9(`ba]T?

FiK<Î?]KLACTÀoTHVGI^CTNDg :9 o' (^nACBEK<H ACX<kluyDP?IDACD/?]KwH K<?IH BETV/^ K<?]yKg

xi > xi−1V?IFF:THUBCTVP^CKL?I¼TX<^CT_ IV?IFaIV/^

Fi(xi−1) = 09]«IG]BnAEa]TUBF]TI?]T

tk =k∑

i=1

Fi(xi).

s%0GIVACK<DP? ]9Q,o' b/KLº/T^ tn = t9(THVGI^CTNAEa]TNg$G]?IH ACK<DP?I^

FiVPBCTNuyDP?IDACD/?]KwH+ACa]Tka(VºPTNK<?0º/TUBW^nT+g$GI?IH ACK<DP?(^

F−1i

V?(FlTdHUVP?F:TI?]Tω : [0; t] → ojk

ω(s) = F−1i (s− ti−1)

g$DPBVX<Xs ∈ [ti−1, ti]

9 TNtKLX<X-\]BED,ºPT/AEaIV,A

ω^CVACKw^ (T^tVX<XACaITdH DP?(F:KAEKLD/?I^g$BEDPu E J9THVGI^CTÀTyaIVºPTti − ti−1 = Fi(xi)

VP?IFAEa0G(^F−1

i (ti − ti−1) = xi = F−1i+1(ti − ti)

ACa]Tg$G]?IH ACK<DP?

ωKw^+TXLX F]TI?]TF¹D/?¹ACaITyHUDP?]?ITHJAEKLD/?M\DPK<?0AE^+V,A+ACK<uyT^ ti V?(F¹Kw^NH DP?0AEKL?jG]D/GIÛ9`ba]Kw^NVXw^nD^Ca]D,^

ωti= xi

g$DPBi = 0, 1, . . . , n

V?(FT^n\THUK<VPXLX<kω0 = x0 = z

VP?IFωt = xn = a

9THVGI^CT+ACa]T

FiVBETÀF:K TBCT?/AEK<VPo]X<TdV,AmVPXLX\(D/KL?0AE^

x^nACBEK<H ACX<koT AxTUT?

xi−1V?IF

xi(ACa]Tdg$G]?IHJAEKLD/?

ωKw^F:K TUBETU?0ACKwVo]X<T+DP?ACa]TNK<?0ACTBCº,VXw^

(ti−1; ti)tKAEaF:TUBEKLº,VACK<ºPT

ωs =1

F ′i (ωs)

= sgn(xi − xi−1)√

2(E + λv(ωs)).

THVGI^CTωKw^H D/?0ACK<?0GIDPGI^VP?IFACa]TÀXLK<uyKAW^

lims→tiωsT:K<^nA ]Td^CTUT+ACa(V,A

ωK<^TUºPT?F:KTUBETU?0AEK<VPo]XLTND/?

[0; t]tKLACa

ω0 = sgn(x1 − x0)√

2(E + λv(0))VP?IF

ωti= 0

g$D/Bi = 1, . . . , n− 1

9

r/q

¡f^CK<?]vACa]TdÊVuyT+³jKL?IFDg|VBEPGIuvT?0AbVP/VK<? :T I?IF

ωs =sgn(xi − xi−1)

2√

2(E − λv(ωs))· 2λv′(ωs) · sgn(xi − xi−1)

√

2(E − λv(ωs)) = λv′(ωs),

IBW^xAOo(TUAxTTU?¢AEa]TtiV?IFACa]T?¹DP?AEa]TÀtaIDPX<TÀK<?/AETUBEº,VX

[0; t]9`bajGI^

ωBETVPXLX<k^CDPX<ºPT^bACaIT¼F:K TBCT?0ACKwVX

T0GIVACK<DP?g$BCD/u E 9¡f^CK<?]vACa]Td^CG]oI^nACKLACG:AEKLD/?1

2

∫ t

0

v(ωs) ds =1√8

∫

T

v(x)√

E + λv(x)dx

VP^bK<?ACa]T (BE^nAt\IVBCA :TNVXw^nDyPTUAtoIVPHW³ :9 _o' %g$BEDPu :9 ,Vd W9 0TF'

D, TdaIVº/TNBCTF:GIH TFACa]T¼\]BEDPo]X<TUuÁDg[uyK<?]KLuyKw^nK<?]It(ω)

g$D/Bm^CDPX<G:ACK<DP?I^ωDg|ACaIT¼^Ckj^nACTu :9 _M ACDvAEa]TN\]BEDPo]X<TUu DPguyK<?]K<uvKw^CKL?]

It(E, λ) = tE + 2λg$DPB^CDPX<G:AEKLD/?I^(E, λ)

DPgAEa]Td^nk:^nACTu ]9QD J9«]D/BtVvACBWVPH TTF:TI?IT

HT =

(E, λ)∣

∣ E ≥ − infx∈T

λv(x)

⊆ 2

V?(FÂg$G]BCACaITUBEuvD/BCTNF:TI?]T+AEa]T+g$G]?IH ACK<DP?I^f, g : Ht → [0;∞]

ojk

f(E, λ) =

∫

T

1√

E + λv(x)dx

V?(Fg(E, λ) =

∫

T

v(x)√

E + λv(x)dx.

«KL/G]BET¼:9Q8yK<XLX<GI^nACBWV,ACT^tAEa]T¼F:D/uVK<?HT

9DPACa¢g$G]?(HJACK<DP?(^mVPBCT (?]KAETÀK<?ACaITÀK<?/AETUBEKLD/BfDPg|ACa]TvF:DPuVK<? o]G:AOHUV?¢oTNKL?I?]KLACTÀVAACa]TNoDPGI?IF]VBEkP9(`ba]TNT0GIVACK<DP?I^ :9 0 bVPBCT+T0G]K<º,VX<TU?0AtACD f(Eλ, λ) =

√2tVP?IF

g(E, λ) =√

89Z«]D/Bm\(V,ACa(^mtaIK<HWa HWaIV?]/TÀF:K<BCTHJAEKLD/?£V,AN^CDPuyTÀ\DPK<?0AOTÀtK<XLX I?(F£^CDPX<G:ACK<DP?(^

(E, λ)Dg ]9QD ta]KwHWa¹X<Vk¢D/?ACa]TvoDPG]?IFIVBEkDPg HT9Z«]D/Bm\(V,ACa(^fta]KwHWaM/D^nACBWVK<Pa0Ag$BEDPu

zACDaTÀtK<X<X

I?IF¢^CDPX<G:ACK<DP?(^(E, λ)

K<?AEa]TNK<?/AETUBEKLD/BbDgHT

93 .j 5¤ & 1 &*^+<-

t > 0 ./21 T 3 + . -7C . 4+ = C`5N z ∈ - 5a ∈ 9U8 4? -%? . - v|T 6 9 / 50-45 /S9 - ./ - QOR ?@+ / -I?S+<C`+B6 9Z. - NT5 9 -O5 / + 9 5 L]8 -J6K5 / (E, λ)

5 = ]9QD Q)+,-¨ & «]D/B

E > − infx∈T λv(x)TyHV?²HWa]DjD/^CTyV?

E∗oT AxTUT? − infx∈T λv(x)

VP?IFE9

`ba]T?v(x)/(E∗ + λv(x))3/2 K<ÔV?K<?0ACTPBWVo]X<TNG]\]\TUBOo(D/G]?IF¢DPg v(x)/(e + λv(x))3/2 g$DPBmVPXLX e K<?MV

(E − E∗) @ TUK<PajoDPG]BEa]DjD:FDg E 9ijDÂTdHUVP?GI^CT+ACa]TNAEa]TUD/BCTu VPo(D/G:AfK<?/AETUBWHWaIV?IPK<?]vACa]TÀcTUoT^CPG]T K<?/AETU/BEVPXtKLACaF:TBCK<º,V,AEKLº/TÂCDy/T A∂

∂Eg(E, λ) = −1

2

∫

T

v(x)(

E + λv(x))3/2

dx < 0.

ijDlg$DPBOTUºPTBCkλACa]TvuV\

E 7→ g(E, λ)Kw^m^nACBEK<H ACX<kF:THUBCTVP^CKL?]ÂVP?IF¢AEa]TUBET¼HV?¹oTvVAmuyD/^nAmD/?]T

EλtKLACag(

Eλ, λ)

=√

89

rj

λ0

E

− infx∈T

λv(x)

HT

«KL/G]BETy]9 8 R ?a6 9 A 8 CE+\6 LIL]89 -JC . -#+ 9 -%?@+ 1 50N . 6 / HT5 = -I?S+ =8a/ 4"-76K5 /@9 f ./21 g QBR ?@+ 1 5N . 6 / 6 98a/ 3 5 8a/H1 + 1 6 / 1 6ICE+4<-J6K5 /S9 λ → ∞ .0/21 E → ∞ Q - 6 9T3 5 8a/H1 + 1B= CE5N 3 + L 5F 3"> λ 7→ − infx∈T λv(x)

:F^?6K4?6 9 +`m 8S.L -#5 −λ supx∈T v(x)= 50C λ ≤ 0 ./H1 - 5 −λ infx∈T v(x)

= 50C λ ≥ 0 Q KLACa*ACaITa]TUX<\²DgACa]TlKLuy\]X<K<HUKAÀg$G]?IHJAEKLD/?*ACa]TDPBETUu TÂHUVP?²HUVXwH GIX<VACTyACaITÂF:TUBEKLº,VACK<ºPTDg

Eλ9

=@?0ACTBEHWaIVP?]PK<?]¼ACa]TNK<?0ACTPBWVXtKAEaACa]TdF:TBCK<º,V,AEKLº/TOV/^tVoD,ºPT+T+/T A∂

∂λEλ = −

∂∂λg(Eλ, λ)∂

∂E g(Eλ, λ)

= − (− 12 )∫

Tv2(x)

(

Eλ + λv(x))−3/2

dx

(− 12 )∫

T v(x)(

Eλ + λv(x))−3/2

dx

= −∫

T v2(x) dµ(x)

∫

T v(x) dµ(x)ta]TBCTµK<ÂEa]TN\]BEDPoIVPo]KLX<KLAxkÂuyTVP^CG]BETjtKLACaF:TU?(^nKLAxk

dµ

dx=

1

Z

(

Eλ + λv(x))−3/2

V?(FÂAEa]TN?]DPBEuVX<K<ÊV,AEKLD/?HUDP?I^nAEVP?/AtKw^Z =

∫

T

(

Eλ + λv(y))−3/2

dy.

«]GIBnAEa]TUBEuyDPBETfg$D/B(E, λ) ∈ (HT )

T+aIVºPT∂

∂Ef(E, λ) = −1

2

∫

T

(

E + λv(x))−3/2

dx = −Z2V?(FÂAEa0G(^

∂

∂λ

(

f(Eλ, λ))

=∂f

∂E(Eλ, λ) ·

∂

∂λEλ +

∂f

∂λ(Eλ, λ)

=Z

2·∫

T v2(x) dµ(x)

∫

Tv(x) dµ(x)

− Z

2

∫

T

v(x) dµ(x)

=Z

2·∫

T v2(x) dµ(x) −

(∫

T v(x) dµ(x))2

∫

Tv(x) dµ(x)

≥ 0.s%0GIVPXLKLAxkD/G]XwFDP?]X<ka]D/X<F¢g$D/BOACa]TyHUV/^nT¼DgHUDP?I^nAEVP?/A

v|T9-ijDACaIT¼uV\

λ 7→ f(Eλ, λ)K<^+^nACBEK<H ACX<k

K<?IH BETV/^nK<?]yV?IFlACa]TBCTdHV?o(TdVAtuvD0^xAD/?]TλtKLACa

f(

Eλ, λ)

=√

2t9I`ba]Kw^ I?]Kw^Ca]T^bAEa]TN\]BEDjDgx9

0TF'

r/

3 .j 5¤)& ¿ & *^+<-T . -JC . 4+ZFG6I-%? m ∈ T .0/21 t ≥ 2|T |/

∫

Tv(x) dx Q R ?@+ / +m 8S. -76K5 / ]9QD ? .9.\9 5 L]8 -J6K5 / (E, λ)

FG6I-%?FG6I-I?E, λ > 0 Q

)+,-¨ & emTI?]Tλ∗ = (

∫

T

√

v(x) dx)2/8V?IFVPÊ^CG]uyT

0 < λ ≤ λ∗9]`ba]T?¢T+aIVº/T

g(0, λ) =

∫

T

v(x)√

λv(x)dx =

1√λ

∫

T

√

v(x) dx ≥√

8.

V?(FÂAEa]TdF:DPuyK<?IV,AETF¢H D/?jºPTUBEPT?IH TfACa]TDPBETUu /KLº/T^lim

E→∞g(E, λ) = 0.

`bajGI^g$DPBVPXLX0 < λ ≤ λ∗

ACa]TBCTNTjKw^nAE^tV?Eλ ≥ 0

tKAEag(Eλ, λ) =

√89

THVGI^CTmDPgg(0, λ∗) =

√8T+aIVº/T

Eλ∗ = 09(«IV,AEDPG- ^bX<TUuyuVÀAEa]TU?¢/KLº/T^

lim infλ↑λ∗

f(Eλ, λ) ≥∫

T

1√

λ∗v(x)dx.

THVGI^CTvKwÔ\(D0^nKLACK<ºPTyV?IF

v(m) = 0TvaIVºPT

v′(m) = 0V?IF

v′′(m) ≥ 09Z`ba]T?£ojk¹`Vk0X<DPB ÂCaITUDPBETUu ACa]TBCTT:K<^nAE^dV

c > 0V?IF*VH X<D/^CTF K<?0ACTUBEº,VX

I ⊆ tKAEam ∈ I ⊆ T

-^nGIHWa AEaIV,Av(x) ≤ c2(x−m)2

g$DPBVPXLXx ∈ I

9]`ba]TUBET g$D/BCT+T I?IF∫

T

1√

v(x)dx ≥

∫

I

1√

c2(x−m)2dx =

∫

I

1

c|x−m| dx = +∞

V?(FÂAEa0G(^λ 7→ f(Eλ, λ)

K<^tVyHUDP?0ACK<?jG]DPGI^g$GI?IHJAEKLD/?tKLACalimλ↑λ∗

f(Eλ, λ) = +∞.

µm?ACa]TNDPACa]TBtaIV?IFoTHVGI^CT+Dgg(E0, 0) =

√8T+aIVºPT

E0 = (∫

Tv(x) dx)2/8

9IijDyg$D/Bλ = 0

TPTUA

f(E0, 0) =

∫

T

1√E0

dx =

√8

∫

Tv(x) dx

|T |.`DPPTUACa]TBACa]Kw^^Ca]D,^jAEaIV,Atg$D/BVX<X

t ≥ 2|T |∫

T v(x) dxACaITUBETOT:K<^nAE^Vy^CDPX<G:ACK<DP?(Eλ, λ)

tKLACaf(Eλ, λ) =

√2t9 0TFS

3 .j 5¢¤ & Ã & R ?@+"CE+ . CE+ /S8 N 3 +<C 9 ε, c1, c2 > 0 :G98 4?j-%? . --%?@+ = 5 LIL 5FG6 / AP?S5 L 19"c 5Ch+"f+"C >-JC . 4+ T 9 - . CU-76 / A6 / K1: + /21 6 / A6 / K2

:O.0/21 f_6 9 6I-J6 / A-%?@+ 3.LIL Bε(m)-%?@+"CE+\6 9.i/ 5 / +"N V - >:4 L 5 9 + 1 6 / -#+"CUf .0L A ⊆ :98 4?-%? . - A ⊆ T : |A| = ε .0/21 F + ? . f+ c1 ≤ v(x) ≤ c2

= 5Ce+"f+"C > x ∈ A Q)+,-¨ & THVGI^CT

m /∈ K1 ∩K2TUKLACa]TB

K1DPBK2

aIV/^tVv\(D0^nKLACK<ºPTNF:Kw^xAWV?IHUTOg$BCD/um9(c-T A

εoTNDP?IT

ACaIKLBWFDgACa]Kw^F:Kw^xAWV?IHUTP9IemT(?]TA′ = x ∈ | ε ≤ |x −m| ≤ 2ε VP?IFXLTUA c1 = inf v(x) | x ∈ A′V?(F

c2 = sup v(x) | x ∈ A′ 9s%V/HWaACBWVPHUTÀ^xAWVBCACK<?]ÂK<?K1

(T?IF:K<?]lK<?K2

V?IFºjKw^nKLACK<?]lACa]TvoIVX<XBε(m)

TUKLACa]TBOHUBCD0^C^CT^[m −

2ε;m − ε]DPB

[m + ε;m + 2ε]9c-TUA

Ao(TvAEa]TvHUBCD0^C^CTF¹KL?0ACTBCº,VPX'9`ba]TU?*H X<TVPBCX<k |A| = ε

V?(FMV?(FoTHUVPGI^CTODPg

A ⊆ A′ ACa]TNT^xAEKLuV,AET^g$D/B v DP? A a]DPXwFZ9 0TFS 3 .j 5¤ & & 50C +<f+<C > η > 0

-%?@+"CE+Z6 9 . t1 > 0 :;9U8 4?-I? . - F^?S+ / +<f+"C t ≥ t1: T 6 9 . -7C . 4+

= CE50N z ∈ K1- 5a ∈ K2

FG6I-%?m ∈ [z; a] ./H1 (E, λ) 9 5 L f+ 9 ]9QD : -I?S+ / F + ? . f+∣

∣

∣It(E, λ) −1

4

(

∫

T

√

v(x) dx)2∣∣

∣ ≤ η.

r/z

)+,-¨ & `ba]Kw^fHVP^CT+K<^fKLX<XLG(^xAEBEVACTF¢K<?¢ACaITdXLTUgAfaIVP?IF¢KLuVPPTNDg I/G]BETÀ:9<;P9ITHVGI^CTm ∈ [z; a]

VP?0k\IVACag$BCD/u

zACDau¼GI^nAº0Kw^CKA

mVP?IFACajGI^tT I?IF

E > −λv(m) = 09]`bajGIÂEa]TNDP?]X<kl\D/Ê^CKLo]X<TOACBWVPHUTK<?AEa]Kw^HUV/^nTOK<^

T = (z, a):oTHUVPGI^CTmAEa]TN\]BED:H TÊ^bH D/G]X<FD/?]XLkÂACG]BE?VA\(D/KL?0AW^

xta]TUBET −λv(x) = E

9 D,XLTUA η > 0

9IemTI?]TL = sup

|a− z|∣

∣ z ∈ K1, a ∈ K2

9]`ba]T?¢T+PTUA√

2t =

∫

T

1√

E + λv(x)dx ≤

∫ a

z

1√Edx ≤ L√

E

V?(FÂAEa0G(Ê ≤ L2

2t2.

ijDyT+HV? I?IF¢Vt1 > 0

tKLACaE · t < η ]9 zd ta]T?]TUº/TUB

t ≥ t19

·a]DjD/^CK<?]Ac1

IV?IFc2V/^bKL?¢X<TUuyuVy]9 zvT+PT A

√8 =

∫

T

v(x)√

E + λv(x)dx ≥

∫

A

c1√E + λc2

dx =c1|A|√E + λc2

V?(FÂAEa0G(^λ ≥ c21|A|2 −E

8c2≥ c21|A|2 − L2/2t2

8c2.

ijDyT+HV?HWa]DjD/^CT+Vv^CuVX<Xc3 > 0

VP?IFKL?IHUBCTVP^CTt1ACDVPHWa]K<TUº/T

λ > c3taITU?]TºPTUB

t ≥ t19

THVGI^CTmDPglimE↓0

∫

T

v(x)√

E + v(x)dx =

∫

T

√

v(x) dx

T+HV? I?IF¢Vc4 > 0

tKLACa∫

T

v(x)√

E + v(x)dx ≥

√

1 − η/J(z, a)

∫

T

√

v(x) dx

g$DPBVPXLXE ≤ c4

9I=@?IH BETV/^nTt1G]?0ACK<X

L2

2t2c3< c4

V?(FÂAEa0G(^√

8 =

∫

T

v(x)√

E + λv(x)dx

≥ 1√λ

∫

T

v(x)√

L2/2t2λ+ v(x)dx

≥ 1√λ

∫

T

v(x)√

c4 + v(x)dx

≥ 1√λ

√

1 − η/J(z, a)

∫

T

√

v(x) dx

g$DPBVPXLXt ≥ t1

9(ijD/XLºjK<?]vACa]Kw^bg$D/BλT+/T A

2λ ≥ (1 − η/J(z, a))J(z, a) = J(z, a) − η. ]9L;~M THVGI^CT

EKw^t\D/^CKAEKLº/TOTNVXw^nDI?IF

√8 =

∫

T

v(x)√

E + λv(x)dx ≤ 1√

λ

∫

T

√

v(x) dx

V?(FÂAEa0G(^2λ ≤ J(z, a). :9<;P;

~

«]D/BACa]TNBWV,AETmg$GI?IHJAEKLD/?ItT/G(V,ACK<DP? ]9L;~M %PK<ºPTÎt(E, λ) = E · t+ 2λ ≥ J(z, a) − η

V?(FlT/G(V,ACK<DP?(^ :9 zM VP?IF :9<;P; [/KLº/TIt(E, λ) = E · t+ 2λ ≤ J(z, a) + η

g$DPBVPXLXt > t1

9 0TF' 3 .j 5¤ & '& 50C +<f+<C > η > 0

-%?@+"CE+Z6 9 . t2 > 0 :;9U8 4?-I? . - F^?S+ / +<f+"C t ≥ t2: T 6 9 . -7C . 4+

= CE50N z ∈ K1- 5a ∈ K2

FG6I-%?m /∈ [z; a] : ./H1 (E, λ) 9 5 L f+ 9 ]9QD : -I?S+ / F + ? . f+∣

∣

∣It(E, λ) −1

4

(

∫

T

√

v(x) dx)2∣∣

∣ ≤ η.

)+,-¨ & `ba]Kw^NHUVP^CTvK<^+K<XLX<GI^nACBWV,AETF£KL? ACaITvBEK<Pa0ANaIV?IF£K<uyVPPTvDg IPG]BET:9<;P9-THVGI^CTÀAEa]Ty\IV,AEaaIV/^fAEDHWa(V?]/T¼F:K<BCTHJAEKLD/?MT¼tK<XLX|a(VºPT

E < 0K<?MAEa]KwÔHVP^CTP9 KLACa]D/G:ANXLD0^CÔDgPTU?ITUBWVX<KAxk¢TvHV?

VPÊ^CG]uyT+ACaIVAm < a, z

9£T¼HVX<XVº,VX<G]Tb ∈ VPF:uyKw^C^CK<o]XLTÀKg|KLAmX<KLT^K<?¢ACa]TdK<?0ACTBCº,VX

(m; min(a, z))V?(F¢Kg%V/F]F:KLACK<DP?IVPXLX<kv(x) > v(b)

g$DPBOVX<Xx > b

a]DPXwF]^9(«IDPBmV/F:uyK<Ê^CKLo]X<Tdº,VX<G]T^bH D/?I^CK<F:TBtACaITdACBWVPHUT

T = (z, b, a)VP?IFF:TI?IT

hz,a(b) = 2

∫

(z,b,a)1√

v(x)−v(b)dx

∫

(z,b,a)v(x)√

v(x)−v(b)dx.

¡f^CK<?]`VkjX<DPBtV\]\IBCD:K<uyVACK<DP?VP^bK<?X<TUuyuVÂ:9 :D/?]TN^nTTÂEaIV,Atg$D/Bb → m

ACa]TN?jG]uyTBEVACDPBtHUDP?jºPTBC/TÂCD+∞ VP?IFojklF:DPuyK<?IV,AETF¢H D/?jºPTUBEPT?IH TmACaITdF:TU?]D/uyKL?IVACD/BH DP?jº/TUBEPT^%ACD ∫

(0,m,a)

√

v(x) dx9i:D

hK<^

VyH D/?/AEKL?jG]D/GI^g$G]?IH ACK<DP?¢tKLACahz,a(b) → ∞ g$DPB

b→ m9

c-TUAεc1

VP?IFc2V?(F

AoTÂVP^+K<?*XLTuyuyV]9 zI9 £TÂDPGIX<F£XLK<³PTvAED I?IF*V

b ∈ Bε(m)tKLACa

hz,a(b) = t]^CDyTN?ITUTFV?G]\]\TUBfo(D/G]?IFDP?

infb∈(m;m+ε)

ha,z(b) :9<;8M ta]KwHWaKw^tG]?IKg$D/BCu K<?

aV?(F

z9 £T (?IF

hz,a(b) ≤ 2supz∈K1,a∈K2

∫

(z,b,a)1√

v(x)−v(b)dx

∫

Ac1√c2dx

. ]9L;_M THVGI^CT

v′′(m) > 0VP?IF

lim inf |x|→∞ v(x) > 0THUVP? F:TH BETV/^nT

εAEDTU?I^CG]BET¼AEaIV,A

v′(x) ≥v′′(m)(x −m)/2

g$D/BfVX<Xx ∈ [m;m + ε]

VP?IFv(x) ≥ v(m + ε)

g$DPBmVX<Xx ≥ m + ε

9¡f^CKL?IÂ`VkjX<DPB ÂCaITUDPBETUu V/VPKL?TN/T A

v(x) − v(b) = v′(ξ)(x − b) ≥ v′′(m)(b−m)

2(x− b)

g$DPB^CDPuyTξ ∈ [b;x]

g$DPBVPXLXx ∈ [m;m+ ε]

9]`bajGI^tT+HV?¢H D/?IH X<GIF:T∫

(z,b,a)

1√

v(x) − v(b)dx

≤ 2

∫ m+ε

b

1√

v′′(m)(b−m)2 (x− b)

dx

+

∫ z

m+ε

1√

v(m+ ε) − v(b)dx+

∫ a

m+ε

1√

v(m+ ε) − v(b)dx

≤ 2

√

2

v′′(m)(b−m)·√m+ ε− b

+ 21

√

v(m+ ε) − v(b)sup

|x−m|∣

∣ x ∈ K1 ∪K2

.

]9L;rd

:;

`ba]T¼BCK<Pa0AfaIVP?IF^CK<F:T¼Dg :9<;Ur K<^mKL?(F:TU\TU?IF]TU?0AmDPg a V?(F z 9ijDÂT¼HV?¢AEVP³PTNAEa]TÀK<? Iu¼G]uD,ºPTBfVX<Xb ∈ (m;m+ ε)

V?(FG(^nT ]9L;_M %ACDy/T AtACaITNG]?]KLg$DPBEu G]\]\TUBoDPGI?IFDP? :9<;8M W9·VPXLXAEa]K<ô(D/G]?IF t2 9 D,´X<T A t > t29Z`ba]TU?Mg$DPBOTUº/TUBEk

z ∈ K1VP?IF

a ∈ K2TyHUV? I?IFMV

b ∈ (m;m + ε)tKLACa

hz,a(b) = t9]«IG]BCACa]TBfF:T(?]T

λ > 0ojk

√λ =

1√8

∫

(z,b,a)

v(x)√

v(x) − v(b)dx

V?(FEojk

E = −λv(b).`ba]T?g$D/BbACa]T+AEBEV/H TT = (z, b, a)

AEa]T^CTNº,VX<G]TÊV?IF

λ^CDPX<ºPT

E + λv(b) = 0,∫

(z,b,a)

v(x)√

E + λv(x)dx =

1√λ

∫

(z,b,a)

v(x)√

v(x) − v(b)dx =

√8

V?(F∫

(z,b,a)

1√

E + λv(x)dx =

1√λ

∫

(z,b,a)

1√

v(x) − v(b)dx =

√2t.

«]D/Bt→ ∞ T+aIVºPT

b→ mG]?]KLg$DPBEuyXLklKL?

aVP?IF

z

λ→ 1

8

(

∫

(z,m,a)

v(x)√

v(x) − v(m)dx)2

=1

8

(

∫

(z,m,a)

√

v(x) dx)2

,

V?(FVP/VK<?E → 0 ACa]Kw^bACK<uyT+g$BCD/u¶oTUX<D, J9]`ba]Kw^t/KLº/T^

It(E, λ) =1

2

∫

T

√

2(E + λv(x)) dx → 1

4

(

∫

(z,m,a)

√

v(x) dx)2

ta]KwHWa\IBCD,º/TÂCa]TNX<TUuyuV]9 0TF' KLACa£VPXLXAEa]T^CTÀ\]BETU\(VBWV,ACK<DP?(^K<?M\]XwVPHUTÀTvVPBCTd?ID, BCTVPF:kACD¢HUVPX<HUG]XwV,ACTÀACa]TyVP^Ckjuy\:ACDPACKwHdX<D,TB

oDPG]?IFg$BEDPu X<TUuyuVÂ:9<;P9)+,-¨ DPg[XLTuyuyV:9<;_ 9(THVGI^CTdDg|X<TUuyuV]9 yTÀHV?¹BET^nACBEK<H AfDPGIBE^CTUX<ºPTÂCDÂAEa]T¼HVP^CT β = 1

K©9 T/9]T+aIVº/TfAED\]BED,ºPT

limt→∞

inf

It(ω)∣

∣ ω ∈ Ma,z,1t

= J(a, z)

X<DjHVX<XLklG]?]KLg$DPBEuvX<kÂK<?a, z ∈ 9c-TUA

K1,K2 ⊆ oT¼HUDPuy\IVPH AmtKLACa0 /∈ K1 ∩ K2

V?IFη > 0

9«IG]BCACa]TBCuyDPBETdX<T Az ∈ K1

VP?IFa ∈ K2

9fÊ^CG]uyT IBE^nAtACaITÀHUV/^nT

m ∈ [z; a]9I«IBCD/u¶X<TUuyuVÂ:9<;~vTN/T AfV

t0 > 0I^nG(HWaAEaIV,Atg$D/BTUº/TUBEk

t >t0AEa]TUBETdT:Kw^xAW^fV^CDPX<G:ACK<DP?

(E, λ)DPg :9 0 bg$DPBtAEa]TdAEBEV/H T T = (z, a)

tKLACa ∣∣It(E, λ) − J(a, z)

∣

∣ ≤ η9

`ba]Kw^t0DP?]X<kF]TU\TU?IF]^bD/?

K1V?IF

K2Io]G:A?]DPADP?

zV?IF

a9

D, V/^C^CG]uyTdAEa]TyHUVP^CT m /∈ [z; a]9Z«]BEDPuX<TUuyuV]9L;/;dTvVP/VK<?¹/T AOV

t0 > 0Z^CGIHWaAEaIV,A+g$DPB

TUº/TUBEkt > t0

ACa]TBCT¼T:Kw^xAW^fVl^CDPX<G:AEKLD/?(E, λ)

Dg :9 0 g$D/BOVACBWVPH T T = (z, x1, a)tKAEa ∣

∣It(E, λ) −J(a, z)

∣

∣ ≤ ηVP?IF

t0DP?]X<klF:T\(T?IF]^tD/?

K1VP?IF

K2]oIG:A?]DAD/?

zVP?IF

a9

=@?*TUKLACa]TBÀHUV/^nTyTÂHUVP? G(^nTX<TUuyuV¹:9 qACD¹H D/?IH X<GIF:TZACa(V,ANACaITUBETT:K<^nAE^ÀV?ω-ta]KwHWa²^nD/XLº/T^

]9 _/VM :9 _o2 ]V?(F :9 _PH_ W9ITHVGI^CTmDPg :9 qM AEa]Kw^b\IV,AEaa(VP^∣

∣It(ω) − J(a, z)∣

∣ ≤ η.

c-TUAc = inf

It(ω)∣

∣ ω ∈ Ma,z,1t

9-THVGI^CTdAEa]Ty\IV,AEaωH DP?(^xAEBCGIH ACTFxGI^nAN?]D,´K<^+oDAEa ZK<?

Ma,z,1t

V?(FVPoI^CDPX<G:ACTXLkH DP?0AEKL?jG]D/GI^:TdaIVº/Tc < ∞ 9(cT A

Mn = Ma,z,1t ∩ ω | It(ω) < c + 1/n 9

P8

THVGI^CTMa,z,1

t

K<^mH X<D/^CTFVP?IFItK<^mVPDjD:FBEVACTNg$G]?(HJACK<DP? ]ACaIT¼^CT AE^

MnVBETdH DPuy\IV/HJA I?]D/? Tuy\:AxkV?(FMÊV,ACKw^ng$k

Mn ⊇ Mn+1g$DPBOTUº/TUBEk

n ∈ g 9-ijDlAEa]TvKL?0ACTBE^CTH ACK<DP? M =⋂

n∈ MnK<ÔV0VK<?¹?]D/? TUuy\:Axk/9-THUVPGI^CTÀTUº/TUBEk

ω ∈ Ma(VP^

It(ω) = cZTy^nTTACaIVA+ACa]TBCTyK<?¹gVPH ANT:K<^nAE^+V\IVACa

ωg$DPB

ta]KwHWa¢ACa]T¼KL?IuÀG]uKw^fVAnAEVPKL?ITFZ9«]BEDPu ACa]T¼s[G]X<TUB cVPBWV?IPTOuvTUACa]D:FTd³j?]D, IACaIVAωVPX<^CDl^CDPX<ºPT^

T0GIVACK<DP?I^ ]9 _/VM :9 _o' VP?IF :9 _PH J9«IBCD/u XLTuyuyV/Ô]9 qV?IF£:9 yT¼³j?]D, (ACaIVAmAEa]Ty^nD/XLG:AEKLD/?¹Kw^G]?]Kw0G]T(^nD

ωu¼GI^xAfHUDPK<?IH KwF:T+tKLACa¢DPG]Bt\(V,ACa

ωH DP?(^xAEBCGIH ACTFVPo(D,º/TOVP?IFlT+PTUA

∣

∣

∣inf

It(ω)∣

∣ ω ∈Ma,z,1t

− J(a, z)∣

∣

∣≤ η

g$DPBVPXLXz ∈ K1

a ∈ K2

V?(Ft ≥ t0

9IijK<?IH Tη > 0

bVP^tVBEo]KLACBWVBEkvACa]Kw^H D/uy\]XLTUACTÂCa]TN\]BEDjDgx9 0TFS

" & " " =@?¢ACaIK<^m^nTHJACK<DP?TNbV?0AACDÂ^nACG(F:klAEa]TÀTºPT?/AtAEaIV,Afg$DPBm^CDPuyTdF:BEKLgAg$G]?IH ACK<DP?

b : → AEa]TdK<?0ACTU/BEVPX12

∫ t

0b2(Bs) ds

KwÔ^nuVX<X©9(=@?MHUDP?0ACBWVP^nAAEDlAEa]TÀ\]BETUºjK<DPGIÔ^CTHJAEKLD/?¹a]TBCTdT¼VPBCTvH D/?I^nKwF:TBCK<?]ÂX<DP?]ÂACK<uvTK<?/AETUBEº,VXw^:o]G]AaIVºPTO?]DÂH D/?IF:KLACK<DP?I^bD/?lAEa]T I?IVPXZ\(D/KL?0A9

`ba]T¼\]BCDjDPg[GI^nT^m`VkjXLD/BfV\I\]BCD:K<uV,ACK<DP?VPBCD/G]?IF¢AEa]T TUBED/^DPgbAEDlBETF:GIHUTdAEa]TÀ\]BEDPoIXLTu AEDÂAEa]T

HUV/^nT¼Dg[X<KL?ITVBbta]KwHWaMbVP^mVX<BCTVPF:k¢^nACGIF:K<TFMKL?²·DPBEDPX<X<VPBCkrI9 _]9=@?MD/BEF:TBAEDluV³/TdACaITv`VkjXLD/BmV\ \]BED:KLuV,AEKLD/?¹DPBE³TvaIVºPTv?]TTF£G]\]\TUBNoDPG]?IFI^+DP?¹AEa]T\]BEDPoIVPo]K<XLKLAxk¢ACaIVA+ACa]Ty\]BED:H TÊÔXLTVºPT^mV?]TKL/a0oDPGIBCa]DjD:FÂDPgAEa]T UTBCDvDg

b9]`ba]Kw^tKw^bPK<ºPT?ojkAEa]T+g$DPX<XLD,tK<?]yX<TUuyuV]9

3 .j 5¤ & ]2)&,*^+<-B 3 + . CE5F / 6 ./ NT50-76K5 /@: a, t > 0 : ./H1 v : → 3 + . =8a/ 4"-76K5 / FG6I-I?

v(x) ≥ x2 ∧ a2 = 50CZ+"f+<C > x ∈ iQR ?S+ / F + ? . f+

lim supε↓0

ε log supx∈

Px

(

∫ t

0

v(Bs) ds ≤ ε, sup0≤s≤t

|Bs| > a)

≤ −1

8

(

t+1

2a2)2

.

=@?£ACa]Tyg$D/XLX<D,tKL?IX<TUuyuVP^+TytKLX<X|DP?]X<k¹?]TTFMACa]TygV/HJA+ACa(V,ANACaITlim sup

g$BCD/u ACaITXLTuvuVKw^^nACBEK<H ACX<k¢^CuVX<XLTBfAEaIV? −t2/8 9ANV IBW^nAmPXwV?(H TÀKLAN^nTTUu^mHUXLTVBfACaIVAmAEa]K<ÔKwÂEBCGITo(THUVPGI^nT¼TvVBETH D/?I^CK<F:TBCK<?]Â^CuVX<X-º,VX<G]T^fDg

ε]ACaITÀTUº/TU?0A ∫ t

0v(Bs) ds ≤ ε g$DPBWH TÂCa]TÀ\]BCD:HUTÊ^bACDl^C\TU?IFuyD0^xADPgACaITNACK<uyTN?]TVPBf~V?IF¢^CDvACa]TdTºPT?/Af^Ca]D/G]X<FAxkj\]KwHUVPXLX<kÂD:HUHUG]BtACD/PTUACa]TBttKAEa sup0≤s≤t |Bs| ≤ a o]G:A?]DPAtKAEa sup0≤s≤t |Bs| > a 9IijDyBETHUVPXLX<K<?]vHUDPBEDPX<X<VPBCkyr(9 _vDP?ITODPGIX<FPG]T^CÂEaIV,AtAEaIV,A

lim supε↓0

ε logP(

∫ t

0

v(Bs) ds ≤ ε)

= lim supε↓0

ε logP(

∫ t

0

v(Bs) ds ≤ ε, sup0≤s≤t

|Bs| ≤ a)

≤ lim supε↓0

ε logP(

∫ t

0

B2s ds ≤ ε

)

= −1

8t2

V?(FACaIVAfACaIT¼H D/?IF:KLACK<DP?sup0≤s≤t |Bs| > a

tK<X<XHVGI^CTdV?MV/F]F:KLACK<DP?IVPXHUD/^nAfta]KwHWauyVP³PT^tACaIK<^mBEVACT^CuyVPXLX<TUBACaIVP? −t2/8 9·DP?jºPTBnAEKL?]vAEa]K<^fK<F]TVyK<?/AEDlVvg$DPBEuVX-\IBCDjDgACGIBC?I^DPG:AAEDo(T¼H G]u¼o(TBE^CDPuyTdV?(FT+F]T g$TUBACa]TN\]BEDjDgACDvACa]TNT?IFDgACa]Kw^^CTHJAEKLD/?-9

IBW^xAfHUDP?I^CT0G]T?IH TODgXLTuyuyV]9L;8dKw^bACaIT+g$DPX<XLD,tK<?]y^xAWV,ACTuyTU?0A93 .j 5¤)&' & 50CZ+"f+"C > a > 0 .0/21 +"f+"C > x ∈ (−a/

√2; +a/

√2)F + ? . f+

limε↓0

ε logPx

(

∫ t

0

B2s ds ≤ ε, sup

0≤s≤t|Bs| ≤ a

)

= limε↓0

ε logPx

(

∫ t

0

B2s ds ≤ ε

)

= −(

t+ x2)2

8.

_

)+,-¨ & `baIT¼^CTH D/?IFT/G(VX<KAxkK<^m\]BED,ºPTFK<?£HUDPBEDPX<X<VPBCklrI9 _]9\]\]X<kjKL?IlX<TUuyuV:9<;8¼AEDÂAEa]TÀg$GI?IH ACK<DP?v(x) = x2 TN^CTUTOACa(V,A

lim supε↓0

ε logPx

(

∫ t

0

B2s ds ≤ ε, sup

0≤s≤t|Bs| > a

)

≤ −1

8

(

t+1

2a2)2

< lim infε↓0

ε logPx

(

∫ t

0

B2s ds ≤ ε

)

.

`bajGI^bT+HV?¢GI^CT+XLTuvuV8:9 _¼ACD\]BED,ºPTmACaIT (BE^nAT0GIVPXLKLAxkP9 0TF' =@?ACa]TN\]BEDjDgDgXLTuvuV:9<;8ÀTNtKLX<XZGI^CT+ACa]T+g$D/XLX<D,tKL?IvHUDPG]\]X<K<?]VBEPG]uyTU?0A93 .j 5¢¤ & j,& 6If+ / x, y ∈ FG6I-I? |x| ≥ |y| F +4 .0/ 4?@5_5 9 +\-7F 5 CE50F / 6 .0/ N5-J6K5 /S9 Bx

./H1 By 5 /. 45NN5 /V C`5 3.D3 6 L 6I- >\9JVS. 4+eFG6I-%? Bx0 = x : By

0 = y : ./H1 |Bxt | ≥ |By

t | = 5C .0LIL t ≥ 0 Q)+,-¨ & c-TUA

Bx oTdV?jklBCD,t?IK<VP?luyDAEKLD/?tKLACa^nAEVPBnAtK<? x VP?IF B o(TÀVP?]DAEa]TUBbDP?ITNDP?ACa]TdÊVuyT\]BEDPoIVPo]K<XLKLAxkÂ^n\(VPH T:o]G:AtKLACa¹^nAEVPBnAtK<?

y9IemTI?IT+ACa]Td^nACD/\]\]K<?]vACK<uvT

Tojk

T = inf

t ≥ 0∣

∣ |Bxt | = |Bt|

V?(F¹AEa]TyBEVP?IF:DPuº,VBEKwVo]X<Tσojk

σ = 1KLgBx

T = BTVP?IF

σ = −1TUXw^nT/9-`ba]TU?£ACa]Ty\]BED:H T^C^

ByF:TI?]TFojkBy

t =

BtKLgt ≤ T

]V?IFBT + σ(Bx

t −BxT )

KLgt > T

Kw^VBCD,t?]KwV?uyDACK<DP?¢tKLACa |Byt | < |Bx

t |g$D/B

t < TV?(FTKAEa]TUB

Byt = Bx

t

D/BBy

t = −Bxt

g$D/Bt ≥ T

9`ba]Kw^t\]BED,ºPT^[AEa]TdH XwVK<u¢9 0TF'

`ba]T+uVK<?¢BCT^nG]XLAtDgACa]Kw^^CTHJAEKLD/?Kw^bAEa]T+g$DPX<XLD,tK<?]vPT?]TUBWVX<K<ÊV,AEKLD/?lDPgX<TUuyuVÂ:9<;_]9)+,I±(&$ ¤ & ]¤)& *^+<-

b : → 3 + .1 6 +<CE+ / -J6 .D3"L + =8a/ 4"-76K5 / FG6I-I? b(0) = 0 : b′(0) 6= 0 .0/21lim inf |x|→∞ |b(x)| > 0 Q R ?@+ / = 5C +"f+"C > η > 0

F + ? . f+

limε↓0

ε · logP(1

2

∫ t

0

b2(Bs) ds ≤ ε, sup0≤s≤t

|Bs| ≤ η)

= limε↓0

ε · logP(1

2

∫ t

0

b2(Bs) ds ≤ ε)

= − |b′(0)|2t216

.

)+,-¨ & ·a]DjD0^nTÀ^CDPuyT0 < δ < |b′(0)| 9¡f^CKL?IACa]Tv`VkjX<DPBbg$D/BCu¼G]X<V b(x) = b′(0) · x + o(x)

TI?IF¢VP?

a > 0tKAEa(

|b′(0)| + δ)2x2 ≥ b2(x) ≥

(

|b′(0)| − δ)2x2 g$D/BVX<X

x ∈ [−a; a] 9 :9<;M KLACa]D/G:AOX<D/Ê^DPg[PT?]TUBWVX<KAxkT¼uVkV/^C^CG]uyTNACa(V,A

aKw^f^CuVX<XLTBACa(V?

ηVP?IF¹VXw^CDl^nuVPXLXTU?]D/G]Pa¢AED

\TUBEuvKLA |b(x)| ≥ a(

|b′(0)| − δ) g$DPBVPXLX

x ∈ tKAEa |x| > a9

TOa(VºPTOACDHUVPX<HUG]XwV,ACTOAEa]TNT:\DP?]T?/AEK<VPXZBEVACT^bDg

P(1

2

∫ t

0

b2(Bs) ds ≤ ε)

= P(1

2

∫ t

0


|Bs| ≤ a)

+ P(1

2

∫ t

0


|Bs| > a)

.

]9L;qM

,r

a]T?]TUº/TUBsup0≤s≤t |Bs| ≤ a

TdHUVP?VP\]\]BEDjK<uV,AETb(x)

ojkb′(0)x

V/^bKL? ]9L;D W9(`ba]K<^b/KLº/T^

P(1

2

∫ t

0

(

|b′(0)| + δ)2B2

s ds ≤ ε, sup0≤s≤t

|Bs| ≤ a)

≤ P(1

2

∫ t

0


|Bs| ≤ a)

≤ P(1

2

∫ t

0

(

|b′(0)| − δ)2B2


|Bs| ≤ a)

.

DAEaoDPG]?(F]^tDgACaIK<^bT^xAEKLuV,AETdHUVP?oTNaIV?(F:XLTFGI^nK<?]

limε↓0

ε logP(

∫ t

0

cB2s ds ≤ ε, sup

0≤s≤t|Bs| ≤ a

)

= −c · t2

8,

ta]KwHWaKw^VyH D/?I^CT0G]TU?(H T+DgXLTuvuVÂ]9L;_¼VP?IFÂAEa]TdÊHUVX<K<?]y\]BCD/\(TBnAxkr(9 8]9«]D/BACa]TNX<D,TBbo(D/G]?IFACa]Kw^t/KLº/T^

lim infε↓0

ε logP(1

2

∫ t

0

b2(Bs) ds ≤ ε)

≥ lim infε↓0

ε logP(1

2

∫ t

0


|Bs| ≤ a)

≥ −(

|b′(0)| + δ)2

16t2

ta]T?]TUº/TUBδ > 0

9]«IDPBbACaITNG]\]\TUBoDPG]?(FT I?IF

lim supε↓0

ε logP(1

2

∫ t

0


|Bs| ≤ a)

≤ −(

|b′(0)| − δ)2

16t2. ]9L;,0

emTI?]Tv(x) = b2(x)/

(

|b′(0)| − δ)2 9-`ba]T? ojk¹D/G]BdHWa]D/K<HUTvDPg

aTaIVºPT

v(x) ≥ x2 ∧ a2 VP?IFX<TUuyuVy]9L;8ÀACDP/T AEa]TUBbtKLACa¢rI9Q8¼PK<ºPT^

lim supε↓0

ε logP(1

2

∫ t

0


|Bs| > η)

≤ lim supε↓0

ε logP(1

2

∫ t

0


|Bs| > a)

≤ −1

8

(

t+1

2a2)2

(

|b′(0)| − δ)2

2

< −(

|b′(0)| − δ)2

16t2.

]9L; M

¡f^CK<?]DP?]X<kÂAEa]TdXwVP^nAACaIBCTTNXLK<?]T^fDgT0GIVACK<DP? ]9L; M Td^nTTNACaIVAACaITdG]\]\TUBoDPGI?IFg$DPB :9<;qd K<^F:D/uvK<?IVACTFojk :9<;D VP?IFlg$BCD/u XLTuyuyVÂ8]9 _¼TNPT A

lim supε↓0

ε logP(1

2

∫ t

0

b2(Bs) ds ≤ ε)

≤ −(

|b′(0)| − δ)2

16t2

g$DPBVPXLXδ > 0

9]c-TUAnACK<?]δ ↓ 0

(?]K<^Ca]TÂCa]TN\]BEDjDgDg

limε↓0

ε · logP(1

2

∫ t

0

b2(Bs) ds ≤ ε)

= − |b′(0)|2t216

.

¡tAEKLX<Kw^nK<?]ÂX<TUuyuV8]9 _VP/VK<? (o]G:AfAEa]KwÂEKLuyTÀtKLACa¹ACa]TÀg$G]XLXT0GIVACK<DP? :9<; M VXw^nD\]BED,ºPT^bACaIT IBW^xAT0GIVPXLKLAxkÂDPgACaITN\]BCD/\(D0^nKLACK<DP? ^tHUX<VPKLu¢9 0TF'

P

KLACa*ACaITa]TUX<\²DgACa]TVoD,ºPTyH DPGI\]XLK<?]¹VPBC/G]uyTU?0ANTÂHV?*PTUANACa]Tg$D/XLX<D,tK<?]¢BCTI?]TUuyT?/AdDPg\]BEDP\D/^CKAEKLD/?]9L;:9

3 .j 5¢¤ & :® &^*^+"-b : → 3 + .1 6 +"CE+ / -J6 .D3"L + =8a/ 4"-76K5 / FG6I-%? b(0) = 0 : b′(0) 6= 0 ./H1

lim inf |x|→∞ |b(x)| > 0 Q R ?@+ / = 5C +"f+"C > η > 0F + ? . f+

limζ↓0

lim infε↓0

ε · log inf−ζ<z<ζ

Pz

(1

2

∫ t

0


|Bs| ≤ η)

= − |b′(0)|2t216./H1

lim supε↓0

ε · log supy∈

Py

(1

2

∫ t

0


|Bs| ≤ η)

= − |b′(0)|2t216

.

)+,-¨ & Td^xAWVBCAojkl\IBCD,ºjK<?]ÀAEa]TdH XwVK<u¶VPo(D/G:AbACa]Tlim inf

9]¡m^nK<?]\]BEDP\D/^CKLACK<DP?]9L;dT I?IF

lim infε↓0


Pz

(1

2

∫ t

0


|Bs| ≤ η)

≤ limε↓0

ε · logP0

(1

2

∫ t

0


|Bs| ≤ η)

= − |b′(0)|2t216g$DPBtTºPTBCk

ζ > 09

D,XLTUA κ > 0V?(FHWaID0D0^nT+V

δ > 0tKAEa

−(

|b′(0)| + δ)2 (t+ δ2)2

16> − |b′(0)|2t2

16− κ.

f^KL?ACa]TN\IBCDjDgDg\]BEDP\D/^CKAEKLD/?:9<;¼TNHUV?GI^CTN`VkjX<DPBtV\]\IBCD:K<uyVACK<DP?lACD I?IF¢V?a > 0

tKAEab2(x) ≤

(

|b′(0)| + δ)2x2

g$DPBVPXLXx ∈ [−a; a] 9 KLACa]D/G:AfX<D/Ê^DPg/TU?]TBEVPXLKLAxkTOuVklV/^C^CG]uyT a ≤ min(2δ, η)

9c-TUA

ζ < a/2V?IF

z ∈ [−ζ; +ζ] 9`baITU?*THV? GI^nTyX<TUuyuV:9<;UrACDHWa]DjD/^CT¼AxDBCD,t?IK<VP?uyDAEKLD/?I^Bζ VP?IF Bz tKLACa Bζ

0 = ζBz

0 = z]V?IF |Bζ

t | ≥ |Bzt |g$D/BVX<X

t ≥ 09 T I?IF

P(1

2

∫ t

0

b2(Bzs ) ds ≤ ε, sup

0≤s≤t|Bz

s | ≤ η)

≥ P(1

2

∫ t

0

b2(Bzs ) ds ≤ ε, sup

0≤s≤t|Bz

s | ≤ a)

≥ P(1

2

∫ t

0

(

|b′(0)| + δ)2

(Bzs )2 ds ≤ ε, sup

0≤s≤t|Bz

s | ≤ a)

≥ P(1

2

∫ t

0

(

|b′(0)| + δ)2

(Bζs )2 ds ≤ ε, sup

0≤s≤t|Bζ

s | ≤ a)

g$DPBtTºPTBCkz ∈ [−ζ; +ζ] IVP?IFÂAEajGI^

lim infε↓0


Pz

(1

2

∫ t

0


|Bs| ≤ η)

≥ lim infε↓0

ε · logPζ

(1

2

∫ t

0

(

|b′(0)| + δ)2B2


|Bs| ≤ a)

=1

2

(

|b′(0)| + δ)2

lim infε↓0

ε · logPζ

(

∫ t

0

B2s ds ≤ ε, sup

0≤s≤t|Bs| ≤ a

)

.

q

THVGI^CTζ < a/2 < δ

TNHUVP?G(^nTNX<TUuyuV:9<;_ÀACDPTUA

lim infε↓0


Pz

(1

2

∫ t

0


|Bs| ≤ η)

≥ −1

2

(

|b′(0)| + δ)2 (t+ ζ2)2

8

≥ −1

2

(

|b′(0)| + δ)2 (t+ δ2)2

8

> − |b′(0)|2t216

− κ

g$DPBVPXLX-^CGa!ÂH K<TU?0AEXLk^nuVX<Xκ > 0

9Ic-TUAnAEKL?]ζ ↓ 0

H D/uy\]XLTUACTÂCa]TN\]BEDjDgDgAEa]T IBW^xAfHUX<VPKLu¢9«]D/BACa]Td^CTHUDP?IFH XwVK<u IBW^nAb?IDACTOAEaIV,A

lim supε↓0

ε · log supy∈

Py

(1

2

∫ t

0


|Bs| ≤ η)

≥ lim supε↓0

ε · logP0

(1

2

∫ t

0


|Bs| ≤ η)

= − |b′(0)|2t216

,

V0VK<?lojkl\IBCD/\(D0^nKLACK<DP?¢:9<;:9c-TUA

κ > 0V?(FHWa]DjD0^nT

δ > 0tKAEa

−(

|b′(0)| − δ)2 t2

16< − |b′(0)|2t2

16+ κ.

¡f^CK<?]`VkjX<DPBtV\]\IBCD:K<uyVACK<DP?TNHUVP? (?IF¢V?a > 0

tKLACab2(x) ≥

(

|b′(0)| − δ)2x2

g$DPBdVPXLXx ∈ [−a; a] VP?IF£o0k£HWa]DjD/^CKL?] a ^CuVX<X|TU?]D/G]Pa TyHV? I?IF*V^CuvDjDPACa -VP?/AEK<^CkjuyuvTUACBEK<H

uyDP?]DPACD/?]TOg$G]?IHJAEKLD/?ϕ : → tKAEa |b(x)| ≥ |ϕ(x)| g$D/BVX<X x ∈ VP?IF

ϕ′(0) = |b′(0)| − δ9

¡f^CK<?]vACa]TdHUDPG]\]X<K<?]VBEPG]uyTU?0AtV?IF\]BEDP\D/^CKLACK<DP?]9L;ÀVP/VPKL? :T+PTUA

lim supε↓0

ε · log supy∈

Py

(1

2

∫ t

0


|Bs| ≤ η)

≤ lim supε↓0

ε · log supy∈

Py

(1

2

∫ t

0

ϕ2(Bs) ds ≤ ε, sup0≤s≤t

|Bs| ≤ η)

≤ lim supε↓0

ε · logP0

(1

2

∫ t

0

ϕ2(Bs) ds ≤ ε, sup0≤s≤t

|Bs| ≤ η)

= −(

|b′(0)| − δ)2

t2

16

< − |b′(0)|2t216

+ κ

g$DPBVPXLXκ > 0

9]`V³jK<?]vACa]TNX<K<uvKLAκ ↓ 0

H D/uv\IXLTUACTÂEa]TN\]BEDjDgx9 0TFS s[º/TUBEk/AEa]K<?]¼X<T gAfACDF:Dy?]D, :Kw^bACDVPFIFlAEa]TN\]BEDjDgDPgX<TUuyuVÂ:9<;8]9)+,-¨ Dg[X<TUuyuV:9<;8M J9 £Tv?]TUTF¢ACD I?IF¹VP?¹G]\]\TUBOoDPG]?IFMDP?AEa]TÀT:\(D/?]TU?0ACKwVXBWV,ACTdg$DPBmACa]T\]BEDPoIVPo]K<XLKLAxkDgACa]TNTºPT?/A

Aε =

∫ t

0

v(Bs) ds ≤ ε, sup0≤s<t

|Bs| > a

,

/

0 0.2 0.4 0.6 0.8 1

−0.5

0

0.5

1

«KL/G]BETN]9 _ 6If+ V@. -I? 9 5 =Z. CE50F / 6 ./ NT50-76K5 / 5 / -%?@+Z6 / - +"CUf .0L [0; 1] : 45 /21 6I-J6K5 / + 1 5 / -%?@+h+<f+ / - -%? . -∫ 1

0 B2s ds ≤ 0.05 ./21 -I? . - F +? . f+ sup0≤s≤1Bs > 1 Q O/ +P4 ./ 9 ++-%? . - -%?@+- >`V 6K4 .0LHV@. -I? 8a/21 +"C-%?@+ 9 +45 /H1 6I-J6K5 /@9 CE+ . 4`?S+ 9 6I- 9 N . M6IN 8 N / + . Ce-I?S+h+ /21 5 = -I?S+e6 / - +"CUf .0L Q R ?6 9h3 +? . f_6K5 8 C - 9 F + LIL FG6I-%?-%?@+ 9JV +`4"6 .L C L + 5 = -I?S+e+ /H1`V 56 / -6 /e= 5CUN 8aL . rI9 _M Q

ta]KwHWaK<^G]?]KLg$DPBEu K<?¢ACa]TÀKL?]KLACKwVX\(D/KL?0AB0 = x

9«K<BE^nAfF:TI?]TNAxDKL?0AETUBEX<V/H TF¢^CT0G]T?IH T^tDg[^nACDP\I\]KL?IACK<uyT^(Sj)j∈

VP?IF(Tj)j∈ 0

ojkÂXLTUAnAEKL?]T0 = 0

VP?IFSj = inf

s > Tj−1

∣

∣ |Bs| ≥ a

Tj = inf

s > Sj

∣

∣ |Bs| = a/2

g$DPBmVX<Xj ∈ g 9]= gACaITdKL?IKAEK<VPX-\(D/KL?0A B0 = x

aIVP^ |x| > aTNaIVº/T

S0 = 0VP?IF |BS0 | > a

9]s ]HUTU\:Ag$D/BACaIK<^mTÀaIVºPT |BSj

| = a9«]D/B

s ∈ [Sj ;Tj ]TÀaIVºPT |Bs| ≥ a/2

VP?IF¢AEa0G(^v(Bs) ≥ a2/4

9µmG:AW^nKwF:TACaIT^CTNKL?0ACTBCº,VPX<^TOa(VºPT |Bs| < a

V?(FÂAEa0G(^v(Bs) ≥ B2

s

9I`baITUBET g$DPBETOTNHUVP?HUDP?IHUXLG(F:T

∫ Tj

Sj

v(Bs) ds ≤ ε

⊆

∫ Tj

Sj

a2/4 ds ≤ ε

=

Tj − Sj ≤ 4ε/a2

V?(FÂg$D/Bd > 0

VPX<^CD

∫ Sj

Tj−1

v(Bs) ds ≤ ε, Sj − Tj−1 ≥ d

⊆

∫ Sj

Tj−1

B2s ds ≤ ε, Sj − Tj−1 ≥ d

⊆

∫ Tj−1+d

Tj−1

B2s ds ≤ ε

.

f^tVP?VPo]o]BETUºjKwV,ACK<DP?¢F:TI?]TJ = d2t/a2e + 1

9 TNVP?/AACDÂ^C\]X<KAtAEa]Td^CT AAε K<?0ACDvACa]T+AxDy\IVBCAE^

Aε =(

Aε ∩ TJ ≤ t)

∪(

Aε ∩ TJ > t)

.

`ba]T IBW^nA+\(VBCANHUDPBEBCT^n\DP?(F]^mACDACa]TÂHVP^CTvACaIVANACa]TBCTVBETV,ANX<TVP^nAJT:HUG]BW^nK<DP?IÔG]\ AED¢ACa]TyX<TUºPTX

|Bs| = aV?(FMAEa]TU?*oIV/HW³ACD |Bs| = a/2

oT g$DPBETvACK<uyTt9«]D/BNACaIK<^dHVP^CTyTtK<XLX%PT AÀVP?£GI\]\(TB

oDPG]?IF D/?MAEa]T\]BEDPoIVPo]KLX<KLAxkg$BCD/u ACaITygVPHJA+AEaIV,ANAEa]T\]BEDjHUTÊ^+aIVPÔAED¢uvD,º/T¼º/TUBEkgVP^nAdF:G]BEKL?IAEa]TK<?/AETUBEº,VXw^[Sj ;Tj ]

9`ba]Tv^nTH DP?(F\IVBCAmHUDPBEBCT^n\DP?IFI^bACDlACa]TvHUVP^CTNAEaIV,AmACa]TBCT¼K<^mV,AmX<TV/^xAODP?]TÀo]G:AOACaIVAACaITUBET¼VBETÀVAfuyD/^nA

J − 1^nG(HWaT]H G]BW^CKLD/?IÛ9`ba]Kw^fHUV/^nTÀK<^muyDPBETÀF:K !HUG]XLA o(THUVPGI^nTÀT¼aIVºPTNAEDÂAWV³PT

ACaITNKL?0ACTBCº,VPX<^boT AxTUT?lAEa]TNT]H GIBE^CKLD/?I^KL?0AEDyV/HUHUDPG]?0A9«KLBW^nAtHUDP?I^CKwF:TUBACaITdHUVP^CT

TJ ≤ t9 TBCTNT+aIVºPT

Aε ∩ TJ ≤ t ⊆

J∑

j=1

∫ Tj

Sj

v(Bs) ds ≤ ε

⊆

J∑

j=1

(Tj − Sj) ≤ 4ε/a2

.

¡f^CK<?]vACa]Td^nACBEDP?]¸¹VPBC³/D,ºy\]BCD/\(TBnAxkyg$D/BBCD,t?]KwV?uyDACK<DP?¢V?(FlAEa]TNBET ITHJACK<DP?\]BEKL?(H K<\]XLT+T I?IFPx

(

Tj − Sj ≤ ε)

≤ P(

sup0≤s≤ε

Bs > a/2)

= 2P(

Bε > a/2)

= 2P(√εB1 > a/2

)

g$DPBdVPXLXx ∈ 9-`ba]To(VP^CK<H¼X<VPBC/TvF]TUºjK<VACK<DP? BCT^nGIXA+g$DPB+AEa]TÂ^nAEV?(F]VBWFM?IDPBEuyVPX|F:K<^nACBEK<o]G:ACK<DP? DP?

HUDPBEDPX<X<VPBCkÂ8]9L;8D ?]D, /KLº/T^

limε↓0

ε · log supx∈

Px

(

Tj − Sj ≤ ε)

≤ −1

2

(

a/2)2

= −a2

8.

=@?ACa]Kw^^CKAEGIV,AEKLD/?TNHUVP?VP\]\]X<kl\]BEDP\D/^CKAEKLD/?8]9QdACDy/T Alim sup

ε↓0ε log sup

x∈ Px

(

Aε ∩ TJ ≤ t)

≤ lim supε↓0

ε logPx

(

J∑

j=1

(Tj − Sj) ≤ 4ε/a2)

=a2

4lim sup

ε↓0ε logPx

(

J∑

j=1

(Tj − Sj) ≤ ε)

≤ −a2

4

(

J∑

j=1

a√8

)2

≤ −1

8

(

t+1

2a2)2.

]9L;zM

D,´HUDP?I^CK<F]TUBAEa]T¼HVP^CT TJ > t9Z·a]DjD0^nT

n ∈ g tKAEan > 2J

V?IFε > 0

tKAEa4ε/a2 < t/n

9emTI?]T

∆t = t/n]ACaIT¼K<?0ACTUBEº,VXw^

I1 = [0; ∆t]VP?IF

Ik =(

(k − 1)∆t; k∆t] g$DPB

k = 2, . . . , n(ACa]T

K<?IF:T¢^nTUAQ =

(k1, . . . , k`) ∈ g `∣

∣

∣ ` ∈ 1, . . . , J, 1 ≤ k1 ≤ · · · ≤ k` ≤ n

,V?(FÂAEa]TNTUº/TU?0AAε

(k1,...,k`)= Aε ∩

Sj ∈ Ikj

g$DPBj = 1, . . . , `

VP?IFS`+1 > t

.`ba]T?T+aIVºPTAε ∩ TJ > t =

⋃

q∈Q

Aεq .

·a]DjD/^CT(k1, . . . , k`) ∈ Q

9ZfÔT¼aIVº/T¼^CTUT?£VoD,ºPTdACa]TH D/?IF:KLACK<DP? ∫ Tj

Sjv(Bs) ds ≤ ε

K<uv\IXLK<T^Tj − Sj ≤ 4ε/a2 ≤ ∆t

9I`bajGI^bDP?Aε

q

T+aIVºPTSj − Tj−1 ≥ max

(

(kj − kj−1 − 2)∆t, 0)

=: dj−1 ]9 8P~M g$DPBj = 1, . . . , ` − 1

Ita]TBCTdTdGI^CTdAEa]T¼HUDP?jºPT?0ACK<DP?k0 = 0

9I= gk` < n

AEa]TU?MTdG(^nTy:9Q8~VXw^CDyg$DPBj = `

VP?IFlT+aIVºPTt− T` ≥ max

(

(n− k` − 2)∆t, 0)

=: d`.«]D/Bk` = n

KA+tK<XLXACG]BE?¹D/G:AACa(V,A+Td?ITUTFACDÂAEBCTV,AAEa]TvBCK<Pa0AfT?IF:\DPK<?/AODg|AEa]TÀK<?0ACTUBEº,VX^n\THUK<VPXLX<ka]TBCT+T+F]TI?]T

d`−1 = max(

(n− k`−1 − 3)∆t, 0) 9

c-TUAδ > 0

VP?IF¢F:T(?]TDδ

2`+1

VP^bK<?¢XLTuyuyV8]9 qI9]«]DPBα ∈ Dδ

2`+1

g$GIBnAEa]TUBfF]TI?]T

Aαε(k1,...,k`)

=

∫ S1

T0

v(Bs) ds ≤ α1ε,

∫ T1

S1

v(Bs) ds ≤ α2ε, S1 ∈ Ik1 ,

999∫ S`

T`−1

v(Bs) ds ≤ α2`−1ε,

∫ T`

S`

v(Bs) ds ≤ α2`ε, S` ∈ Ik`,

∫ t

T`

v(Bs) ds ≤ α2`+1ε, S`+1 > t

z

KLgk` < n

V?IF

Aαε(k1,...,k`)

=

∫ S1

T0

v(Bs) ds ≤ α1ε,

∫ T1

S1

v(Bs) ds ≤ α2ε, S1 ∈ Ik1 ,

999∫ S`

T`−1

v(Bs) ds ≤ α2`−1ε, S` ∈ In, S`+1 > t

TUXw^CTP9]`ba]T?T+aIVºPTAε ∩ TJ > t =

⋃

q∈Q

Aεq ⊆

⋃

q∈Q

⋃

α∈Dδ2`+1

Aαεq .

fÊ^CG]uyT (BE^nAbACa]TdHVP^CTk` < n

9]`baITU?¢T+/T A

Px

(

Aαε(k1,...,k`)

)

≤ Px

(

∫ T0+d0

T0

B2s ds ≤ α1ε, T1 − S1 ≤ 4α2ε/a

2, S1 ∈ Ik1 ,

999∫ T`−1+d`−1

T`−1

B2s ds ≤ α2`−1ε, T` − S` ≤ 4α2`ε/a

2, S` ∈ Ik`,

∫ T`+d`

T`

B2s ds ≤ α2`+1ε, S`+1 > t

)

.

D,TNG(^nTOACaITd^xAEBCD/?]¸¹VBE³PD,ºy\]BEDP\TUBCAxkDg|BCD,t?]KwV?uyDACK<DP?g$D/BbACa]Td^nACD/\]\]K<?]vACK<uvT^ SjV?IF

Tj9

THVGI^CT |BTj| = a/2

V?IF |BSj| = a

VBETdF:T AETUBEuvK<?]Kw^xAEK<H¼V?IFAEa]T¼BED,t?]KwV?¢uyDACK<DP?Kw^f^CkjuyuvTUACBEK<HTO/T A

Px

(

Aαε(k1,...,k`)

)

≤ Px

(

∫ T0+d0

T0

B2s ds ≤ α1ε, T1 − S1 ≤ 4α2ε/a

2, S1 ∈ Ik1 ,

999∫ T`−1+d`−1

T`−1

B2s ds ≤ α2`−1ε, T` − S` ≤ 4α2`ε/a

2, S` ∈ Ik`

)

·P a2

(

∫ d`

0

B2s ds ≤ α2`+1ε

)

≤ Px

(

∫ T0+d0

T0

B2s ds ≤ α1ε, T1 − S1 ≤ 4α2ε/a

2, S1 ∈ Ik1 ,

999∫ T`−1+d`−1

T`−1

B2s ds ≤ α2`−1ε, S` ∈ Ik`

)

·P0

(

sup0≤s≤4α2`ε/a2

Bs > a/2)

·P a2

(

∫ d`

0

B2s ds ≤ α2`+1ε

)

.

qP~

ªtT\(TV,ACK<?]vACaIT^CT+AxD^nACTU\(^g$DPBj = `− 1, . . . , 0

I?IVPXLX<kÂPK<ºPT^

Px

(

Aαε(k1,...,k`)

)

≤ Px

(

∫ d0

0

B2s ds ≤ α1ε

)

·∏

j=1

P a2

(

∫ dj

0

B2s ds ≤ α2j+1ε

)

·∏

j=1

P0

(

sup0≤s≤4α2jε/a2

Bs > a/2)

.

=@?DPBWF:TBACDGI^CTdXLTuyuyV8:9QyTdaIVºPTdACDlHVXwH G]XwV,AETdACaITÀK<?IF:K<º0KwF:GIVPXBWV,AET^bg$D/BfAEa]TdgV/HJAEDPBW^tDP?AEa]TBEKL/a/A a(V?IF^nKwF:TP9(¡f^CKL?]HUDPBEDPX<X<VPBCkyrI9Q¼T+/T A

limε↓0

ε · log supx∈

Px

(

∫ d

0

B2s ds ≤ ε

)

= −1

8d2. ]9 8];_

¡f^CK<?]vACa]TNBET ITHJAEKLD/?\]BEK<?IH K<\]X<T+VP?IFACa]T+oIV/^nKwH+^CHVX<KL?Iv\IBCD/\(TBnAxkDPgBED,t?]K<VP?luyDAEKLD/?T I?IFP0

(

sup0≤s≤4ε/a2

Bs > a/2)

= 2P(

B4ε/a2 > a/2)

= 2P(√

4ε/a2B1 > a/2)

= 2P(√εB1 > a2/4

)

.`ba]T¼X<VPBC/TdF:TUºjKwV,ACK<DP?M\]BCK<?IHUKL\]X<TNg$D/BfAEa]T¼^nAEVP?IF]VBWF¢?]D/BCuVXF:Kw^xAEBCK<o]G:AEKLD/?DP? H D/BCD/XLXwVBEk8:9<;8M b?]D,PK<ºPT^limε↓0

ε · logP0

(

sup0≤s≤4ε/a2

Bs > a/2)

= −1

2

(

a2/4)2

= −1

8

(a2

2

)2

. ]9 8/8D D,TNHV?¢V\]\]X<kÂX<TUuyuVÂ8:9QNAEDPTUAAEa]TdH D/uÀo]K<?]TFlBWV,AETP9]`ba]TNBET^CG]XLAbKw^

limε↓0

ε log supx∈

Px

(

Aαε(k1,...,k`)

)

≤ − 1

1 + δ

1

8

(

∑

j=0

dj + n1a2

4+ `

a2

2

)2

,

ta]TBCTn1 =

∣

∣

j = 1, . . . , `∣

∣ dj > 0∣

∣

9THUVPGI^CTdTV/HWaDg|ACaIT¼K<?0ACTUBEº,VXw^[Sj ;Tj ]

HUV?MaIVºPTÀVÂ?]DP? TUuy\:AxkKL?0AETUBW^nTHJACK<DP?¢tKLACa¹VAuvD0^xAtAxDDgACaITnK<?0ACTBCº,VXw^

IkTNaIVºPT ∑`

j=0 dj ≥ n − 2JV?IFAEa0G(^

n1 ≥ 19IijDyT I?(F

limε↓0

ε log supx∈

Px

(

Aαε(k1,...,k`)

)

≤ − 1

1 + δ

1

8

(n− 2J

nt+

a2

4+ `

a2

2

)2 ]9 8P_M g$DPBVPXLX

α ∈ Dδ2`+1

V?IFVX<Xδ > 0

9 D, VPÊ^CG]uyT k` = n

9I`ba]Kw^fHVP^CT+K<^m^nK<uyKLXwVB :o]G:Af?ITUTFI^fV?V/F]F:KLACK<DP?IVPXVBEPG]uyT?/AbAEDAEVP³PTNHUVPBCTNDPgACaITÀHUV/^nT

t ∈ [S`;T`)9 TUBETNT+HV?¢?]DX<DP?]/TUBtGI^CT ]9 8/8D %g$DPBACa]TNK<?0ACTUBEº,VX [S`;T`)

9`DyDPBE³lVBEDPGI?IFACaIK<^tTNF:T(?]TdVy^xAEDP\]\]K<?]vACK<uyT

Rojk

R = inf

s ≥ max(T`−1, (n− 2)∆t)∣

∣ |Bs| = a/2

.ROK<ºPTU?¹ACa]T¼TUºPT?0AAαε

(k1,...,k`)

ACa]T¼\]BCD:HUTÊ^fHUVP?]?]DPAfaIVºPT |Bs| > a/2g$D/BmVÂ\TUBEK<DjF¹DgACK<uyT¼DPgX<TU?IACa

∆tVP?IFGI^nK<?]vACaITd^n\THUK<VPXF:T(?]KAEKLD/?DPg

d`−1g$DPBbAEa]Kw^HUV/^nTOTN/T A

T` − 1 + d`−1 ≤ R ≤ S`9

ijK<uyKLXwVBbAED¼AEa]TNDAEa]TUBHVP^CT+T+PTUAbACa]T?

Px

(

Aαε(k1,...,k`)

)

≤ Px

(

∫ T0+d0

T0

B2s ds ≤ α1ε, T1 − S1 ≤ 4α2ε/a

2, S1 ∈ Ik1 ,

999∫ T`−2+d`−2

T`−2

B2s ds ≤ α2`−3ε,

T`−1 − S`−1 ≤ 4α2`−2ε/a2, S`−1 ∈ Ik`−1

,∫ T`−1+d`−1

T`−1

B2s ds ≤ α2`−1ε, S` −R ≤ 4α2`ε/a

2, S` ∈ In

)

.

q];

¡f^CK<?]vACa]Td^nACBEDP?]¸¹VPBC³/D,ºy\]BCD/\(TBnAxkyg$D/BbACa]Td^nACD/\]\]K<?]vACK<uyTR

IBW^xAt/KLº/T^

Px

(

Aαε(k1,...,k`)

)

≤ Px

(

∫ T0+d0

T0

B2s ds ≤ α1ε, T1 − S1 ≤ 4α2ε/a

2, S1 ∈ Ik1 ,

999∫ T`−2+d`−2

T`−2

B2s ds ≤ α2`−3ε,

T`−1 − S`−1 ≤ 4α2`−2ε/a2, S`−1 ∈ Ik`−1

,∫ T`−1+d`−1

T`−1

B2s ds ≤ α2`−1ε

)

·P0

(

sup0≤s≤4α2`ε/a2

Bs > a/2)

.

D,TNHV?¢H DP?0AEKL?jG]TN^C\]XLKLAnAEKL?IDgACTUBEu^tVP^bK<?ACa]T IBW^nAtHVP^CTOACDyPTUA

Px

(

Aαε(k1,...,k`)

)

≤ Px

(

∫ d0

0

B2s ds ≤ α1ε

)

·`−1∏

j=1

P a2

(

∫ dj

0

B2s ds ≤ α2j+1ε

)

·∏

j=1

P0

(

sup0≤s≤4α2jε/a2

Bs > a/2)

.

¡f^CK<?]yT0GIV,AEKLD/?I^ ]9 8];_ :9Q8P8M VP?IFXLTuvuVÂ8]9 vVP^bK<?ACa]T IBW^nAfHUVP^CTOTN/T A

limε↓0

ε log supx∈

Px

(

Aαε(k1,...,k`)

)

≤ − 1

1 + δ

(

`−1∑

j=0

dj + n1a2

4+ `

a2

2

)2

≤ − 1

1 + δ

1

8

(n− 2J − 1

nt+ `

a2

2

)2

]9 8rd

g$DPBOVX<Xα ∈ Dδ

2`+1

V?IFMVX<Xδ > 0

9 DPACTdAEaIV,AOKL?MACa]KwÔHUVP^CT n1 = 0K<^m\D/Ê^nK<o]X<T

ACa]Kw^fD:HH G]BW^K<?ACaIT

HUV/^nT` = 1

V?IFS1 ∈ In

]o(THUVPGI^nTInbVPÂCaITNKL?0ACTBCº,VPXT+ACBETVACTF^C\(TH KwVX<XLk/9

`DT^xAEKLuVACTyACaITG]\]\TUBdTj\DP?ITU?0ACKwVX|BWV,AETvDPgAε ∩ TJ > t tKLACa*XLTuyuyV¹8]9 8Ty?]TUTFMAEDH D/uy\IVBETOVPXLX-ACa]T+BWV,ACT^g$BCD/u :9Q8_vV?(F¢]9 8rI9 TNPT A

lim supε↓0

ε log supx∈

Px

(

Aε ∩ TJ > t)

= maxq∈Q

maxα∈Dδ

2`+1

lim supε↓0

ε logPx

(

Aαεq

)

≤ − 1

1 + δ

1

8

(n− 2J − 1

nt+

a2

2

)2

g$DPBOVX<Xδ > 0

V?IF¢XwVBEPTNT?]DPGIPan(taITUBET+ACa]T¼X<VPBC/T^nAo(D/G]?IFHUVPuyTNg$BCD/u¶AEa]TÀHVP^CT

` = 1k1 = n

9c-TUAnAEKL?] (BE^nA

δ ↓ 0V?IFlACaITU?

n→ ∞ ^na]D,^

lim supε↓0

ε logPx

(

Aε ∩ TJ > t)

≤ 1

8

(

t+a2

2

)2. ]9 8/D

?IDACaITUBV\I\]XLKwHUVACK<DP?DgXLTuvuVÂ8]9 8ÀPK<ºPTÂCa]TNG]\I\(TBto(D/G]?IFg$DPBP (Aε)

9I¡f^CKL?IyACa]T+T^nACK<uV,AET^ :9<;zM V?(F :9Q8PD T I?IFlim sup

ε↓0ε log sup

x∈ Px(Aε) ≤ 1

8

(

t+a2

2

)2.

`ba]Kw^H D/uy\]XLTUACTÂCa]TN\]BEDjDgDgAEa]TNX<TUuyuVÂ:9<;8:9 0TF'

q/8

& # =@? ACa]Kw^N^CTHJAEKLD/?£TyGI^CT¼AEa]TBET^CG]XAW^+DgACa]T\]BETUºjK<DPGI^+^CTH ACK<DP?£ACD I?IVPXLX<kMF:TBCK<ºPTvACaITycefY g$D/BOACa]TTU?(F:\(D/KL?0A

Xϑt

DgACa]Td^CDPX<G:ACK<DP?DgdXϑ

s = ϑb(Xs) ds+ dBsDP?

[0; t]

Xϑ0 = z ∈ ]9 8PqM

g$DPBϑ→ ∞ 9(`ba]TNuVK<?BET^CG]XLAbKw^bAEa]TUD/BCTu :9<;zvV,AtAEa]T+TU?IFDgACaIK<^^CTH ACK<DP?-9 TdVPÊ^nG]uyT

b = −Φ′ g$D/BV C2 g$G]?IH ACK<DP? Φ: → tKLACa¢oDPG]?(F:TF¢^CTH D/?IF¢F:TUBEK<ºVACK<ºPT Φ′′ 9]`ba]TU?ACaITdF:BCKLgAbK<^cKL\(^CHWa]KLANHUDP?0ACK<?jG]DPGI^tVP?IFlACa]TÀi:ems ]9 8PqM %aIVP^tVvG]?IK<0G]Td^CDPX<G:ACK<DP? Xϑ 9

( 5 `DÂVº/DPKwFHUDPuy\]X<K<HV,AETFV?IFaIVPBEFlACDBETV/FlTj\IBCT^C^CKLD/?I^bK<?¢^CuVX<X-\]BEKL?0AtTd^nD/uvTUACK<uyT^tBEKAET

(A)g$D/BbACa]TNK<?IF:KwHUVACDPBbg$GI?IHJAEKLD/?DPgAEa]TNTUº/TU?0A

AF:G]BEKL?]vAEa]K<^f^nTHJACK<DP?9

3 .j 5M¤)&' 1 &*^+<-Φ: → 3 + . C2 =8a/ 4<-J6K5 / FG6I-I? 3 5 8a/21 + 1 Φ′′ ./21L +"- b = −Φ′ Q 99U8 N+Z-I?S+<C`+ 6 9 ./ m ∈ FG6I-I?

b(x) = 06 = ./21 5 /'L]> 6 = x = m .0/21 lim inf |x|→∞ |b(x)| > 0 Q 8 CU-I?S+<C .99U8 NT+Z-%? . -G-I?S+<C`+Z6 9Z. C . - + =8a/ 4"-76K5 / I : → [0;∞]

FG6I-I?

lim infϑ→∞

1

ϑlogE

(

exp(−ϑ2

2

∫ t

0

b2(ωs) ds)1O(Bt))

≥ − infx∈O

I(x)

= 5C +"f+"C > 5 V + /j9 +"- O ⊆ .0/21

lim supϑ→∞

1

ϑlogE

(

exp(−ϑ2

2

∫ t

0

b2(ωs) ds)1K(Bt))

≤ − infx∈K

I(x)

= 5Ch+"f+<C > 45N VS. 4"- 9 +"- K ⊆ iQ 50C ϑ > 0 L +"- Xϑ 3 + .9 5 L 8 -76K5 / 5 = -%?@+ WXZY ]9 8PqM FG6I-%? 9 - . CU-6 / Xϑ0 = 0 Q R ?@+ /\= 5C ϑ → ∞ -I?S+ =<. N\6 L]> (Xϑ

t )ϑ9". -J6 9 + 9 -%?@+ F + . *,X FG6I-%?C . - + =8a/ 4"-76K5 / J :F^?@+"CE+

J6 9Z1 + / + 13<>

J(x) = Φ(x) − Φ(0) − 1

2· t · Φ′′(m) + I(x).

)+,-¨ & «]BEDPu c-TUuyuV;P9Q¼T+³j?]D,AEa]TdF:T?I^nKLAxkÂDPgACaITdF:K<^nACBEK<o]G:ACK<DP?¢DPgACaIK<^^CDPX<G:AEKLD/?Xϑ

t

P (Xϑt ∈ A) =

∫

1A(ωt) exp(

ϑF (ω) − ϑ2G(ω))

d (ω) :9Q8/D ta]TBCT

F (ω) = Φ(ω0) − Φ(ωt) +1

2

∫ t

0

Φ′′(ωs) dsV?IF

G(ω) =1

2

∫ t

0

b2(ωs) ds.

«KLBW^nAXLTUAOo(TdD/\(T?

x ∈ OV?IF

δ > 09I`baITU?TdHUV? (?IFV?

ηtKLACa

0 < η < δBη(x) ⊆ O

V?(F |Φ(y) − Φ(x)| ≤ δ

g$DPBVPXLXy ∈ Bη(x)

9(emTI?]T

F ∗(x) = Φ(0) − Φ(x) +1

2tΦ′′(m).

qP_

`ba]T?T I?IF

lim infϑ→∞

1

ϑlogP (Xϑ

t ∈ O)

≥ lim infϑ→∞

1

ϑlogP (Xϑ

t ∈ Bη(x))

= lim infϑ→∞

1

ϑlog

∫

1Bη(x)(ωt) exp(

ϑF (ω) − ϑ2G(ω))

d (ω)

≥ lim infϑ→∞

1

ϑlog

∫

1Bη(x)(ωt) exp(

ϑ(F ∗(x) − 2δ) − ϑ2G(ω))

·(

|F (ω) − F ∗(x)| ≤ 2δ)

d (ω)

= F ∗(x) − 2δ + lim infϑ→∞

1

ϑlog

∫

1Bη(x)(ωt) exp(

−ϑ2G(ω))

·(

|F (ω) − F ∗(x)| ≤ 2δ)

d (ω).

klF:TI?]KLACK<DP?¢DgF ∗(x)

TOa(VºPT∣

∣F (ω) − F ∗(x)∣

∣ =∣

∣Φ(0) − Φ(ωt) +1

2

∫ t

0

Φ′′(ωs) ds

− Φ(0) + Φ(x) − 1

2tΦ′′(m)

∣

∣

≤∣

∣Φ(x) − Φ(ωt)∣

∣+1

2

∫ t

0

∣

∣Φ′′(ωs) − Φ′′(m)∣

∣ ds.

`bajGI^bta]T?]TUº/TUBωt ∈ Bη(x)

V?IF ∣∣F (ω) − F ∗(x)

∣

∣ ≥ 2δT I?IF

1

2

∫ t

0

∣

∣Φ′′(ωs) − Φ′′(m)∣

∣ ds ≥ 2δ − δ = δ.

THVGI^CTΦ′′ K<^boDPG]?(F:TFACaITdVoD,ºPTOT^nACK<uyVACT+K<uv\IXLK<T^bAEaIV,ATNHUVP? I?(FVP?

ε > 0tKLACa

∣

∣

∣

s ∈ [0; t]∣

∣ |ωs −m| ≥ δ/t

∣

∣

∣> ε

g$DPBmVX<X-\IV,AEaI^ωtKAEa

ωt ∈ Bη(x)V?IF ∣

∣F (ω) − F ∗(x)∣

∣ ≥ 2δ9]THUVG(^nT

mKw^bAEa]TNDP?]X<k TUBEDyDg

bVP?IF

oTHUVPGI^CTlim inf |x|→∞ |b(x)| > 0

TNaIVºPTinf

b2(x)∣

∣ |x−m| ≥ δ/t

> 0,

K©9 T/9ITdHUV? I?(FVg > 0

tKAEaG(ω) > g

g$DPBmVX<X-\IV,AEaI^ωtKAEa

ωt ∈ Bη(x)V?(F ∣

∣F (ω) − F ∗(x)∣

∣ ≥2δ9]`DP/T ACaITUBbACaIK<^b/KLº/T^

lim supϑ→∞

1

ϑlog

∫

1Bη(x)(ωt) exp(

−ϑ2G(ω))

(|F (ω) − F ∗(x)| > 2δ) d (ω)

≤ lim supϑ→∞

1

ϑlog

∫

exp(

−ϑ2g)

d (ω)

= −∞.

ijDyT+HV?¢GI^CT+XLTuvuV8:9 _¼ACDÂHUDP?IHUXLGIF]T

lim infϑ→∞

1

ϑlog

∫

1Bη(x)(ωt) exp(

−ϑ2G(ω))

d (ω)

= lim infϑ→∞

1

ϑlog

∫

1Bη(x)(ωt) exp(

−ϑ2G(ω))

(|F (ω) − F ∗(x)| ≤ 2δ) d (ω)

qr

V?(Fl/T A

lim infϑ→∞

1

ϑlogP (Xϑ

t ∈ O)

≥ F ∗(x) − 2δ + lim infϑ→∞

1

ϑlog

∫

1Bη(x)(ωt) exp(

−ϑ2G(ω))

d (ω)

≥ F ∗(x) − 2δ − infy∈Bη(x)

I(y)

≥ F ∗(x) − 2δ − I(x)

g$DPBVPXLXδ > 0

9]c-TUAnACK<?]δ ↓ 0

/KLº/T^

lim infϑ→∞

1

ϑlogP (Xϑ

t ∈ O) ≥ F ∗(x) − I(x)

V?(FÂAWV³jKL?I¼AEa]Td^CG]\]BETUuÀGIu D,ºPTBbVPXLXx ∈ O

DP?ACa]TNBEK<Pa0AtaIV?IF^CK<F:TN\IBCD,º/T^%ACa]TNX<D,TBbo(D/G]?IFZ9 D, XLTUA K ⊆ oTÂH D/uv\(VPHJAdVP?IF

δ > 09-«]DPBNTVPHWa

x ∈ KTyHV? I?IF*V?

η > 0tKLACa

|Φ(y) − Φ(x)| ≤ δta]TU?]TºPTB

y ∈ Bη(x)9-THUVPGI^CT

IK<ÔX<D,TBm^CTUuyK HUDP?0ACK<?jG]DPGI^mT¼HUVP?£V/^C^CG]uyT

I(y) ≥ I(x) − δg$D/BNTUºPTBCk

y ∈ Bη(x)ojk¹HWa]DjD/^CKL?I

η^CuVX<X|TU?]D/G]Pa9¡m^nK<?]ACaITyHUDPuy\IV/HJAC?ITÊ^Dg

KTvHV?MHUD,ºPTUB

KtKLACa*V I?IKAETv?jG]u¼o(TBmDPg^CGIHWa¹oIVPXLXwÂEa]TUBETvVPBCT

x1, . . . , xn ∈ KV?(F

0 <η1, . . . , ηn < δ

tKLACaK ⊆

n⋃

k=1

Bηk(xk)

V?(F¢ACa]TVPo(D,º/TÀVPÊ^nGIuv\]ACK<DP?¹D/?ΦV?(F

IaIDPXwFg$DPBOTV/HWa

k9«]D/B

k = 1, . . . , nHUDP?I^CKwF:TUB

F ∗(xk)VP^

F:TI?]TF¢VPo(D,º/TP9:`ba]Kw^bACK<uyTNT I?IFlim sup

ϑ→∞

1

ϑlogP (Xϑ

t ∈ K)

≤ lim supϑ→∞

1

ϑlog

n∑

k=1

P (Xϑt ∈ Bηk

(xk))

= maxk=1,...,n

lim supϑ→∞

1

ϑlog

∫

1Bηk(xk)(ωt) exp

(

ϑF (ω) − ϑ2G(ω))

d (ω).

THVGI^CTFKw^bo(D/G]?IF:TFD/? ωt ∈ Bηk

(xk) T+HV?GI^CTNXLTuvuV8:9 _yVP^tVPo(D,º/TfAEDyHUDP?IHUXLG(F:T

lim supϑ→∞

1

ϑlog

∫

1Bηk(xk)(ωt) exp

(

ϑF (ω) − ϑ2G(ω))

d (ω)

= lim supϑ→∞

1

ϑlog

∫

1Bηk(xk)(ωt) exp

(

ϑF (ω) − ϑ2G(ω))

· (|F (ω) − F ∗(xk)| ≤ 2δ) d (ω)

g$DPBk = 1, . . . , n

9]`ba]K<^PK<ºPT^

lim supϑ→∞

1

ϑlogP (Xϑ

t ∈ K)

≤ maxk=1,...,n

lim supϑ→∞

1

ϑlog

∫

1Bηk(xk)(ωt) exp

(

ϑF (ω) − ϑ2G(ω))

· (|F (ω) − F ∗(xk)| ≤ 2δ) d (ω)

≤ maxk=1,...,n

lim supϑ→∞

1

ϑlog

∫

1Bηk(xk)(ωt) exp

(

ϑ(F ∗(xk) + 2δ) − ϑ2G(ω))

·(

|F (ω) − F ∗(xk)| ≤ 2δ)

d (ω)

≤ maxk=1,...,n

F ∗(xk) + 2δ

+ lim supϑ→∞

1

ϑlog

∫

1Bηk(xk)(ωt) exp

(

−ϑ2G(ω))

d (ω).

q/

D,TNHV?GI^nTOAEa]TNG]\]\TUBoDPGI?IFDP?ACa]TNBWV,AETODPgAEa]TNKL?0AETUPBWVX-VP?IFDPG]BHWa]D/K<HUT+Dg ηkAEDyPT A

lim supϑ→∞

1

ϑlogP (Xϑ

t ∈ K)

≤ maxk=1,...,n

F ∗(xk) + 2δ − infy∈Bδ(xk)

I(y)

≤ maxk=1,...,n

F ∗(xk) + 2δ − I(xk) + δ.

V?(FlX<T ACACK<?]δ ↓ 0

I?]Kw^Ca]TÂEa]TN\]BED0DPgg$D/BH DPuy\IV/HJA^CT AE^9 0TFS `DlPTUAfAEa]T¼GI\]\(TB+o(D/G]?IFg$D/B+PTU?ITUBWVXHUXLD0^nTF¹^CT AE^mT¼aIVº/TNACD¢^Ca]D,´T:\DP?]T?/AEK<VPXAEKL/a0AC?]T^C^mDg

ACaIT L(Xϑt )

]K©9 T/9]T+aIVº/TfAEDÂ^naID, ACaIVAtg$DPBtTºPTUBEkα ∈ ACa]TBCT+T:Kw^xAW^tVH D/uv\(VPHJA^CT A

KαI^CGIHWalAEaIV,A

lim supϑ→∞

1

ϑlogP (Xϑ

t /∈ Kα) < −α.

`ba]T¼G]\]\TUBOo(D/G]?IF¢g$D/BOVPBCo]KLACBWVBEkHUXLD0^nTF^nTUAE^mD/G]XwFACa]T?¹g$DPX<X<D,g$BEDPuX<TUuyuVl8:9<;P9-·DPBEDPX<XwVBEk:9Q8~oTUX<D,^Ca]D,^tVy^nKLACGIVACK<DP?ta]TUBETOACa]Kw^tDPBE³:Û9

= A+K<^mKL?0AETUBET^nACK<?]ÂAED?]DPACT¼ACaIVAOT¼HV?¹X<TVPBC?£^nD/uvTv\]BEDP\TUBCACK<T^fDgIg$BEDPuÁACa]TvgV/HJAOACaIVA

JKw^mV

BWV,ACTÀg$G]?IH ACK<DP?-9Zf^+VlBEVACTdg$GI?IHJAEKLD/?JKw^m\D/^CKLACK<ºPTP9ZijDTvHUV? H D/?IH X<GIF:TÀACaIVAmAEa]TÀg$GI?IHJAEKLD/?

Ig$BCD/u

ACaITNXLTuvuVvuÀG(^xAfÊV,AEK<^ng$kI(x) ≥ −Φ(x) + Φ(0) +

1

2· t · Φ′′(m)

g$DPBVPXLXx ∈ 9`ba]Tvg$DPX<X<D,tKL?]X<TUuyuVKw^NVPT?]TUBWVX<K<ÊV,AEKLD/?¹DgbH D/BCD/XLXwVBEkr(9 _I9Z= ANa]TUX<\IÔACDF:TUACTBCuyK<?]TyACa]TyBWV,AET

g$G]?IH ACK<DP?Ita]KwHWaK<^t?ITUTF]TFlACDÂV\I\]XLklXLTuvuVÂ]9L;,j9

3 .j 5¤ & ¿ &^*^+<-m ∈ .0/21 b : → 3 + . C2 =8a/ 4<-J6K5 / FG6I-I? b(x) = 0

6 = ./21 5 /'L]> 6 =x = m : b′(m) 6= 0 : ./H1 lim inf |x|→∞ |b(x)| > 0 Q R ?@+ / = 5C ./'> 450N VS. 4"- 9 +<- K ⊆ F + ? . f+

lim supε→0

ε logP(1

2

∫ t

0

b2(Bs) ds ≤ ε,Bt ∈ K)

≤ −1

4infa∈K

(

∣

∣

∫ m

0

|b(x)| dx∣

∣ +1

2|b′(m)|t+

∣

∣

∫ a

m

|b(x)| dx∣

∣

)2

./H1 = 5C ./'> 5 V + /j9 +<- O ⊆ F + ? . f+

lim infε→0

ε logP(1

2

∫ t

0

b2(Bs) ds ≤ ε,Bt ∈ O)

≥ −1

4infa∈O

(

∣

∣

∫ m

0

|b(x)| dx∣

∣ +1

2|b′(m)|t+

∣

∣

∫ a

m

|b(x)| dx∣

∣

)2

.

)+,-¨ & m^V?VoIo]BCTºjK<VACK<DP?¢F:TI?ITv(x) = b2(x)/2

g$DPBfVPXLXx ∈ 9I«]D/BtACaITd\]BED0DPgDPgAEa]TdG]\I\(TBoDPG]?IF¢HWa]DjD0^nTNVH D/uy\IVPH At^CT A

K]X<T A

δ, τ > 0VP?IF¢HWa]DjD/^CT

Dδ3

VP^tK<?¢XLTuyuyVÂ8]9 qI9]`ba]TU?¢g$D/Bε < t/2τTOa(VºPT

∫ t

0

v(Bs) ds ≤ ε,Bt ∈ K

⊆⋃

α∈Dδ3

∫ ετ

0

v(Bs) ds ≤ α1ε,

∫ t−ετ

ετ

v(Bs) ds ≤ α2ε,

∫ t

t−ετ

v(Bs) ds ≤ α3ε,Bt ∈ K

.

qPq

BCKLACK<?](A)

g$DPBtAEa]TNKL?(F:K<HV,AEDPBbg$G]?IH ACK<DP?DPgAVP?IFGI^nK<?]vACaITÀ^xAEBCD/?]y¸¹VBE³PD,º\]BEDP\TUBCAxkÂDPgBED, ?]KwV?uyDACK<DP?AEa]K<^PK<ºPT^

P(

∫ t

0

v(Bs) ds ≤ ε,Bt ∈ K)

≤∑

α∈Dδ3

E(

(∫ ετ

0 v(Bs) ds ≤ α1ε)(∫ t−ετ

ετ v(Bs) ds ≤ α2ε)

E(

(∫ t

t−ετv(Bs) ds ≤ α3ε,Bt ∈ K)

∣

∣ Ft−ετ

)

)

=∑

α∈Dδ3

E(

(∫ ετ

0 v(Bs) ds ≤ α1ε)(∫ t−ετ

ετ v(Bs) ds ≤ α2ε)

EBt−ετ

(

(∫ ετ

0v(Bs) ds ≤ α3ε,Bετ ∈ K)

)

)

=:∑

α∈Dδ3

p(α, ε)

D, X<T A α ∈ Dδ3

oT :TF£V?IFa > 0

9£Ty^C\]XLKLAOACa]TyH D/BCBET^C\(D/?IF:K<?]TUº/TU?0Afg$GIBnAEa]TUBOojk¢F]K<^nACK<? PGIK<^Ca]K<?]yACa]TdAxDÂHVP^CT^ supετ≤s≤t−ετ |Bs −m| > a

VP?IF supετ≤s≤t−ετ |Bs −m| ≤ a

9i:KL?IHUTDPuyKLAnAEKL?]Â^CDPuyTNH D/?IF:KLACK<DP?I^buVP³PTÂEa]TN\]BEDPoIVPo]KLX<KLAxkDP?]X<klXwVBEPTBT+PT A

p(α, ε) ≤ p1(α, ε) + p2(α, ε)

tKLACap1(α, ε) = sup

y∈ Py

(

∫ t−2ετ

0

v(Bs) ds ≤ α2ε, sup0≤s≤t−2ετ

|Bs −m| > a)

V?(F

p2(α, ε) = P(

∫ ετ

0

v(Bs) ds ≤ α1ε, |Bετ −m| ≤ a)

· supy∈

Py

(

∫ t−2ετ

0

v(Bs) ds ≤ α2ε, sup0≤s≤t−2ετ

|Bs −m| ≤ a)

· sup|z−m|≤a

Pz

(

∫ ετ

0

v(Bs) ds ≤ α3ε,Bετ ∈ K)

.

`DlHUVPX<HUG]X<VACTÀACa]T¼BEVACTNg$D/BfAEa]T¼^CG]up1(α, ε) + p2(α, ε)

T¼aIVºPTNAEDHVXwH G]XwV,AETNACa]TvBWV,ACT^DPg|ACa]TK<?IF:K<º0KwF:GIVPXZACTBCu^9]c-TUA

η > 09]«]D/B

p1TdHUV?GI^CT+XLTuvuVÂ]9L;8NAEDv/T A

lim supε→0

ε log p1(α, ε)

≤ lim supε→0

ε log sup|y−m|<a/2

Py

(

∫ t−η

0

v(Bs) ds ≤ α2ε, sup0≤s≤t−η

|Bs −m| > a)

,

≤ − 1

8α2

(

t− η +1

2a2)2

.

ijK<?IH T+g$D/B :TFηAEa]Kw^BWV,ACT+oTHUDPuyTdVBEo]KLACBWVBEky^CuVX<X-ta]T?

ao(TH DPuyT^bX<VPBC/T:T+HV?HWa]DjD/^CT

aXwVBEPT

TU?IDPG]/alAEaIV,AtAEa]TNBEVACT+Dgp1(α, ε) + p2(α, ε)

Kw^F:D/uyKL?IVACTFlojkp29

`DACBETVAfACaITp2 AETUBEuTvV\]\]X<kXLTuvuV8:9Qyg$DPBfACa]TvBWV,ACTÀDg%VÂ\IBCD:F:GIH A9«]BEDPuÁ\]BEDP\D/^CKAEKLD/?M:9 _

TO³j?]D, ACa]TNK<?IF:K<ºjK<F]GIVX-BWV,AET^

lim supε→0

ε logP(

∫ ετ

0

v(Bs) ds ≤ ε, |Bετ −m| ≤ a)

≤ −1

4

(

∫ m

0

|b(x)| dx)2

· r21(τ)

q0

V?(F

lim supε→0

ε log sup|z−m|≤a

Pz

(

∫ ετ

0

v(Bs) ds ≤ ε,Bετ ∈ K)

≤ −1

4inf

a∈K

(

∫ a

m

|b(x)| dx)2

· r22(τ)

ta]TBCTlimτ→∞ r1(τ) = limτ→∞ r2(τ) = 1

(V?IFXLTuyuyV]9L;q¼/KLº/T^

lim supε→0

ε log supy∈

Py

(

∫ t−2ετ

0

v(Bs) ds ≤ ε, sup0≤s≤t−2ετ

|Bs −m| ≤ a)

≤ lim supε→0

ε log sup|y−m|<a/2

Py

(

∫ t−η

0

v(Bs) ds ≤ ε, sup0≤s≤t−η

|Bs −m| ≤ a)

≤ −|b′(m)|2(t− η)2

16.

¡f^CK<?]yXLTuvuVÂ8]9 ÀTN/T AbACaITdH DPu¼o]K<?]TFBEVACTlim sup

ε→0ε log p2(α, ε)

≤ − 1

1 + δ

(1

2

∣

∣

∫ m

0

|b(x)| dx∣

∣ · r1(τ)

+1

4|b′(m)| · (t− η) +

1

2inf

a∈K

∣

∣

∫ a

m

|b(x)| dx∣

∣ · r2(τ))2

g$DPBVPXLXα ∈ Dδ

3

9`ba]T+BWV,ACTOg$D/BbACa]Td^CG]u¶D,º/TUBbVX<X

α ∈ Dδ3

HUVP?oTNT^nACK<uV,AETFtKAEaX<TUuyuVÂ8:9Q8:9]`baITOBET^CG]XLAK<^

lim supε→0

ε logP(

∫ t

0


≤ − 1

1 + δ

(1

2

∣

∣

∫ m

0

|b(x)| dx∣

∣ · r1(τ)

+1

4|b′(m)| · (t− η) +

1

2inf

a∈K

∣

∣

∫ a

m

|b(x)| dx∣

∣ · r2(τ))2

g$DPBVPXLXη > 0

δ > 0

]V?IFτ > 0

9]c-T ACACK<?] I?IVPXLX<kτ → ∞

δ ↓ 0]V?IF

η ↓ 0PK<ºPT^

lim supε→0

ε logP(1

2

∫ t

0

b2(Bs) ds ≤ ε,Bt ∈ K)

≤ −1

4

(1

2

∣

∣

∫ m

0

|b(x)| dx∣

∣ +1

2|b′(m)|t+ inf

a∈K

∣

∣

∫ a

m

|b(x)| dx∣

∣

)2

.

`ba]Kw^t\]BED,ºPT^[AEa]TNG]\]\TUBoDPGI?IFZ9«]D/BtACaITÀX<D,TBo(D/G]?IF c-TUA

ζ, η, τ > 0VP?IF

α1, α2, α3 ∈ tKAEaα1 + α2 + α3 = 1

9`ba]TU?g$D/Bε < t/2τ

T+aIVº/T

∫ t

0

v(Bs) ds ≤ ε,Bt ∈ O

⊇

∫ ετ

0

v(Bs) ds ≤ α1ε, |Bετ −m| < ζ

∩

∫ t−ετ

ετ

v(Bs) ds ≤ α2ε, |Bt−ετ −m| < η

∩

∫ t

t−ετ

v(Bs) ds ≤ α3ε,Bt ∈ O

qP

V?(FÂAEa0G(^tT+/T A

P(

∫ t

0

v(Bs) ds ≤ ε,Bt ∈ O)

≥ E(

(

∫ ετ

0

v(Bs) ds ≤ α1ε, |Bετ −m| < ζ

)

·(

∫ t−ετ

ετ

v(Bs) ds ≤ α2ε, |Bt−ετ −m| < η)

·E(

(

∫ t

t−ετ

v(Bs) ds ≤ α3ε,Bt ∈ O)

∣

∣

∣ Ft−ετ

))

≥ E(

(

∫ ετ

0

v(Bs) ds ≤ α1ε, |Bετ −m| < ζ)

·E(

(

∫ t−ετ

ετ

v(Bs) ds ≤ α2ε, |Bt−ετ −m| < η)

∣

∣

∣Fετ

))

· infm−η<y<m+η

Py

(

∫ ετ

0

v(Bs) ds ≤ α3ε,Bετ ∈ O)

≥ P0

(

∫ ετ

0

v(Bs) ds ≤ α1ε,Bετ ∈ (m− ζ;m+ ζ))

· infm−ζ<z<m+ζ

Pz

(

∫ t−2ετ

0

v(Bs) ds ≤ α2ε, |Bt−2ετ −m| < η)

· infm−η<y<m+η

Py

(

∫ ετ

0


.

«KLBW^nAAWV³/TOX<D,TUBbT:\DP?]T?/AEK<VPX-BWV,ACT^g$DPBε ↓ 0

9]`ba]TNX<D,TBbT:\(D/?]TU?0AEK<VPXZBEVACT+DgACa]TNX<T gA a(V?IF¢^CK<F:TKw^f/BCTV,ACTBtDPBmT0GIVXACDlACa]Tv^nG]uDPg|ACa]T¼XLD,TUBBWV,AET^tDPg|ACa]T¼BCK<Pa0A aIVP?IF^CK<F:T/9(`ba]Kw^mKL?]T0GIVX<KAxka]D/X<F]^g$DPBVPXLXη, τ > 0

VP?IFα1, α2, α3 ∈ tKLACa

α1 + α2 + α3 = 19

`ba]T?X<T Aτ → ∞ 9 TNACBETVAtACa]TdACa]BETUT+AETUBEuy^fDP?ACaITdBCK<Pa0AaIVP?IF^CK<F:TNK<?IF:K<ºjK<F]GIVX<XLk/9(«K<BE^nAtACTBCu

g$BEDPu¶cTUuyuVy]9L;+T+³0?ID,

limτ→∞

lim infε↓0

ε logP0

(

∫ ετ

0

v(Bs) ds ≤ α1ε,Bετ ∈ (m− ζ;m+ ζ))

≥ − 1

α1

1

4inf

m−ζ<a<m+ζ

(

∣

∣

∫ m

0

|b(x)| dx∣

∣ +∣

∣

∫ a

m

|b(x)| dx∣

∣

)2

= − 1

α1

1

4

(

∣

∣

∫ m

0

|b(x)| dx∣

∣

)2

· r1(ζ)

ta]TBCTlimζ↓0 r1(ζ) = 1

9ijTH DP?(F¢ACTBCu TvHUV?MuyVP³PTÀACa]Tv\]BEDPoIVPo]KLX<KLAxk^CuVX<XLTBfojk¢BCT\]X<V/H K<?]

t − 2ετtKLACa

t9`baITU?¹AEa]T

ACTBCu K<^b?IDvX<DP?IPTUBτ F:TU\TU?(F:TU?0AmVP?IFlG(^nK<?]X<TUuyuVy]9L;qÀT+PTUA

lim infε↓0

ε log infm−ζ<z<m+ζ

Pz

(

∫ t−2ετ

0

v(Bs) ds ≤ α2ε, |Bt−2ετ −m| < η)

)

≥ − 1

α2

|b′(m)|216

t2 · r2(ζ)ta]TBCT

limζ↓0 r2(ζ) = 19

`ba]K<BEFlACTBCu :GI^CK<?]ÂH D/BCD/XLXwVBEkÂ:9 r¼TNPTUA

lim infε↓0

ε log infm−η<y<m+η

Py

(

∫ ετ

0


≥ − 1

α3

1

4infa∈O

(

∫ a

m

|b(x)| dx)2

· r3(η)

ta]TBCTlimη↓0 r3(η) = 1

9

qPz

·DPu¼o]K<?]KL?IÀAEa]T+ACa]BETUTNBWV,AET^TO/T A

lim infε↓0

ε logP(

Bt ∈ O,

∫ t

0

v(Bs) ds ≤ ε)

≥ − 1

α1

1

4

(

∣

∣

∫ m

0

|b(x)| dx∣

∣

)2

· r1(ζ)

− 1

α2

|b′(m)|216

t2 · r2(ζ)

− 1

α3

1

4infa∈O

(

∣

∣

∫ a

m

|b(x)| dx∣

∣

)2

· r3(η).V?(FlX<T ACACK<?] IBW^xA

ζ ↓ 0V?IFlACaITU?

η ↓ 0k0K<TUXwF]^

lim infε↓0

ε logP(

Bt ∈ O,

∫ t

0

v(Bs) ds ≤ ε)

≥ − 1

α1

(1

2

∫ m

0

|b(x)| dx)2

− 1

α2

( |b′(m)|4

t)2

− 1

α3

(1

2infa∈O

∫ a

m

|b(x)| dx)2

g$DPBVPXLXα1, α2, α3 ∈ tKLACa α1 + α2 + α3 = 1

9·a]DjD/^CK<?]¼D/\:ACK<uVX

α1α2

]V?IFα3VP^tF:T^CHUBCK<o(TFlK<?¢XLTuyuyV8]9 ryTO/T A

lim infε↓0

ε logP(

Bt ∈ O,1

2

∫ t

0

b2(Bs) ds ≤ ε)

≥ −(1

2

∣

∣

∫ m

0

|b(x)| dx∣

∣+|b′(m)|

4t+

1

2infa∈O

∣

∣

∫ a

m

|b(x)| dx∣

∣

)2

= −1

4

(

∣

∣

∫ m

0

|b(x)| dx∣

∣+|b′(m)|

2t+ inf

a∈O

∣

∣

∫ a

m

|b(x)| dx∣

∣

)2

.

`ba]Kw^H D/uy\]XLTUACTÂCa]TN\]BEDjDgx9 0TF' `ba]TduVPKL?BET^CG]XLAfDgACaIK<ÔHWaIV\]ACTUBfK<ÂCa]TÀg$DPX<XLD,tK<?]yACaITUDPBETUu ACD/PTUACa]TBtKAEaACaIT¼H D/BCD/XLXwVBEK<T^t:9Q8~

V?(F]9 8];P9d.jI,.j ¤ & Ã ( *^+<-

Φ: → 3 + . C3 =8a/ 4"-76K5 / FG6I-I? 3 5 8a/21 + 1 Φ′′ .0/21 b = −Φ′ Q 99U8 N+-%?@+"CE+Z6 9Z.0/ m ∈ FG6I-I?b(x) = 0

6 =e./H1 5 /'L]> 6 = x = m : b′(m) 6= 0 : ./21 lim inf |x|→∞ |b(x)| > 0 QR ?@+ / = 5C +"f+"C > t > 0

-%?@+ 9 5 L]8 -76K5 / Xϑ 5 =dXϑ

s = ϑb(Xϑs ) ds+ dBs

= 50C s ∈ [0; t] : ./H1Xϑ

0 = z ∈ 9". -J6 9 + 9 -%?@+ = 5 LIL 5FG6 / APF + . * X cD= 50C +<f+<C > 45N V@. 4<- 9 +<- K ⊆ F + ? . f+

lim supϑ→∞

1

ϑlogP (Xϑ

t ∈ K) ≤ − infx∈K

Jt(x)

./H1 = 5C +"f+"C > 5 V + /j9 +<- O ⊆ F + ? . f+

lim infϑ→∞

1

ϑlogP (Xϑ

t ∈ O) ≥ − infx∈O

Jt(x),

F^?@+"CE+Z-%?@+eC . -#+ =8a/ 4"-J6K5 / 6 9Jt(x) = V m

z (Φ) − Φ(z) + t(Φ′′(m))− + V xm(Φ) + Φ(x). :9Q8 d

,~

=@?¹AEa]T¼ACa]TDPBETUuV b

a (Φ)F:TU?IDACTÂEa]TvACDPAEVXº,VBEK<VACK<DP?¹DPg

ΦoT AxTUT?

aV?IF

b9= A+HV?¹oTyKL?0ACTB \]BET AETFVPÂCaIT nHUD/^nA ODPgAEa]TN\]BEDjHUTÊ^PDPK<?]vg$BEDPu

aAEDb9]THUVPGI^CT

b = −Φ′ TOa(VºPT

V ba (Φ) =

∣

∣

∫ b

a

|b(x)| dx∣

∣

g$DPBmV?jka, b ∈ 9I`ba]TN?]DPAEV,AEKLD/? (Φ′′(m))−

F:TU?]DPACT^bACa]TN?ITU/VACK<ºPT+\IVPBnADPgΦ′′(m)

]K©9 T/9(Φ′′(m))− =

0KLg

Φ′′(m) ≥ 0V?(F

(Φ′′(m))− = |Φ′′(m)| KLg Φ′′(m) < 09I`ba]Kw^fHV?oTÀKL?0AETUBE\]BCTUACTFV/^tAEa]TCH D/^nA Dgt^nAEVkjKL?I?]TVB

mg$D/BdVGI?]KA¼Dg%AEKLuyT/9`ba]Kw^+ACTBCu D/?]X<kMD:HUHUG]BE^ KgACaITT0G]K<XLK<o]BEKLG]u \(D/KL?0A

mKw^

G]?I^nAEVPo]X<TP9)+,-¨ & ijKL?(H TvACa]TyBWV,AETdg$G]?IH ACK<DP?

JtKwÔKL?jº,VBEK<VP?0AmG]?IF]TUBd^C\IVPHUT¼^Ca]KgAW^+TyHUVP?MtKLACa]D/G:ANXLD0^C^mDg

PT?]TUBWVX<KLAxkV/^C^CG]uyTz = 0

ojk¢BETU\]XwVPHUKL?]ΦtKAEa¹ACaITv^Ca]KLgACTF¹g$G]?IH ACK<DP?

Φ( · + z)V?IFM^xAWVBCACK<?]ÂAEa]T

i:emsK<?¹~]9i:KL?IHUTduyD/^nADgACaITdD/BC³V/^tVX<BCTVPF:kF:DP?ITNKL?¢AEa]TÀ\]BETUºjK<DPGI^t^CTH ACK<DP? IACa]TÀ\]BCDjDPgHUDP?I^CK<^nAE^DP?IXLkÂDPgAEa]BETUTN^xAETU\I^9«KLBW^nAtF]TI?]T

H(x) =1

4

(

∣

∣

∫ m

0

|b(y)| dy∣

∣+1

2|b′(m)|t+

∣

∣

∫

[m;x]

|b(y)| dy∣

∣

)2

=1

4

(

V m0 (Φ) +

1

2|b′(m)|t+ V x

m(Φ))2

V?(Fv(x) = b2(x)/2

g$DPBNVX<Xy ∈ 9-«]BEDPu X<TUuyuV:9<; Tv³j?]D, AEaIV,A+g$DPBNTºPTBCk¹HUDPuy\IVPH Ad^nTUA

K ⊆ TOa(VºPT

lim supε→0

ε logP(

∫ t

0


≤ − infa∈K

H(a)

V?(FÂg$D/BtTUºPTBCklDP\TU?¢^nTUAO ⊆ T+aIVºPT

lim infε→0

ε logP(

∫ t

0

v(Bs) ds ≤ ε,Bt ∈ O)

≥ − infa∈O

H(a).

ijTH DP?(F :XLTUAI(x) = 2

√

H(x) = V m0 (Φ) +

1

2|b′(m)|t+ V x

m(Φ)

g$DPBVPXLXx ∈ 9(`ba]TU?g$DPBtTºPTUBEkl^CT A A ⊆ T I?IF

−2

√

∣

∣− infx∈A

H(x)∣

∣ = −2√

infx∈A

H(x) = − infx∈A

I(x)

V?(FHUDPBEDPX<X<VPBCkyrI9 yVX<XLD,^GI^tACDH D/?IH X<GIF:T

lim supϑ→∞

1

ϑlogE

(

exp(−ϑ2

∫ t

0

v(ωs) ds)1K(Bt))

≤ − infx∈K

I(x)

g$DPBtTºPTBCkÂHUDPuy\IVPH A^nTUAK ⊆ VP?IF

lim infϑ→∞

1

ϑlogE

(

exp(−ϑ2

∫ t

0

v(ωs) ds)1O(Bt))

≥ − infx∈O

I(x)

g$DPBtTºPTBCkD/\(T?¢^CT AO ⊆ 9«KL?(VX<XLkÂTNHUVP?G(^nT+X<TUuyuVÂ:9<;NACDÂH D/?IH X<GIF:TOAEaIV,AtAEa]T+gVuyK<XLk

(Xϑt )ϑ>0

^CVACKw^ (TÂCaITNTV³ÂcemYtKLACa¢BEVACTOg$G]?IH ACK<DP?

Jt(x) = Φ(x) − Φ(0) − 1

2· t · Φ′′(m) + I(x)

= Φ(x) − Φ(0) + V m0 (Φ) + t(Φ′′(m))− + V x

m(Φ).

`ba]Kw^H D/uy\]XLTUACTÂCa]TN\]BEDjDgx9 0TF'

j;

ROK<ºPTU?¢AEa]T¼^CK<P?¹DPgb′(m)

ACa]T¼BEVACTNg$G]?(HJACK<DP?Mg$BCD/u¶AEa]T¼ACa]TDPBETUuHUV?oTy^nK<uy\]XLK ITFo(THUVPGI^nTdACa]TF:BEKgA

ba(VP^tD/?]XLkDP?IT UTUBEDI9]`baITNg$DPX<XLD,tK<?]H D/BCD/XLXwVBEkÂF:T^CHUBCK<o(T^bACaITÀHUV/^nTNDPg

b′(m) < 0]ta]KwHWa¢HUDPBEBCT ^C\(D/?IF]^bAEDÂV,AnAEBEV/HJAEKL?]ÂF]BCKLgA9(=@?AEa]Kw^fHUV/^nTOAEa]TdTV³cemYg$BCD/u AEa]TNACaITUDPBETUu HUV?¢oTÀ^nACBETU?]PACa]T?IFACD

ACaIT+g$G]XLXcemY9»¼I, 5 ° ¤)&2 ,&[*,+"-

Φ: → 3 + . C3 =8a/ 4"-76K5 / FG6I-%? 3 5 8a/21 + 1 Φ′′ .0/21 b = −Φ′ Q 99U8 N+T-I?S+<C`+6 9P./ m ∈ FG6I-I?b(x) = 0

6 =P.0/21 5 /SL]> 6 = x = m : b′(m) < 0 : .0/21lim inf |x|→∞ |b(x)| > 0 Q 8 CU-I?S+<CUN5CE+ L +<- Xϑ 3 +Z-%?@+ 9 5 L]8 -76K5 / 5 =

dXϑt = ϑb(Xt) dt+ dBt,

Xϑ0 = z ∈ . ]9 8PzM

R ?@+ / -I?S+ = 5 LIL 5FG6 / A4 L . 6IN 9 ?S5 L 1Mc. 50C +<f+<C > t > 0-%?@+ =<. N\6 L > (Xϑ

t )ϑ>0= 50C ϑ → ∞ 9". -J6 9 + 9 -%?@+ F + . *,X 5 / FG6I-%?C . - +

=8a/ 4"-J6K5 /Jt(x) = 2

(

Φ(x) − Φ(m)) = 50C .LIL x ∈ iQ ]9 _/~M

3 = b 6 9 N5 / 50- 5 / + : -%?@+ / -%?@+ =<. N6 L]> (Xϑt )ϑ>0

9". -76 9 + 9 -%?@+ =8aLIL *,X[FG6I-%?C . -#+ =8a/ 4"-J6K5 / Jt Q)+,-¨ & VM bijK<?IHUTNTNVPÊ^CG]uyTmAEaIV,A

mKw^bAEa]TNDP?]X<k TUBEDvDPgAEa]TdF:BEKgA

b]g$DPB

b′(m) < 0AEa]T+\(D/KL?0A

mKw^bACa]TÀuvK<?]K<uÀG]uÁDgΦ9]=@?¢AEa]K<^fHUVP^CT+TNa(VºPT

V mz (Φ) = Φ(z) − Φ(m)

V x

m(Φ) = Φ(x) − Φ(m)V?(F

Φ′′(m) > 0]^CDvACa]TNBWV,AETfg$GI?IHJAEKLDP?¹^CK<uv\IXLK ITÂEDyACa]TNT:\]BCT^C^CK<DP?l/KLº/TU?KL?g$DPBEuÀGIX<V ]9 _/~M W9o' b`D^xAEBCT?]AEa]TU?ACaITNTV³ÂcemYACDyAEa]T+g$G]X<XcemY TOa(VºPTOACDÂHWa]THW³ACaITNT:\(D/?]TU?0AEK<VPXACK<Pa0AC?]T^C^

H D/?IF:KLACK<DP?g$BCD/u X<TUuyuVÂ8:9<;:K©9 T/9:TNaIVº/TfAED^na]D, ACaIVAtg$DPBtTºPTUBEkc > 0

ACaITUBETOKw^VP?a > 0

tKAEa

lim supϑ→∞

1

ϑlogP

(

|Xϑt −m| > a

)

< −c. :9 _]; TNGI^nTNVHUDPuy\IVBEKw^nD/?lVBEPG]uyT?/AbAEDvD/o:AEVPKL?ACaIK<^tT^xAEKLuVACTP9

¡f^CK<?]yACa]T¼VPÊ^nG]uy\:AEKLD/?lim inf |x|→∞ |b(x)| > 0

VP?IFACaITUDPBETUu ;P9 rT (?IFACaIVAACaIT¼i:ems ]9 8PzM aIV/^V^xAWV,AEKLD/?IVBEklF:Kw^xAEBCK<o]G:AEKLD/?tKAEa¹F:TU?(^nKLAxkexp(

−2ϑΦ(x)) 9(c-T A

Xϑ oT¼V^CDPX<G:ACK<DP?¢DPg :9Q8zd tKAEa^nAEVBCAfK<?zV?IF

Y ϑ o(TvVl^nAEV,AEKLD/?IVBEk^CDPX<G:AEKLD/? (oDAEa¹tKLACa¹BET^C\(THJAmACDÂAEa]T¼ÊVuyTvBED,t?]KwV?¢uyDAEKLD/?-9`ba]T?T+PTUAAEa]TdF:TUACTUBEuyKL?IK<^nACKwHNF:K TBCT?/AEK<VPXZT0GIVACK<DP?d

dt(Xϑ

t − Y ϑt ) = ϑ

(

b(Xϑt ) − b(Y ϑ

t ))

g$DPBbAEa]TÀF:K TUBETU?IHUTOoT AxTUT?ACa]Td\IBCD:H T^C^CT^9:«KLBW^xAfV/^C^CG]uyTXϑ

0 − Y ϑ0 ≥ 0

9ITHUVG(^nTOg$DPBXϑ

t − Y ϑt = 0ACaITNBCK<Pa0AtaIV?(F^CK<F]TNºVP?]Kw^na]T^jAEa]TN\]BED:H TÊ^

Xϑt − Y ϑ

t

HUV?¢?]TºPTBbHWa(V?]/TOKLAE^t^CK<P?¢V?IF¢^nAEVk:^\(D0^nKLACK<ºPT/9ijK<?IH T

bKw^tF:THUBCTVP^CKL?]vTNaIVºPT

b(Xϑt ) − b(Y ϑ

t ) ≤ 0V?(FlTdHUVP?HUDP?IHUXLGIF]T

0 ≤ Xϑt − Y ϑ

t ≤ Xϑ0 − Y ϑ

0 .

«]D/BbACa]TdHVP^CTXϑ

0 − Y ϑ0 ≤ 0

TdHV?KL?0ACTBEHWa(V?]/TfAEa]TNBCD/XLT^DgXV?(F

YAEDDPo:AWVK<?lAEa]TNT^nACK<uV,ACT

0 ≤ Y ϑt −Xϑ

t ≤ Y ϑ0 −Xϑ

0 .

·DPu¼o]K<?]KL?I¼AEa]T^CTOAxDHVP^CT^bPK<ºPT^|Y ϑ

t −Xϑt | ≤ |Y ϑ

0 −Xϑ0 | = |Y ϑ

0 − z|.¡f^CK<?]

|Xϑt −m| ≤ |Xϑ

t − Y ϑt | + |Y ϑ

t −m|≤ |z − Y ϑ

0 | + |Y ϑt −m|

≤ |z −m| + |Y ϑ0 −m| + |Y ϑ

t −m|

8

xmϕ

b

lim infx→−∞

b(x) > 0

lim supx→∞

b(x) < 0

«KL/G]BET¼:9 r R ?6 9 Â 8 CE+h6 LIL]89 -7C . -#+ 9 -%?@+ V 50- + / -J6 .L^89 +5 =\. 45N VS. CU6 9 5 / -I?S+5CE+"N = 5C 9 5 L 8 -76K5 /@9 5 = -%?@+WXZY ]9 8PzM Q R ?S+h-%?6K4 PL 6 / +h6 9 -I?S+\50CU6 A06 /H.L1 CU6 = - b Q R ?S+\-I?a6 /L 6 / + 6 9 -%?@+ / +<F 1 CU6 = - ϕ Q R ?@+ 9 5 L]8 -J6K5 /= 5C 1 CU6 = - b 9 ?S5 8aL 1P3 + 4 L 5 9 +"CB- 5 m -%? .0/ -I?S+ 9 5 L]8 -J6K5 / = 50C 1 CU6 = - ϕ QT+HV?¢H D/?IH X<GIF:T

P(

|Xϑt −m| > a

)

≤ P(

|Y ϑ0 −m| + |Y ϑ

t −m| > a− |z −m|)

≤ P(

|Y ϑ0 −m| > a− |z −m|

2

)

+ P(

|Y ϑt −m| > a− |z −m|

2

)

= 2P(

|Y ϑ0 −m| > a− |z −m|

2

)

.

D,XLTUA c > 09]`ba]T?¢GI^nK<?]vACaITUDPBETUuÁ8:9<;_¼TdHV? I?IF¢V?

a > 0tKLACa

limϑ→∞

1

ϑlogP

(

|Y ϑ0 −m| > a− |z −m|

2

)

≤ −cV?(FlGI^CK<?]yACa]TdVPo(D,º/TmT^xAEKLuVACT+T+/T A

limϑ→∞

1

ϑlogP

(

|Xϑt −m| > a

)

≤ −c.ijK<?IH T+AEa]Kw^Kw^bACa]TÀT:\(D/?]TU?0AEK<VPXZACK<Pa0AC?ITÊ^tH DP?(F:KAEKLD/? :9 _]; TdHV?¢GI^CTNXLTuyuyVÂ8]9L;OAEDlF:TUBEK<ºPTOACa]Tdg$G]XLXcemY V?IFAEDyHUDPuy\]X<T AET+ACa]TN\]BEDjDgx9 0TF'

l.j 5 -& & ;_ DAETACa(V,A¼K<?²AEa]K<^¼HUV/^nTAEa]TlBWV,AETg$G]?IH ACK<DP?Kw^dK<?IF:T\(T?IF:T?/AvDgtAEa]TlK<?0ACTUBEº,VXX<TU?]PACat9]THUVPGI^CTOT+aIVºPT

lim inf |x|→∞ |b(x)| > 0ACaITN\(DPACTU?0AEK<VPX

ΦHUDP?jºPTBC/T^%ACD

+∞ g$DPB |x| → ∞V?(FJtKw^fVvPDjD:FBEVACTOg$G]?IH ACK<DP?-9(=@?gV/HJAtAEa]TdBWV,ACTOg$G]?(HJACK<DP?¹HUDPK<?IH KwF:T^ttKAEaACa]TdBWV,AETOg$G]?IHJAEKLD/?DgACa]T

cemYg$D/BAEa]T¼^nAEVACK<DP?IVPBCkF:K<^nACBEK<o]G:ACK<DP?¹g$BCD/u¶AEa]TUD/BCTu8]9L;_]9(`baIK<^muV³PT^^CTU?I^CT(o(THUVG(^nTÀg$DPBm^nACBEDP?IF:BEKgAT+D/G]XwFlTj\TH AtACa]TN\IBCD:H T^CÂEDBCTVPHWaÂAEa]TNT0G]K<X<KLo]BEK<G]u¶º/TUBEkÂ/GIK<HW³jX<kP9

8D b¡m^nK<?]AEa]TÀV/^C^CG]uy\:ACK<DP?(^tDP?bTdHUVP? I?IFVuyDP?IDACD/?]KwHUVX<X<klF:THUBCTVP^CKL?I]F:K TBCT?0ACKwVo]X<TNg$G]?IH ACK<DP?

ϕ : → taIK<HWaÊV,ACKw^ IT^|b(x)| ≥ |ϕ(x)| g$DPBVPXLX

x ∈ V?(FlaIV/^

ϕ′(m) < 09]`ba]Kw^tKw^bKLX<X<GI^xAEBEVACTFK<? (PG]BETd:9 rI9

THVGI^CTvACa]TÂF:BEKLgAb\]GI^Ca]T^+AEa]TÂ\]BED:H T^C^N^nACBEDP?]/TUB+AED,VPBEF]^

mAEaIV?*AEa]TÂF:BEKgA

ϕF:DjT^ -D/?]T

H D/G]XwF¢PG]T^C^bAEaIV,Amta]TU?Y ϑ Kw^fVÂ^CDPX<G:ACK<DP?¢DPgAEa]T¼i:emstKLACa¹F:BEKLgA ϑϕ K<?I^nACTVPFDg ϑb TNDPG]XwF¢aIVº/T

P(

|Xϑt −m| > a

)

≤ P(

|Y ϑt −m| > a

) 9`ba]K<^fD/G]XwF^na]D,ACa(V,Afg$D/B I?(F:KL?IlV?G]\I\(TBmoDPG]?(F¢DP?P(

|Xϑt −m| > a

) TNH DPGIX<FtKLACa]D/G:AXLD0^C^bDPg/TU?]TBEVPXLKLAxklVPÊ^CG]uyTbACDoTNuyDP?]DPACD/?]K<HVX<XLkÂF]TH BETV/^nK<?](9

`ba]T?M\IVPBnA+o' fDgHUDPBEDPX<X<VPBCk:9Q8~DPGIX<F£V\]\]X<k¹V?(F¹TÀDPG]XwF¹/T AOACa]T¼g$G]X<X[cemYK<? ]9 8P~ÂTUº/TU?g$D/B?]D/? uyD/?]DAEDP?]T+F:BEKgAtg$G]?(HJACK<DP?(^ b 9

,_

½m¾ 5 ²±. ¤)&' '& «IDPBmACa]TlµmBE?I^nACTUK<? ¡a]X<TU?joTHW³\]BEDjHUTÊ^fTÀa(VºPT Φ(x) = αx2/29`bajGI^ (GI^CK<?]

H D/BCD/XLXwVBEk ]9 8P~M TO/T AbACaITNBEVACTOg$G]?IH ACK<DP?Jt(x) = αx2

ta]KwHWaHUDPK<?IH KwF:T^btKAEaAEa]TN\]BETUºjK<DPGI^bBET^CG]XLAg$BEDPu g$DPBEuÀG]XwV _]9 rd J9`ba]TNHUV/^nT+DPgBETU\TUX<XLK<?]F:BEKgA IK©9 T/9:Dg

b′(m) > 0K<^tF:T^CHUBCK<oTFKL?ACaIT+g$DPX<XLD,tK<?]yH DPBEDPX<XwVBEkP9

»¼I, 5 ° ¤)&2Z 2&[*,+"-Φ: → 3 + . C3 =8a/ 4"-76K5 / FG6I-%? 3 5 8a/21 + 1 Φ′′ .0/21 b = −Φ′ Q 99U8 N+T-I?S+<C`+6 9P./ m ∈ FG6I-I?

b(x) = 06 =P.0/21 5 /SL]> 6 = x = m : b′(m) > 0 : .0/21

lim inf |x|→∞ |b(x)| > 0 Q R ?@+ / = 5C +"f+"C > t > 0-%?@+ 9 5 L]8 -76K5 / Xϑ 5 =

dXϑs = ϑb(Xs) ds+ dBs

= 5C s ∈ [0; t] :G.0/21Xϑ

0 = z ∈ 9". -J6 9 + 9 -%?@+ZF + . * X 5 / FG6I-%?j45 /S9 - ./ -GC . -#+ =8a/ 4<-J6K5 /

Jt(x) = 2(

Φ(m) − Φ(z))

− tΦ′′(m). :9 _/8M )+,-¨ & =@?¢AEa]T¼HVP^CT

b′(m) > 0AEa]Td\DPK<?/A

mK<ÂCa]T¼uV :KLu¼G]u¶DPg

ΦVP?IF¢oTHUVPGI^CTdDg

V mz (Φ) =

Φ(m) − Φ(z)V x

m(Φ) = Φ(m) − Φ(x)V?IF

Φ′′(m) < 0T+PTUA

Jt(x) =(

Φ(m) − Φ(z))

− Φ(z) − tΦ′′(m) +(

Φ(m) − Φ(x))

+ Φ(x)

= 2(

Φ(m) − Φ(z))

− tΦ′′(m)

g$DPBVPXLXx ∈ 9 0TF'

l.j 5 -& & `ba]TyH D/BCD/XLXwVBEk^Ca]D,^fACaIVAOK<?£ACa]TyHUV/^nT¼Dg%BETU\TUX<XLK<?]¢F:BEKgAOACa]TvBWV,ACT¼g$G]?IH ACK<DP? F:DjT^?]DPAmF:TU\TU?(F¢DP?

x9=@?\IVBCACKwH GIX<VPBtKAfKw^?IDAfV/D0D:FBWV,ACT+g$GI?IHJAEKLD/?-9(fX<^CDÂK<?AEa]Kw^fHUV/^nTNKLAfKw^K<uy\(D0^C^CKLoIXLT

ACD^nACBETU?]PACa]T?ACa]T+TV³cemY AEDyACa]T+g$GIXLXcefYoTHVGI^CTNT+aIVº/T

limϑ→∞

1

ϑlogP (Xϑ

t ∈ ) = 0 6= 2(

Φ(m) − Φ(z))

− tΦ′′(m).

½m¾ 5 ²±. ¤)&2)& «IDPBϑ > 0

H DP?(^nKwF:TUBAEa]Td^nD/XLG]ACK<DP?DgACa]Tdi:emsdXϑ

t = ϑXϑt dt+ dBt,

Xϑ0 = z ∈ . ]9 _/_M

`ba]Kw^mACK<uvTvAEa]TyT0G]K<XLK<o]BEKLG]u \(D/KL?0Ad~lKwÔG]?I^nAEVPo]XLT-VPÔ^nDjD/?£VP^mAEa]Ty\]BCD:HUTÊ^fX<TVº/TÔ~ÂAEa]TF:BCKLgANtKLX<XF:BEKLº/T+KAg$GIBnAEa]TUBfVP?IFg$G]BCACa]TBfVVk/9 TUBETNTNaIVºPT

Φ(x) = −x2/2VP?IFGI^nK<?]ÂHUDPBEDPX<X<VPBCkÂ:9Q8:;+TdHUVP?

F:TUACTUBEuyKL?IT+ACa]TNBWV,AETfg$G]?(HJACK<DP?g$D/BbACa]TNXwVBEPTNF:Tº0KwV,AEKLD/?oTUa(Vº0K<DPGIBDPgXϑ

t

VP^Jt(x) = t+ z2. ]9 _Prd

f^fKL?¹ACa]TyHUV/^nTdDPgACa]TÂµmBC?(^xAETUK<? ¡fa]X<TU?jo(THW³l\]BED:H T^C^ ^CTUTÀ\(VPTv8,rd tTÀHV?MVP?IF¢tK<XLXT:\]X<K<HUKAEXLkF:TUACTUBEuyKL?IT¼AEa]TF:Kw^xAEBCK<o]G:AEKLD/?£DPgXϑ

t

VP?IFMº/TUBEKg$k¢AEa]T^nD/uyTUtaIVA+^CG]BE\]BCKw^CKL?]BET^CG]XAd:9 _ruV?jGIVPXLX<kP9ijK<?IH T+AEa]TdF:TBCK<º,V,ACK<DP?Dgg$DPBEuÀG]XwV _I9 8M F]K<F?]DPAtF]TU\TU?IF¢D/?lAEa]Td^nK<P?DgACa]TdF]BCKLgATN/T A

Xt = ϑtz +

∫ t

0 ϑ(t−s) dBs. ]9 _0D

THVGI^CTfAEa]TÀi:ems ]9 _/_M K<^bX<K<?]TVPB :T+³0?ID,ACa(V,A Xϑt

a(VP^tVR+VGIÊ^CK<VP?F:Kw^nACBEKLo]G]ACK<DP?¢V?IFg$BEDPu :9 _/D T (?IFACa]TNTj\TH AEVACK<DP?µ = E(Xϑ

t ) = ϑtz

r

V?(FÂAEa]TNº,VBEK<VP?IH T

σ2 = E(

(Xϑt − µ)2

)

= E(

(

∫ t

0 ϑ(t−s) dBs

)2)

=

∫ t

0 2ϑ(t−s) ds =

1

2ϑ( 2ϑt − 1).

`bajGI^g$DPBtTºPTBCkÂuyTVP^CG]BEVPo]X<TO^CT AA ⊆ T+aIVºPT

P (Xϑt ∈ A) =

1√2πσ2

∫

A

exp(

− (x− µ)2

2σ2

)

dx

=

√

ϑ

π

(

2ϑt − 1)−1/2

∫

A

exp(

−ϑ ( −ϑtx− z)2

1 − −2ϑt

)

dx.

D,TNHV?HUVPX<HUG]X<VACTOACaITNT:\(D/?]TU?0AEK<VPXZBEVACT^bDPgAEa]K<^fT:\]BETÊ^nK<DP?lg$DPB ϑ → ∞ 9I«]D/BbACa]TN?]D/BCuVPXLKw^ K<?]H DP?(^xAWV?0AtT I?IFlim

ϑ→∞

1

ϑlog

√

ϑ

π

(

2ϑt − 1)−1/2

= −1

22t = −t.

= gA ⊆ K<^toDPGI?IF:TF :ACa]T?

supx∈A

∣

∣

∣

( −ϑtx− z)2

1 − −2ϑt− z2

∣

∣

∣ −→ 0

g$DPBϑ→ ∞ V?(FÂAEa0G(^bg$DPBHUDPuy\IVPH A^nTUAE^

K ⊆ T I?IF

limϑ→∞

1

ϑlogP (Xϑ

t ∈ K)

= −t+ limϑ→∞

1

ϑlog

∫

K

exp(−z2) dx = −(t+ z2).

«]D/BNDP\TU?*^nTUAEÔ ⊆ THUV?*HWa]DjD/^CTyV?jko(D/G]?IF:TF*^nG]o(^nTUA

A ⊆ OtKAEa*?]DP? UTBCDc-To(T^n/G]TuyTV/^nG]BETjAEDyPT A

lim infϑ→∞

1

ϑlogP (Xϑ

t ∈ O)

≥ lim infϑ→∞

1

ϑlogP (Xϑ

t ∈ A)

= −t+ limϑ→∞

1

ϑlog

∫

A

exp(−z2) dx = −(t+ z2).

`ba]Kw^tBETU\]BED:F:GIH TÂCa]TNBET^CG]XLAg$BEDPu ACa]TdHUDPBEDPX<X<VPBCkvg$DPBbAEa]Kw^^n\THUK<VPX-HUV/^nT/9

b Á

µm?]TyV\]\IXLKwHUVACK<DP?MDPg%XwVBEPT¼F]TUºjK<VACK<DP?£BCT^nG]XLAE^mKwÔACD¢F:T AETUBEuvK<?]T¼ACa]TyTj\DP?ITU?0ACKwVX|F:THUVk¢BEVACTvDg[AEa]TbVkPT^tBCKw^C³lg$D/BACaIT¼^CTU\IVPBEVACK<DP?¢Dg|AxDÂ\]BED:H TÊ^CT^tta]KwHWaMVPBCTND/oI^nTBCº/TF¢D,º/TUBtX<DP?]ÂK<?0ACTBCº,VXw^DPg|ACK<uyTP9`ba]Kw^BWV,AETNK<^mVuyTVP^CG]BETNDg|a]D, TV/^nkKAmK<ÂCDlF:Kw^nACK<?]PG]Kw^Ca¹o(TUAxTTU?¢AxD\]BEDjHUTÊ^nT^bta]K<XLT¼DP?]X<klX<DjDP³ K<?]V,AtACaITN\IV,AEaIÛ9=@?¹AEa]T IBW^xAN^CTHJAEKLD/?MDPg%ACaIK<^NHWaIVP\:ACTB+TvtKLX<X|T:\]XwVK<?£ACa]TyPT?]TUBWVX\]BEDPo]X<TUu VP?IF£BCTF:GIH TvKLAOAED

?]D/? AEBCK<ºjK<VPXXwVBEPTÀF]TUºjK<VACK<DP?\]BEDPoIXLTu9=@?AEa]TÀg$D/XLX<D,tK<?]l^CTHJAEKLD/?¹TdtKLX<XuV³PTÀGI^nT¼Dg[XwVBEPTÀF]TUºjK<V ACK<DP?¢BET^CG]XLAE^g$BEDPu «c-Y%z/zV?(F Qi:HWaIzj"-ACDÂ^CDPX<ºPTmACaITN\]BCD/o]X<TUu K<?AxDy^n\THUK<VPXHUV/^nTÛ9

TvHUDP?I^CK<F]TUBOAxDuyTHWaIVP?]K<^CuÂED/TU?]TBEVACTvV^nACD:HWaIV/^xAEK<HÀ\]BCD:HUTÊ^TP9 I9(AxD¢F:K TBCT?0A+F:BEKLgA ITX<FI^ta]KwHWa¹VBET+KL?IF]T:TF¢ojkVÂ\IVBWVuyT AETUB

ϑ ∈ Θ = 0, 1 9]«IDPB ϑ ∈ ΘX<T A

P ϑt = L(X)

∣

∣

Ft

o(TÀACa]TdXwV DgACaITdH DPBEBET^C\(D/?IF:K<?]À\]BED:H T^C^

Xϑ DPoI^CTUBEºPTFÂG]\ACDvAEKLuyT t 9`ba]T Ä 5 °.0&À,&B(λ, t)

tKAEaV¼\IBCK<DPBEkÂF:Kw^xAEBCK<o]G:AEKLD/?λ ∈ Prob(Θ)

K<^F]TI?]TFlojk

B(λ, t) =

∫

minϑ∈Θ

dλϑPϑt

dPtdPt

ta]TBCTPt = λ1P

1t + λ0P

0t

9fÊ^nG]uyK<?]ÂAEaIV,AmACa]TvF:Kw^xAEBCK<o]G:AEKLD/? L(Xt)∣

∣

Ft

D/?ACa]T¼\IV,AEaM^C\IVPHUTdaIVPÔVF:T?I^nKLAxk

ϕϑt

tKAEaBET^C\THJAbAED K<TU?ITUBuyTV/^nGIBCT T I?IFB(λ, t) = λ1P

1t (λ1ϕ

1t < λ0ϕ

0t ) + λ0P

0t (λ1ϕ

1t ≥ λ0ϕ

0t ). qI9L;

`bajGI^B(λ, t)

HV?£o(TÂ^CTUT?£VP^mAEa]TvACDAWVX\]BEDPoIVPo]K<XLKLAxkDgTUBEBCD/Bg$DPBdVX<KL³/TUX<KLa]DjD:F¹BWV,AEKLDACT^xA+g$D/BmAEa]T\IVPBEVPuvTUACTB

ϑ(taITUBET

ϑK<ÔHWa]D0^nT?¹BEVP?IF:D/uvX<k¢VPHUHUDPBWF:K<?]yACDlACa]TyF:Kw^xAEBCK<o]G:AEKLD/?

λ9Z«K<PG]BET¼qI9L;d/KLº/T^mV

PTDPuyT AEBCKwHmKL?0ACTBC\IBCTUAEV,AEKLD/?DPgAEa]TdbVkPT^BEK<^C³ÂK<?ACa]Kw^t^nKLACG(V,ACK<DP?9 TObV?0AbACDÂHVXwH G]XwV,AETOACa]TNTj\DP?ITU?0ACKwVX-F:THUVkBWV,AET^

limt→∞

1

tlogB(λ, t) q]9Q8D

DgACaITÀbVkPT^BEK<^C³9I`ba]Kw^tBWV,ACTNKw^tuyTV/^nGIBCT+Dg|a]D, gVP^nAtACa]T¼Vk/T^bBEK<^C³lF:THUVk:^bta]T?¢ACa]TND/oI^nTBCº,VACK<DP?ACK<uyT

tAETU?IF]^fACDlKL? (?]KAxk/9Z`ba0G(^KLAmKw^mVuyTVP^CG]BCTÀDg|a]D, TV/^nkACa]TÀAxDl\]BCD:HUTÊ^nT^HUVP?o(Tv^nT\IVBWV,AETF

ojklX<DjDP³jKL?IvD/?]X<kV,AfACa]Td\(V,ACa(Û9I ºPTUBEkÂ?]T/VACK<ºPTNbVkPT^bBEKw^n³ÂK<?IF:KwHUVACTÎACaIVAtACa]Td\IBCD:H T^C^CT^tVPBCT+º/TUBEkF:K TBCT?0ANVP?IFAEajGI^+TV/^nk¢AEDF:Kw^nACK<?]PG]Kw^Ca -VbVk/T^mBCKw^n³¢AEaIV,ANKw^+H X<D/^CTÀAED TUBEDlK<?IF:KwHUVACT^+\IBCD:H T^C^CT^ta]KwHWaVPBCTN^CKLuyK<X<VPB9

=@?*H DPBEDPX<XwVBEk¢]9 qDgB zj [KLA+Kw^N^Ca]D,t?MAEaIV,ANKLgAEa]TyXLK<uyKAdK<? q]9Q8D mTjKw^nAE^ ACa]T? KLAdF:DjT^+?]DPAF:T\(T?IFDP?λ T]H T\:Ag$DPBfACa]T¼\IV,AEa]DPX<DP/K<HÀHUVP^CT^ λ0 = 0

D/Bλ1 = 0

J9ijDÂT¼HUVP?¹HWa]DjD/^CT+g$DPBmT]Vu \]X<Tλ = (1/2, 1/2)

AEDHUVXwH GIX<VACTOACa]TNBWV,AET:ta]KwHWa¢PK<ºPT^

B(t) := B(

(1/2, 1/2), t)

=1

2P 0

t (ϕ0t ≤ ϕ1

t ) +1

2P 1

t (ϕ0t > ϕ1

t )

,q

Ω

λ1ϕ1t

λ0ϕ0t

B(λ, t)

«KL/G]BET+qI9L; R ?6 9 Â 8 CE+Z6 LIL]89 -JC . -#+ 9 -%?@+ Ad+5N+<-JCU6K4e6 / - +"C V CE+"- . -J6K5 / 5 = -%?@+ .0> + 9 CU6 9 Q R ?@+ -7F 5i4 8 CUf+ 9. C`+-I?S+F +<6 A?-#+ 1 1 + /@9 6I-76K+ 9 5 = -%?@+ 1 6 9 -JCU6 3"8 -76K5 /@9 5 / -I?S+ V CE5 3`.M3 6 L 6I- >9JV@. 4`+ Ω Q R ?@+ .> + 9 CU6 9 6 9 -%?@+9 6 b_+ 5 = -%?@+ ? . -#4`?S+ 1i. CE+ . Q - 6 9 L . C7AM+e6 = -%?@+ 1 6 9 -7CU6 3"8 -76K5 /@9 . CE+ 9 6IN6 L . C QV?(FlGI^CK<?]XLTuyuyV8]9 8¼TO/T A

limt→∞

1

tlogB(λ, t) = max

(

limt→∞

1

tlogP 0

t (ϕ0t ≤ ϕ1

t ),

limt→∞

1

tlogP 1

t (ϕ0t > ϕ1

t )) qI9 _d

g$DPBVPXLXλ ∈ Prob(ϑ)

9TBCTNTdH D/?I^CK<F:TBtAxDBCTºPTUBW^CKLo]X<TNF:K GI^CK<DP?I^ttKLACa¹F]KTUBETU?0AmF:BEKLgA ITUXwF]^9(«IDPB

ϑ ∈ Θ = 0, 1 XLTUAΦϑ : d → oTNAxDvACK<uvT¼H DP?0AEKL?jG]D/GI^nX<klF:K TUBETU?0ACKwVo]X<T

bϑ = − gradΦϑ (V?IF Xϑ o(TdV^CDPX<G:ACK<DP?¢DPgACaITÀi:efsdXϑ = bϑ(Xϑ) dt+ dB

Xϑ0 = 0.

qI9 r

«]BEDPu¶XLTuvuV;P9Q¼T+³j?]D, ACa]TNTj\IXLKwH KLAbg$DPBEu¶DgACa]TdF]TU?I^CKAEKLT^ϕϑ

t

9I`ba]TUkHUV?¢oTNF:TI?ITFojk

ϕϑt (ω) = exp

(

−Φϑ(ωt) + Φϑ(0) −∫ t

0

vϑ(ωs) ds)

g$DPBmVX<Xω ∈ C

(

[0;∞), d) ta]TBCT

vϑ =(

(∇Φϑ)2 − ∆Φϑ)

/29]THUVPGI^CT+ACa]TNTj\DP?ITU?0ACKwVXZg$GI?IHJAEKLD/?K<^

uyDP?]DPACD/?]K<HVX<XLkK<?IHUBCTVP^CKL?] 0ACa]TNTºPTU?0Aϕ0

t (X0) ≤ ϕ1

t (X0)K<?¢T0GIV,AEKLD/? q]9 _M HUVP?oTNT:\]BETÊ^CTFlV/^

−Φ0(X0t ) + Φ0(X0

0 ) −∫ t

0

v0(X0s ) ds ≤ −Φ1(X0

t ) + Φ1(X00 ) −

∫ t

0

v1(X0s ) ds

DPBtT0G]KLº,VPXLT?/AEXLkVP^1

t

∫ t

0

(

v1 − v0)

(X0s ) ds+

1

t

(

Φ1 − Φ0)

(X0t ) − 1

t

(

Φ1 − Φ0)

(X00 ) ≤ 0. q]9QVd

`ba]TND/\]\(D0^nKLACT+TºPTU?0Aϕ0

t (X1) > ϕ1

t (X1)oTH D/uyT^

1

t

∫ t

0

(

v1 − v0)

(X1s ) ds+

1

t

(

Φ1 − Φ0)

(X1t ) − 1

t

(

Φ1 − Φ0)

(X10 ) > 0. q]9Q,o2

=@?¹D/BEF:TBAEDHVXwH G]XwV,AETNACa]TvBWV,ACTÀg$DPBmACa]TybVkPT^tBCKw^C³TÀaIVº/TNACD¢H D/?I^nKwF:TBfXwVBEPTÀF:TºjK<VACK<DP?I^fg$DPBACaITTUºPT?0AE^ qI9 PVM OV?(F q]9Q,o' 9-=@?*PT?]TUBWVXACaIK<^NKw^dV¢F:K !ÂH G]XLAÀ\]BEDPo]X<TUu¢9-`ba]Tyg$D/XLX<D,tK<?]¢T]Vuy\]X<TK<XLX<GI^nACBWV,ACTÂCa]TN\IBCD:H TF:G]BETfg$D/BVyºPTBCkÂ^CKLuy\]X<TdHUV/^nT/9

½m¾ 5 ²±. ®,& ·D/?I^nAEV?0AÀF:BEKLgA J9fÊ^nG]uyTyAEaIV,AdTaIVºPT :TF*ºPTH ACD/BE^b0, b1 ∈ d tKLACa

bϑ(x) = bϑg$DPBNVX<X

x ∈ d 9Z«]BEDPu T]Vuy\]X<T;/9L;¼Tv³j?]D,ÁCa(V,A+TvaIVºPTΦϑ(x) = −bϑ · x VP?IF

vϑ(x) = |bϑ|2/2 a]TBCT/9(`bajGI^bAEa]T¼F:T?I^CKAEKLT^ ϕϑ DP?]X<kF]TU\TU?IF¢D/?¢ACa]TdT?IF:\DPK<?0A Xϑt

DgACa]TÀ\IV,AEa¹V?IFTO/T A

ϕ0t (X

0) ≤ ϕ1t (X

0) ⇐⇒ b1 + b0

2· (b1 − b0) − 1

tX0

t · (b1 − b0) ≤ 0.

P

THVGI^CTfAEa]TdF:BEKgAKw^HUDP?I^nAEVP?/AbϑACa]TNº,VX<G]T

X0t

Kw^ N (t ·b0, t) F:Kw^xAEBCK<o]G:AETF ]ACajGI^ X0t /t

Kw^ N (b0, 1/t) F:Kw^xAEBCK<o]G:AETF²VP?IF£ACa]TlVPo(D,º/TH DP?(F:KAEKLD/?*uvTV?I^NACaIVAX0

t /tKw^dH D/?/AWVK<?]TF K<? ACa]TÂa(VXLg \]XwV?ITtKAEa?]D/BCuVXZº/TH ACDPB

b1 − b0ta]KwHWa¢F:DjT^b?]DPAfH DP?0AWVK<?lAEa]TNºPTHJACD/B

b09(·D/BCD/XLXwVBEkÂ8:9<;8¼PK<ºPT^

limt→∞

1

tlogP

(

ϕ0t (X

0) ≤ ϕ1t (X

0))

= −1

2

∣

∣

∣

b1 + b0

2

∣

∣

∣

2

= −|b1 − b0|28V?(FVyº/TUBEkÂ^nK<uyKLXwVBHVXwH G]XwV,AEKLD/?VXw^CDy^Ca]D,^

limt→∞

1

tlogP

(

ϕ0t (X

1) > ϕ1t (X

1))

= −|b1 − b0|28

.

`bajGI^boDACa¢BWV,AET^g$BEDPu AEa]TNBCK<Pa0AtaIVP?IF^CKwF:TNDg qI9 _d H DPK<?IHUK<F:TNVP?IFT+PTUAbACa]TNBET^CG]XLA

limt→∞

1

tlogB(λ, t) = −|b1 − b0|2

8.

emT AEVPKLXw^VPo(D/G:AbACa]Kw^HUVP?oT+g$DPGI?IFKL? |D /zj"©9

! " &! "# =@?¢ACaIK<Ô^nTHJAEKLD/?TÀF:TUACTBCuyK<?]TNACaITdT:\DP?]T?/AEK<VPXBWV,AET+g$DPBAEa]TÀF:THUVkDgACa]TvbVkPT^bBEKw^n³ta]TU?MF]K<^nACK<? PGIK<^Ca]K<?]vAxDlµmBE?I^xAETUK<? ¡a]X<TU?joTHW³\]BED:H TÊ^CT^tKAEaF:KTUBETU?0A\IVPBEVPuyT ACTBE^ α0

V?IFα19

f^mTdaIVºPT¼^nTTU?¹K<?MHWaIVP\:ACTBm_(ACaIT¼F:T?I^nKLAxk¢Dg%Vd F:K<uvT?I^nK<DP?(VXµmBE?I^nACTKL? ¡aIXLT?]o(THW³\IBCD:H T^C^tKLACa¢\IVBWVuyTUACTUB

αϑD/?lAEa]TN\IV,AEa^n\IV/H T+Kw^

ϕϑt (ω) = exp

(

−αϑω2

t

2− 1

2

∫ t

0

α2ϑω

2s − αϑ ds

)

.

`ba]T+BETU\]BET^CTU?0AEVACK<DP?DgAEa]TNTUº/TU?0Aϕ0

t (X0) ≤ ϕ1

t (X0)g$BEDPu q]9QVM oTHUDPuyT^

α21 − α2

0

2

1

t

∫ t

0

(X0s )2 ds− α1 − α0

2d+ (α1 − α0)

(X0t )2

2t≤ 0

V?(FV/^C^CG]uyKL?Iα0 > α1 > 0

TNHUVP?¢F]KLºjKwF:T+o0kα1 − α0 < 0

AEDv/T AbACaITdH DP?(F:KAEKLD/?α1 + α0

2

1

t

∫ t

0

(X0s )2 ds− d

2+

(X0t )2

2t≥ 0. q]9 qPVd

`ba]TF:Kw^xAEBCK<o]G:AEKLD/?MDPg%ACaITv\IBCD:H T^C^XH DP?jº/TUBEPTÂCD¢V

d F]KLuyTU?(^nK<DP?IVPX[R+VGIÊ^nKwV? F:Kw^xAEBCK<o]G:AEKLD/?£tKLACaT:\TH AEV,AEKLD/?~yVP?IF¢H D,º,VBEK<VP?IH TmuV,ACBEK 12αId

9]fXLuyD/^nAf^CG]BCTXLklT+aIVº/T

1

t

∫ t

0

(Xs)2 ds→ d

2α

V?(F (Xt)2

t→ 0

g$DPBt→ ∞ I^nDvAEa]TNXLTUgAaIV?(F^CKwF:T+Dg qI9 q/VM V]9 Û9]HUDP?jºPTBC/T^%ACD

α1 + α0

2

d

2α0− d

2+ 0 <

2α0

2

d

2α0− d

2= 0

V?(FlTdHUVP?^CTUTOACa(V,AtACaITN\]BCD/oIVoIKLX<KAxkDPgAEa]TNTUº/TU?0A qI9 q/VM V,AX<TVP^nAH D/?0º/TUBEPT^[AEDy~I9ijK<uyKLXwVBEXLkT I?IFAEaIV,Atg$D/Bα0 > α1 > 0

ACa]TNTºPTU?0Aϕ0

t (X1) > ϕ1

t (X1)Kw^tT0G]K<º,VX<TU?0AbACD

α1 + α0

2

1

t

∫ t

0

(X1s )2 ds− d

2+

(X1t )2

2t< 0. q]9 qo2

`ba]TOg$DPX<X<D,tKL?]vAEa]TUD/BCTu¶^xAWV,AETÂCaITNuyVPKL?BET^CG]XAtDPgAEa]Td^nTHJAEKLD/?-9

,

−12

0 α0−α1

4α1

y

+∞Ic(y)

«KL/G]BETq]9Q8 R ?6 9 Â 8 CE+ 9 +<-#4?@+ 9 -I?S+C . -#+ =8a/ 4"-76K5 / Ic = 5C Yt/t= CE50N = 5CUN 8aL . qI9QD QhR ?S+ V CE5_4+ 9945 / f+"C7Ad+ 9 - 5 −(c · 1/2α+ 1/2) = (α0 − α1)/4α1 Q +BFG6 LIL 45 /S9 6 1 +"CB-%?@+h+<f+ / - Yt/t < 0 Q

d.jI,.j ®,& ( R ?S++ V 5 / + / -J6 .L,1 +`4 .0> C . -#+\5 = -I?S+ .> + 9 CU6 9 B(λ, t) = 50Ce-I?S+ 1 6 9 -J6 / 4"-76K5 /3 +<-JF ++ / -JF 55 / + 1 6INT+ /@9 6K5 /H.L C /@9 -#+<6 / ? L + /H3 +`4 V C`5 4+ 9`9 + 9 FG6I-%? VS. C . NT+"- +"C 9 α0, α1 > 06 9

limt→∞

1

tlogB(λ, t) = − (α1 − α0)

2

8(α1 + α0).

= 5C +"f+"C >. V CU6K5C >P1 6 9 -JCU6 3"8 -76K5 / λ ∈ Prob(

0, 1) FG6I-%?

λ0 6= 0 ./H1 λ1 6= 0 Q)+,-¨ & TyHUVXwH GIX<VACTdAEa]TdAxDlBWV,AET^

R1V?IF

R0KL?¹ACa]TvuV :KLu¼G]uÁg$BCD/uÁg$DPBEuÀG]XwV q]9 _M ^CTU\(V BWV,ACTXLk/9 KLACa]D/G:AmX<D/Ê^DPg[PT?]TUBWVX<KAxkTÀuyVk¢VPÊ^CG]uyTα0 > α1 > 0

V?(F¢GI^nK<?]T0GIV,AEKLD/? qI9 qPo' bTPTUA

R1 := limt→∞

1

tlogP 1

t

(

ϕ0t > ϕ1

t

)

= limt→∞

1

tlogP 1

t

(α1 + α0

2

1

t

∫ t

0

X2s ds−

1

2+X2

t

2t< 0)

= limt→∞

1

tlogP 1

t

( X2t

2t− 1

t

∫ t

0

−α1 + α0

2X2

s +1

2ds < 0

)

`ba]T+XwVBEPT+F:TUºjKwV,ACK<DP?oTUaIVºjK<DPG]BbDgACaITNBEVP?IF:DPu¶ºVPBCKwVoIXLT^Yt(c)/t

tKAEa

Yt(c) =1

2X2

t −∫ t

0

cX2s +

1

2ds

ta]TBCTXK<^¼V?§µmBE?I^xAETUK<? ¡fa]X<TU?jo(THW³M\]BED:H T^C^+Kw^dT:VPuyKL?]TF o0k «XLD/BCT?I^ cVP?IF]VPK<ÔV?IF*Y%aIVu KL? «cY%zPz0©9 £TGI^CTACa]T IBE^nA¼HVP^CTÂDgbACaITUDPBETUup8:9Q8g$BEDPu ACa]TKLBvVBCACKwH X<TcT A

XoTlVP? µmBC?I^nACTKL? ¡aIXLT?0oTHW³\IBCD:H T^CÔtKLACa£\IVBWVuyT AETUB

α9-`baITU?£g$DPB+TUº/TUBEk

c ≤ −α/2 ACa]TvgVPuvK<X<k Yt(c)/tÊV,AEK<^ IT^

ACaITNX<VPBC/TOF:TºjK<VACK<DP?\]BCK<?IHUKL\]X<TNDP? tKAEaAEa]TNPDjD:FBEVACTOg$G]?IH ACK<DP?IcF]TI?]TFo0k

Ic(y) =

−α2

c

(y + c+α2α )2

2y + 1

KLgy > − 1

2

IVP?IF+∞ TX<^CTP9 qI9QD

`ba]TNBWV,AETfg$G]?(HJACK<DP?¢g$D/BbACa]Kw^tHUV/^nT+Kw^^C³PT AWHWa]TFlKL? I/G]BET+qI9 8]9ijK<?IH T+K<?¢DPG]B^CKAEGIV,AEKLD/?¢T+aIVº/T

α = α1V?IF

c = −(α1 + α0)/2ACaITNBEVACTOg$G]?IH ACK<DP?¢K<^tF]TH BETV/^nK<?]

ACDlACa]T¼XLTUgAODPg −(c + α)/2α = (α0 − α1)/4α1 > 0V?(FK<^mK<?IH BETV/^nK<?]yACDlACa]TvBEKL/a/AmDg|ACaIK<Ô\DPK<?/A9

THVGI^CT(−∞; 0)

K<^tVyHUDP?0ACK<?jG]KAxk^nTUADgIcTNPTUA

R1 = limt→∞

1

tlogP 1

t

(Yt(c)

t< 0)

= −Ic(0) = − (c+ α)2

4c= − (α0 − α1)

2

8(α0 + α1).

,z

D, TNaIVºPTOACDHUVXwH GIX<VACT+ACaITdDAEa]TUBBWV,AET+g$BCD/u g$D/BCu¼G]X<V q]9 _M 9(fÊ^CG]uyKL?] α0 > α1 > 0VP/VK<?

TO/T A

R0 = limt→∞

1

tlogP 0

t

(

ϕ0t ≤ ϕ1

t

)

= limt→∞

1

tlogP 0

t

(Yt(c)

t≥ 0)

,

o]G:AfACa]Kw^bACK<uyTNtKAEaα = α0

V?IFc = −(α1 + α0)/2

9I`baITNBEVACTOg$G]?IH ACK<DP?¢K<^fF:THUBCTVP^CKL?]vAED¼AEa]TdX<T gADPg−(c+ α)/2α = (α1 − α0)/4α0 < 0

V?IFKw^K<?IH BETV/^nK<?]¼ACDAEa]TNBEKL/a/ADPgAEa]Kw^\DPK<?/A9ijDTNPTUAfV0VK<?ACaITNBCT^nG]XLA

R0 = −Ic(0) = − (α0 − α1)2

8(α0 + α1).

`ba]Kw^\]BED,ºPTÂCaIVAfoDACa¹ACTUBEu^tK<?ACa]TÀuyV :KLu¼G]u D/?¢ACa]TÀBCK<Pa0AfaIVP?IF^CK<F:TÀDgg$DPBEuÀG]XwV qI9 _d V?IFACajGI^tACa]TNuV :K<uÀG]u¶KAW^nTXg IVPBCT+T/G(VXACDvAEa]TNBEVACTOg$BEDPu D/G]BH XwVK<u¢9 0TF'

! " &! # #! " & # "

=@?£ACa]Kw^+^nTHJACK<DP?£TyF:TUACTUBEuyKL?ITÀAEa]TyT:\(D/?]TU?0AEK<VPXF]THUVk¢BWV,AETÀg$D/BmAEa]TyVk/T^mBCKw^C³ta]TU? ^nT\IVBWV,AEKL?]AxDH D/?0ACK<?0GIDPGI^fACK<uyTy¸MVBE³PD,º¢HWaIVK<?I^9`ba]T^nTv\]BED:H TÊ^CTÔVBETÀ?]D¢F:K GI^CK<DP?M\IBCD:H T^C^CT^mK<?¹ACaITy^CTU?(^nTDg%HWa(V\:AETUBv;o]G:AOACa]TyH D/?IH T\:AmDPg|ACa]TybVkPT^fBCKw^C³DPgH D/G]BE^CTÀVPX<^CDÂuV³/T^f^CTU?(^nT¼a]TUBETP9£TvtK<XLX|^CTUTACa(V,AfAEa]TÀT:\(D/?]TU?0ACKwVXBEVACTdDPg|ACa]TÀbVk/T^tBEK<^C³K<^

r − 1ItaITUBET

rK<^fACa]T¼BEVACT+g$D/Bm^CTU\IVPBEVACK<DP?¢DgACaIT

TUu¼o(TF]F:TF¸MVBE³PD,ºHWaIVK<?I^9c-TUA

XoTÂVHUDP?0ACK<?jG]DPG(^fAEKLuyT¸¹VPBC³/D,º¢HWa(VK<?£tKLACa I?]KLACTÂ^nAEVACT^n\(VPH T

SV?IF£PT?]TUBWV,AEDPB

q ∈ S×S 9I«IDPBtAEa]TNACTHWa]?]KwHUVPXF:TUAEVK<Xw^VoDPG:AOH DP?0AEKL?jG]D/GIÂCK<uyTÀ¸¹VBE³PD,ºHWaIVK<?I^tV?(FACa]TKLB/TU?]TBEVACD/BE^bTBET g$TUBACD cVz0 ©9THUVPGI^CT

qKw^fVy/TU?]TBEVACD/BtTda(VºPT

qij ≥ 0g$D/B

i 6= jV?IF

qii = −∑j 6=i qij < 0g$DPBmVX<Xi ∈ S

9I`ba]Td\IBCD:H T^C^XHV?¢oTÀF:TÊH BEK<o(TF¢tKAEa¢ACa]TdaITUX<\Dg[V?¢TUu¼o(TF]F:TF¸¹VBE³PD,ºlHWa(VK<?

Y

ta]T?]TUº/TUBXBETV/HWa]T^mVÂ^nAEV,AET

i ∈ S(ACa]T¼\]BEDjHUTÊ^f^nAEVk:^tAEa]TUBETdg$D/BmVP?

Exp(−qii) F:Kw^nACBEKLo]G]ACTF¹ACK<uyTV?(F¹AEa]TU?xG]uy\I^NK<?0ACDVBEVP?IF:DPuyX<kMHWa]D0^nT?M?]T ^nAEV,AETP9-`ba]T?ITU ^nAEVACTyKw^j 6= i

tKLACa \]BCD/oIVoIKLX KLAxkqij/

∑

k 6=i qik9

TBCTÀT¼BCT^xAEBCKwHJAfD/G]BW^nTXLº/T^bAEDÂACaIT¼HUV/^nTqii = −1

g$D/BOVPXLXi ∈ S

K©9 T/9IAEDÂACaIT¼HUV/^nTÀDg[T0GIVPXVP?IFa]D/uvD/PT?]TUD/GI^xG]uy\BEVACT^9IcT A

T (t)oTNAEa]TN?jG]uÀoTUBDPg(xGIuv\(^G]\¢AEDyACK<uvT

tVP?IF

Yng$DPB

n ∈ g 0oT

ACaITd^xAWV,ACTNDPgXVgACTUBbAEa]T

nAEa¼xGIuv\9(`ba]T?

T (t)K<^YD/K<Ê^CDP?¢F:K<^nACBEK<o]G:ACTFtKAEa\IVBWVuyT AETUB

tV?IF

YKw^

Vy¸¹VBE³PD,ºHWaIVPKL?tKLACa¢ACBWV?(^nKLACK<DP?uyVACBEK

πij =

qij ,Kgi 6= j

]VP?IF0

TUXw^nT/9 D,´HUDP?I^CK<F]TUBtAxDÂK<BCBETF:G(H K<o]XLT¼¸¹VBE³PD,ºlHWa(VK<?I^ X0 V?IF X1 tKAEaMF:K TBCT?0AACBWV?I^CKLACK<DP?BWV,ACT^ q0V?(Fq19ZµmG]BAWVP^C³Kw^tAEDÂDPoI^CTUBEºPTND/?]Td\IVACa¹VP?IF¢ACDF:T AETUBEuvK<?]TdtaIK<HWaAEBEVP?I^CKAEKLD/?uyTHWaIV?]Kw^Cu /TU?]TB V,AETFlACa]Kw^t\IVACa-9

emK<^nACK<?]PGIK<^Ca]K<?]ÂAxD?IDP? T0G]K<º,VX<TU?0AO¸¹VPBC³/D,ºHWaIVPKL?(^fKwÔTVP^Ck V/^m^CDjDP? VPÔVACBWV?(^nKLACK<DP?£DjHH G]BW^ ta]KwHWaMKwÔDP?]X<k\(D0^C^CK<o]XLT¼g$DPB+D/?]TvHWa(VK<?Mo]G:AN?]DPAmg$D/BOACa]TyDAEa]TUB TvaIVºPT¼K<F]TU?0ACK ITFMACaIT¼AEBEVP?I^CKAEKLD/?uyTHWaIVP?]Kw^nuÁtKAEa¹\]BEDPoIVPo]K<XLKLAxklD/?]TP9ijDÂa]TBCTdTÀV/^C^CG]uyT+ACaIVAfAEa]Td\]BED:H TÊ^CT^VPBCTNT/GIKLº,VX<TU?0A (K©9 T/9ITH D/?I^CK<F:TBACa]TdHVP^CT

q0ij > 0 ⇔ q1ij > 09

0VK<?TNH D/?I^nKwF:TBAEa]TdbVkPT^BEKw^n³

B(t) =1

2P 0

t

(dP 1t

dP 0t

≥ 1)

+1

2P 1

t

(dP 1t

dP 0t

≤ 1)

, qI9 d ta]TBCT

P ϑt

KwÔACaITÂF:K<^nACBEK<o]G:ACK<DP?*DPgAEa]T\IV,AEaXϑ

s

∣

∣

0≤s≤t

9 £TbV?0A+ACDHUVPX<HUG]XwV,ACTvAEa]TT:\DP?]T?/AEK<VPXF:THUVklBWV,AET

limt→∞1t logB(t)

9¡m^nK<?]ÂX<TUuyuVl8:9Q8yTÀHUVP?BCTF:GIHUTNACa]Kw^\IBCD/o]XLTu¶AEDACaIT¼HUVPX<HUG]XwV,ACK<DP?DgACaITOBWV,AET^g$D/BbACa]TNK<?IF:K<ºjK<F:G(VXZAETUBEuy^bK<?ACa]Td^CG]u q]9 M IVP^bTNF:K<FK<? q]9 _M J9

P~

THVGI^CTNACaITÀ\]BED:H TÊ^CTâ(VºPTÀHUDPK<?IH KwF:K<?]dxG]uy\£BEVACT^ (ACa]T¼ta]DPX<TÀK<?:g$DPBEuV,AEKLD/?MVoDPG:AϑK<ÔH D/? AEVPKL?ITFKL?AEa]TNACBWV?(^nKLACK<DP?g$BCT0G]TU?IHUKLT^bo(TUAxTTU?ACaITdF:KTUBETU?0Af^nAEVACT^9I`bajGI^F]TI?]T+AEa]TNTUuy\]K<BEK<HVX-\IVPKLBuyTV/^nG]BET

mn ojk mnij = 1

n

∑nk=1 1(i,j)(Yk−1, Yk)

g$DPBVPXLXi, j ∈ S

9]`ba]TU?¢TOa(VºPT

dP 1t

dP 0t

=π1

Y0Y1π1

Y1Y2· · ·π1

YN(t)−1YN(t)

π0Y0Y1

π0Y1Y2

· · ·π0YN(t)−1YN(t)

=∏

i,j∈S

(π1ij

π0ij

)N(t)·mN(t)ij

,

ta]TBCT+TOG(^nT+AEa]TdH D/?0º/TU?0ACK<DP?(0/0)0 = 1

9]emT(?]KL?I

A1 =

a ∈ S×S∣

∣

∑

i,j

aij log(π1ij/π

0ij) ≤ 0

T+HV?T:\]BETÊÂCa]TN\IBCD/oIVo]K<X<KAEKLT^%g$BEDPu qI9 d V/^

P 0t

(dP 1t

dP 0t

≥ 1)

= P 0t

(

∑

i,j∈S

mN(t)ij log

π1ij

π0ij

≥ 0)

= P 0t

(

mN(t) ∈ A1)

.

THVGI^CTvACa]T+xG]uy\I^ÀDgbACaITTUuÀoTFIF:TF²¸¹VPBC³/D,º¹HWaIVK<?YVBETyKL?IF]TU\TU?IF:T?0AÀDgbACaIT+xG]uy\]KL?I

ACK<uyT^ IV?(FloTHVGI^CTN(t)

Kw^tYDPKw^C^CDP?F:K<^nACBEK<o]G:ACTF ]T I?IF

P 0t

(dP 1t

dP 0t

≥ 1)

=∞∑

n=0

P(

N(t) = n)

P 0(

mn ∈ A1)

=∞∑

n=0 −t t

n

n!P 0(

mn ∈ A1)

.

«]BEDPu YTUACTUBdi:HWaITTUX© ^mACaIT^CK<^Qi:HWaIz0_[Tv³j?]D, ACa]TyTj\DP?ITU?0ACKwVXBEVACT¼g$DPBOACaITv^CTU\IVPBEVACK<DP?¹DPg[AxD¸¹VPBC³/D,º¹HWaIVK<?I^9Z= AÀHUV?*oTÂT:\]BETÊ^CTF¹ojkMACa]Tl^C\THJAEBEVPX|BEV/F:K<GI^

ρ K©9 T/9ojk¹AEa]TuV :K<uÀG]u DgACa]TVo(^nD/XLG:AET¼º,VPXLG]TÔDg[AEa]TyTUK<PTU?jº,VX<G]T^` fDg[AEa]TyuV,ACBEKwH T^

π(λ) tKAEa π(λ)i,j = (π1

ij)λ(π0

ij )(1−λ) g$DPBNVPXLX

λ ∈ [0; 1]9]«IDPBtK<BCBETF:G(H K<o]XLT:T0G]K<º,VX<TU?0A¸¹VBE³PD,ºHWaIVPKL?IÂEa]T+g$DPX<X<D,tKL?]vBET^CG]XAta]D/X<FI^bACBEG]T

limn→∞

1

nlogP 0

(

mn ∈ A1)

= log inf0<λ<1

ρ(π(λ)). qI9 zd KLACaACa]TNaITUX<\DPgAEa]T+g$DPX<X<D,tKL?]vTUX<TUuyT?/AWVBEkÂX<TUuyuV¼TNHUVP?lAEBEVP?I^xg$TBbACa]Kw^tBET^CG]XLAAEDyDPG]B^CKAEGIV,AEKLD/?-9

3 .j 5l® &2)& *^+<-(an) 3 + .\9 +m 8 + / 4`+h5 = V 5 9 6I-J6If+eCE+ .LH/'8 N 3 +"C 9 QOR ?@+ /

lim inft→∞

1

tlog

∞∑

n=1 −t t

n

n!an ≥ exp

(

lim infn→∞

1

nlog an

)

− 1

./H1

lim supt→∞

1

tlog

∞∑

n=1 −t t

n

n!an ≤ exp

(

lim supn→∞

1

nlog an

)

− 1.

)+,-¨ & tHUHUDPBWF:K<?]ÀAEDÂi0ACK<BCX<K<?]I ^g$DPBEuÀG]XwVvT+aIVº/T

n! ∼√

2πnn+1/2

n,

ta]TBCT ∼ K<?IF:KwHUVACT^:AEaIV,AbAEa]Td0G]DAEKLT?/ADPgoDAEa¢^CKwF:T^tHUDP?jºPTBC/T^%ACD1VP^n→ ∞ 9I`bajGI^bT+PTUA

−t tn

n!an ∼ −ttn n

nn

an√2πn

= exp(

−t− n logn+ n log t+ n+ logan√2πn

)

= exp(

t(

−1 − n

tlog

n

t+n

t+n

tcn)

)

qI9L;~M

];

ta]TBCTcn =

1

nlog

an√2πn

.

TBCTvAEa]Tl0G]DAEKLT?0A+DPgoDAEa²^nKwF:T^ÀF:DjT^+?]DPAÀF:TU\TU?IF*D/?tK'9 TP9ZAEa]TlHUDP?jºPTBC/TU?IHUTdg$DPB

n → ∞ K<^G]?]KLg$DPBEu K<?

t9

D,=bV?0AbACDT:\]BCT^CÂEa]TNBEKL/a/Aa(V?IF¢^CK<F:TNV/^V¼g$G]?IHJAEKLD/?Dg n/t 9:=@?¢DPBWF:TUBbAEDÂF:DÂ^CDyF]TI?]T+AEa]Tg$G]?IH ACK<DP?gojk

gc(x) = −1− x log x+ x+ x · cg$DPBNVPXLXx > 0

9-`ba]Kw^mg$G]?IH ACK<DP?*K<^+uyD/?]DAEDP?]KwHUVPXLX<kKL?(H BETVP^CK<?]lK<?c9 £TytK<XLX|G(^nTyKLANACDPT ANoDPG]?(F]^

DP? qI9L;~M K<?AEa]Td^CKAEGIV,AEKLD/?ta]T?ACa]Td^CT0G]T?IH T (cn)n∈ Kw^bo(D/G]?IF:TFZ9(ijK<?IHUT

g′c(x) = − logx− x

x+ 1 + c = c− logx,

ACaIT¼F:TBCK<º,V,ACK<ºPTg′cKw^m^nACBEK<H ACX<kF]TH BETV/^nK<?]ytKLACaMV TUBEDlV,A

x = c9`bajGI^

gcV,ACAEVK<?I^fKAW^f/XLD/oIVX-uV jK uÀGIu V,AbAEa]TN\(D/KL?0A

c9]`baITNºVPXLGITODPgAEa]TNuV :K<uÀG]u Kw^

gc( c) = −1 − c log c + c + c · c = c − 1.

D,XLTUA a = lim infn→∞1n log an

VP?IFε > 0

9]`ba]TU?AEa]TUBET+K<^VP?N ∈ g tKAEa

1

nlog

an√2πn

> a− εg$DPBVPXLX

n ≥ N.

THVGI^CTÀX<TUuyuV8]9 8ÂDP?]X<kV\]\]X<K<T^mACD I?]KLACT^nGIuy^ TvtKLX<X|^n\IXLKLAmAEa]TvKL? (?]KAETv^CG]ug$BEDPuÁACaITyHUX<VPKLuK<?/AEDvACa]Td^CG]uyuV?IF]^tg$DPB

n = 1, 2, . . . , N − 1V?IFlACa]TNBETUuVPKL?]K<?]vAEVPKLX©9]«]D/B

n < NT I?(F

limt→∞

1

tlog −t t

n

n!an = −1 + lim

t→∞n

tlog t+ lim

t→∞1

tlog

an

n!= −1 + 0 + 0 = −1

V?(FltKLACant = b a−ε · tc TNH DP?(H X<GIF:TOg$DPBbAEa]T+AEVPKLX

lim inft→∞

1

tlog

∞∑

n=N −t t

n

n!an

≥ lim inft→∞

1

tlog(

−t tnt

nt!ant

)

= lim inft→∞

1

t

(

t(

−1− nt

tlog

nt

t+nt

t+nt

t

1

ntlog

ant√2πnt

)

)

≥ lim inft→∞

ga−ε

(

nt/t)

= ga−ε

(

a−ε)

= a−ε − 1g$DPBVPXLX

ε > 0

K©9 T/9lim inft→∞

1

tlog

∞∑

n=N −t t

n

n!an ≥ a − 1.

c-TuyuyV8]9 8¼PK<ºPT^?]D,AEa]TNBCT^nGIXA9ITHUVPGI^CTmDPgea−1 > −1

ACa]T IBW^nAN−1

AETUBEuy^tVPBCT+?]DPAtKLuy\DPBCAEV?0AV?(FÂAEa]TNBEVACT+Kw^F:T AETUBEuyKL?]TFlojklACa]T+AWVK<X'9

D,XLTUA b = lim supn→∞1n log an

V?IFε > 0

9]`ba]T?AEa]TUBET+K<^tVP?N ∈ g tKAEa

1

nlog

ann2

√2πn

< b+ εg$DPBVPXLX

n ≥ N9

«]D/BtACaITdG]\]\TUBmo(D/G]?IFTNaIVºPT+AEDÂH D/?I^nKwF:TBfVX<X-AETUBEuy^KL?¢AEa]TNAEVPKLX Iu¼G]XLACK<\]XLkjK<?]AEa]TÀ?jG]uyTUBWV,AEDPBVP?IFF:T?]DPuyK<?IV,AEDPBbDgT0GIVACK<DP? q]9<;~d tKAEa n2 ^Ca]D,^ jACaIVADP?]TNHUVP?HWaID0D0^nT N X<VPBC/TfAEDyDPo:AWVK<?

−t tn

n!an ≤ 1

n2exp(

t(

−1− n

tlog

n

t+n

t+n

t

1

nlog

ann2

√2πn

)

)

· (1 + ε)

/8

g$DPBmVX<Xn ≥ N

9 =@?I^xAETV/FDPg 1/n2 TdH D/G]XwFa(VºPTNHWa]D/^CTU?¢V?jkDACaITUB^CG]uyuyVPo]X<TÀ^nT0G]TU?IHUTP9 b`ba]TU?¢TaIVº/T

lim supt→∞

1

tlog

∞∑

n=N −t t

n

n!an

≤ lim supt→∞

1

tlog

∞∑

n=N

1

n2exp(

tgb+ε(n/t))

(1 + ε)

≤ lim supt→∞

1

tlog(

(

∞∑

n=N

1

n2

)

exp(

tgb+ε( b+ε))

(1 + ε))

= b+ε − 1g$DPBVPXLX

ε > 0

K©9 T/9lim sup

t→∞

1

tlog

∞∑

n=N −t t

n

n!an ≤ b − 1.

¡f^CK<?]yXLTuvuVÂ8]9 8¼V0VK<? :TNVPX<^CDyPT AbAEa]Td^nTH D/?IFl\(VBCAbDPgAEa]TdH XwVK<u¢9 0TF' KLACaACa]TNaITUX<\DPgAEa]TNXLTuyuyVyT+PTUAAEa]TNuVK<?¢BCT^nG]XLAtDgACa]Kw^^CTHJAEKLD/?-9d.jI,.j ®,& ( *,+"-

X0, X1 3 +Z-JF 56ICUCE+ 18 4"6 3<L + : +m 8 6If .L + / -O45 / -76 /'8 5 89 -J6INT+ . C 50f4? . 6 /@9FG6I-%? / 6I-#+ 9 - . -#+ 9JVS. 4+ S .0/21 AM+ / +"C . -#5C 9 q0, q1 ∈ S×S Q 8 CU-I?S+<C .9`9U8 N+ qϑii = −1 = 50C .LIL

i ∈ S Q 50C λ ∈ [0; 1] L +"--%?@+\N . -JCU6 π(λ) 3 + 1 + / + 1.9\.M3 50f+ .0/21iL +"- ρ 1 + / 50- +\-%?@+ 9JV +4"-7C .LC .M1 6 89 Q R ?S+ / -%?@+ .0> + 9 CU6 9 B(t) = 50CZ-I?S+ 9 + VS. C . -76K5 / 5 = X0 .0/21 X1 ? .9 + V 5 / + / -J6 .L 1 +4 .> C . - +

limt→∞

1

tlogB(t) = inf

0<λ<1ρ(π(λ)) − 1.

)+,-¨ & ijG]o(^xAEKAEG:ACK<?]YT AETUBfi:HWa]TTX' ^tBET^CG]XLA q]9 zM %g$DPB¸MVBE³PD,ºÂHWaIVPKL?I^bK<?0ACDyX<TUuyuVvqI9 8¼/KLº/T^

limt→∞

1

tlogP 0

t

(dP 1t

dP 0t

≥ 1)

= exp(

log inf0<λ<1

ρ(π(λ)))

− 1

= inf0<λ<1

ρ(π(λ)) − 1.

?(VX<DPPD/GI^DP?]TNVPX<^CDyPT AW^

limt→∞

1

tlogP 1

t

(dP 1t

dP 0t

≤ 1)

= exp(

log inf0<λ<1

ρ(π(λ)))

− 1

= inf0<λ<1

ρ(π(λ)) − 1.

¡f^CK<?]yXLTuvuVÂ8]9 8 I?]Kw^naITFACa]TN\IBCDjDgx9 0TF' «KLBW^nAb?IDACKwH T+AEaIV,A IK<?H DP?0AEBEV/^xAbACDyAEa]Td¸¹VPBC³/D,ºÂHWaIVK<?HVP^CTjAEa]TdBWV,ACTNHV?]?]DPAfF:BEDP\o(TXLD, −1

9(µm?VÂHUXLD0^nTBXLDjD/³lAEa]K<^fK<^m?]DPAm^nGIBC\]BEKw^nK<?]I9(f^mTÀÊV KL?AEa]TÀ\IBCDjDgDg[X<TUuyuVÂqI9 8]ACaITdº,VX<G]T −1

Kw^[xGI^nAACaITÀT:\DP?]T?0ACKwVXBWV,AET+g$DPBfACa]T¼TUºPT?0A ]AEaIV,AfACa]T¼\]BCD:HUTÊ^taIVP^f?]DNxG]uy\ DPBOV,AmuyD0^xA N xGIuv\(^` G]\ACDACK<uyT

t9]=@?ACaIK<^mHUVP^CT+Dg|H D/G]BE^CT+TdHV?]?IDA/VACa]TBV?jkÂK<?:g$DPBEuV,AEKLD/?lAEDÂF:K<^nACK<?]/G]K<^CaoT AxTUTU?AEa]T+AxD

\]BED:H TÊ^CT^9`bajGIÂCaITd^nKLACGIVACK<DP?K<^fVP^g$DPX<X<D,^:ta]T?lAEa]TN\]BED:H TÊ^CT^tVBETOºPTUBEkl^CKLuyK<X<VPB jAEa]TU?AEa]TNBEVACT

ρg$DPBtAEa]T

^CTU\IVPBEVACK<DP?¢Dg|AEa]TÀTuÀoTF]F:TF¹¸¹VPBC³/D,ºHWaIVPKL?I^fK<ÔH X<D/^CTNAEDl~lVP?IFACa]TvBWV,ACTdg$DPBAEa]T¼HUDP?0ACK<?jG]DPG(^tACK<uyTHUV/^nT+Kw^ ρ − 1 ≈ ρ

]K'9 TP9IKAKw^tuyD/^nACX<kF:TUACTBCuyK<?]TFojkÂACaIT+ACBWV?I^CKAEKLD/?uyTHWaIV?]Kw^Cu9I=@?AEa]TdHUV/^nT+DPgº/TUBEkF:K TBCT?0Af¸¹VBE³PD,ºÂHWaIVPKL?(^:D/?lAEa]TNDAEa]TUBa(V?IF

ρK<^t^CK<P?]K (HUVP?0ACX<kl^nuVPXLX<TUBbACaITU?~V?IFg$D/BH DP?0AEKL?jG]D/GI^

ACK<uyTNT+/T AtACaITNBEVACT ρ − 1 ≈ −1

9 fTUBET+ACa]T+BWV,AETOKw^tuVK<?]X<klF:TUACTUBEuyKL?ITFojkACa]TbxGIuv\¢uyTHWa(V?]Kw^nu¢9

P_

1

2

3

0 10 20 30 40

«KL/G]BETÂq]9 _ R ?@+ 1 6 . A0C . N 9 ?S5F 9 5 / + VS. -%? 5 =P. 45 / -J6 /S8 5 89 -J6INT+ . C 5f 4`? . 6 /@: AM+ / +"C . -#+ 1 3"> -%?@+-JC ./S9 6I-J6K5 / N+4`? ./ 6 9 N = CE5N + . N V2L + QQ R ?S+Pm 8 + 9 -J6K5 / 6 9: F^?@+"-I?S+<CT-%?@+ V C`5 4+ 9`9 ? .9 Ad+ / +<C . -#5C q05Cq1 Q½m¾ 5 ²±. ® &2)& `ba]K<ÔT:VPuy\]XLTyF:TuvD/?I^nACBWV,ACT^(ACa(V,ANPK<ºPT?ACa]TvAEBEVP?I^nKLACK<DP?£uV,ACBEKwH T^mKLA+KwÔTV/^nk

ACDT:\]X<K<HUKAEXLkHVXwH G]XwV,AETNACa]TvT:\DP?]T?0ACKwVXF]THUVkBWV,ACT^tg$D/BfAEa]T¼bVkPT^tBEK<^C³9·DP?(^nKwF:TUBfACa]T¼ACBWV?(^nKLACK<DP?uV,AEBCKwH T^

π1 =

0 1/3 2/32/3 0 1/31/3 2/3 0

V?(Fπ0 =

0 2/3 1/31/3 0 2/32/3 1/3 0

.

emTI?]K<?]π(λ) VPÔVoD,ºPTdTÀ/T AmACa]Tv^n\TH ACBWVXBEV/F:KLG(^ ρ(π(λ)) = (21−λ + 2λ)/3

VP?IF¹HUDP?I^CT0G]T?/AEXLkinf0<λ<1 ρ(π

(λ)) = ρ(π1/2) =√

8/39I«IBCD/u YTUACTUBNi:HWa]TTX' ^BET^CG]XLAmTdPTUAACaIT¼F:THUVkBEVACTdDPgAEa]T

bVkPT^BEKw^n³g$D/BbACa]Td^CTU\(VBWV,ACK<DP?lDgACa]T+AxDH DPBEBET^C\(D/?IF:K<?]¼F:KwÊH BET ACTOAEKLuyTd¸¹VPBC³/D,ºÂHWaIVK<?I^

limn→∞

1

nlogB(λ, n) = log(

√8/3) ≈ −0.059.

`ba]Td/TU?]TBEVACD/BE^g$D/BACaIT¼H D/BCBET^C\DP?IF:K<?]ÂHUDP?0ACK<?jG]DPGI^tACK<uvTv¸¹VBE³PD,ºHWaIVK<?I^VPBCTqϑ = πϑ − I

g$DPBϑ ∈ 0, 1 9]«K<PG]BETÀqI9 _yK<XLX<GI^nACBWV,AET^D/?]TdK<?I^nAEV?0AEK<VACK<DP?¢DgACaIK<^m\]BEDjHUTÊÛ9(«]BEDPu ACa]TDPBETUu qI9 _ÂTd³j?]D,ACaITNBEVACTOg$DPBbAEa]Td^nT\IVBWV,AEKLD/?lDPgACaITdH DP?0AEKL?jG]D/GIÂCK<uyTd¸¹VBE³PD,ºHWaIVPKL?I^9:= AKw^

limt→∞

1

tlogB(t) = inf

0<λ<1ρ(π(λ)) − 1 =

√8/3− 1 ≈ −0.057.

r

b Á

=@?¢ACaITd\]BEDjHUTÊ^tDg|G]?(F:TUBW^xAWV?IF]KL?]ÂHUDPuy\]X<K<HV,AETF¢^xAED:HWaIVP^nACKwH+uyTHWaIVP?]K<^Cu^H D/uy\]G:ACTBm^CKLu¼G]XwV,ACK<DP?(^HUVP?oT¼VGI^CT g$GIXAED0D/X'9`ba]T¼VPBCTVyDg[XwVBEPTdF:TºjK<VACK<DP?\]BEDPo]X<TUu^t\IX<V/H T^m^n\THUK<VPXHWa(VX<XLT?]PT^ta]TUBETP9THUVPGI^CTACaIT¼ta]D/XLTÀ\(D/KL?0A+Dg[XwVBEPTÀF:TºjK<VACK<DP?\]BEDPo]X<TUu^mK<^fACDaIV?(F:XLT¼TjACBETUuyTXLk^CuyVPXLX\]BEDPoIVPo]K<XLKLACK<T^ I?IVPKLº/TV\I\]BCD0VPHWa]T^tAETU?IF¹ACDgVK<XaITUBETP9ZijDPuyTy^nD/XLG:AEKLD/?I^fACDACa]TvBCT^nGIXAEKL?]\]BEDPo]X<TUuÔVBETÀK<XLX<GI^nACBWV,ACTF¹K<?¹AEa]Tg$DPX<X<D,tKL?]vT]VPuv\IXLTÛ9

`ba]TNTVP^CkÂVkÂACDT^nACK<uV,AETNACa]TN\IBCD/oIVo]K<X<KAxkÂDPg[V?¢TUº/TU?0AtK<ÂCDPT?]TUBWV,AET+uyVP?jkÂBEVP?IF:D/u ÊVuy\]X<T^ ACDHUDPG]?0A+ACa]T?jG]u¼o(TBNDgbD:HUH GIBCBETU?IHUT^mDgACa]TTºPT?/ANK<? 0G]T^xAEKLD/? -V?(F (?IVX<XLk¹ACD¢GI^CT¼AEa]TXwV DgXwVBEPTd?jG]u¼o(TBE^fACDT^nACK<uyVACTÀACa]Tv\]BEDPoIVPo]KLX<KLAxktKLACaMACa]T¼BCTX<VACK<ºPTNg$BET0G]T?IH k/9`ba]KwÔD/BC³:^tTUX<XKg[AEa]T\]BEDPoIVPo]K<XLKLAxkÂKw^tBCTVP^CDP?IVPo]X<kÂX<VPBC/TP9]G:AKLg

nKw^bACaITduV :KLu¼G]u¶?jG]u¼o(TBDg[ÊVuy\]X<TÂEa]TÀH D/uy\]G:ACTBtK<XLX

PT?]TUBWV,AETdK<?ACa]TÀACK<uyT¼T¼VPBCTdtK<X<XLK<?]ÂAEDlbVKLA IACa]KwÔuvTUACa]D:FDP? AOD/BC³lg$DPBO\]BCD/oIVoIKLX<KAEKLT^f^CuVX<XLTBACa(V?

1/njoTHVGI^CTOAxk0\IK<HVX<XLkÂT+D/G]XwFD/oI^CTUBEºPTf?IDDjHH G]BEBETU?IHUT^DgACa]TNTºPTU?0A9

?IDACaITUBf\IBCD/o]XLTu D:HH G]BW^tta]T?¹DP?]TdACBEKLT^tACD^CVPuv\IXLTÀV/HUHUDPBWF:KL?IyACDlVÂHUDP?IF]KAEKLD/?IVXF:K<^nACBEK<o]G:ACK<DP?9`ba]TdTVP^CkbVkAEDF:DAEa]Kw^Kw^tACDdxGI^xA+^CVPuy\]XLTdg$BCD/u¶AEa]TNg$G]X<XF]K<^nACBEKLoIG:ACK<DP? V?IFACDÂBETxxTHJAfTºPTUBEklº,VPXLG]Tta]KwHWaMF:DjTÔ?]DPAmuyTUTUAmAEa]TyH DP?(F:KAEKLD/?-9G]AmAEa]TU? ZVP/VK<? IACa]KwÔtKLX<XDP?]X<k¢D/BC³Kg%ACa]Tv\]BEDPoIVPo]KLX<KLAxkKw^?]DPAmAED0D¢^CuyVPXLX o(THUVPGI^nT¼DAEa]TUBEtK<^CTÀTvtKLX<XxGI^xANaIVº/TdAEDBETxxTHJA+TUºPTBCk¢ÊVuy\]X<TvVP?IF¹/T A+?]Dlº,VX<G]T^ta]KwHWauyTT AtACaITdH DP?(F:KAEKLD/?-9

`ba]Kw^tHWaIV\]ACTUBf\]BCT^nT?0AE^t^CDPuyT+uvTUACa]D:F]^tg$BCD/u ACaITNXLKLACTBEVACG]BET+ta]K<HWaVPBCT+GI^CT g$G]XACDD,º/TUBWH DPuyTfACa]T^nT\]BEDPo]X<TUu^9

& `ba]TNoIV/^nKwHOuyT ACaIDjFAED?jG]uyTUBEK<HVX<XLkl^CDPX<ºPTN^nACD:HWaIV/^xAEK<H+F:K TBCT?0ACKwVXZT0GIVACK<DP?I^K<^bAEa]TNs[G]X<TUB ¸¹VBEG]k/VPuVuyT AEa]D:F£D/BN^xAEDjHWa(VP^nACKwHÀs[GIXLTBNuvTUACa]D:FZ9`ba]TyuvTUACa]D:F Kw^g$D/BNT]Vuy\]X<TZF:T^CHUBCK<o(TFMK<?l +YzPz 9 £THUVP?¢GI^nTdACa]KwûyT AEa]D:F¢ACD/TU?]TBEVACTNBWV?IF]DPu¶\(V,ACa(^bg$BCD/u¶AEa]TÀ^CDPX<G:ACK<DP?¢DPg|Vi:efsf9`ba]TÀBCT^nG]XLAE^DgACa]T^CG]oI^CT0G]TU?0Am^CTH ACK<DP?IÔHUVP?¢ACa]T?¹oT¼GI^CTFAEDlT^xAEKLuV,AETÀ\]BEDPo(Vo]K<XLKLACK<T^fDPBfACDlÊVuy\]X<Tdg$BEDPuH D/?IF:KLACK<DP?IVPXF:Kw^xAEBCK<o]G:AEKLD/?I^9

·DP?(^nKwF:TUBAEa]Td^xAED:HWaIVP^nACKwH+efK TBCT?0ACKwVXZT0GIVACK<DP?dXt = b(Xt, t) dt+ σ(Xt, t) dBt

g$DPB0 < t ≤ T j9<;_

X0 = z ∈ d,

ta]TBCTBKwÔV

d F:KLuyT?I^nK<DP?(VXBED,t?]KwV?uyDAEKLD/? b : d × + → d Kw^m^CDPuyTvF:BCKLgAfg$GI?IHJAEKLD/? ZV?IFσ : d × +→ d×n K<^tACa]TdF:K GI^nK<DP?¢H DjT"!HUKLT?0A9ijD/XLG:AEKLD/?IÂCDÂACa]Kw^tuyT AEa]D:FHUV?¢oTÀV\I\]BCD:K<uV,ACTFV/^bg$DPX<X<D,^:X<T A

N ∈ g VP?IF∆t = T/N

9I`ba]T?F:TI?]T

X0 = z

/

Pq

V?(FlKLACTBEVACK<ºPTUX<k

Xn = Xn−1 + b(

Xn−1, (n− 1)∆t)

· ∆t+ σ(

Xn−1, (n− 1)∆t)

ξn ·√

∆t

g$DPBn = 1, . . . , N

(ta]TUBETξ1, . . . , ξN

VPBCTd F]KLuyTU?(^nK<DP?IVPX IK'9 K'9 FZ9^xAWV?IF]VPBEF?]D/BCuVXBEVP?IF:DPu¶ºVPBCKwVoIXLTÛ9

`ba]T?AEa]TÀF:Kw^nACBEKLo]G]ACK<DP?¢DgX0, X1, . . . , XN

K<^VP?V\]\]BED:KLuVACK<DP?g$DPBbACaITÀF:Kw^xAEBCK<o]G:AEKLD/?DPgAEa]Tdº,VX<G]T^X0, X1·∆t, . . . , XN ·∆t

9µm?]TyDgACaIToIVP^CK<HvBET^CG]XLAE^NVoDPG]ANACa]Kw^NuyT AEa]D:F*K<ÔAEa]Tg$DPX<X<D,tKL?]ACaITUDPBETUu -taIK<HWa K<^ÀVF:KLBETH A

H D/?I^CT0G]TU?(H TODgACa]TDPBETUu ;~]9Q8:9Q8Ng$BEDPu +Y%z/z©9d.jI,.j 1 &' )(;*^+<-

X 3 + .i9 5 L]8 -J6K5 / 5 = j9<;_ = 50C . C`5F / 6 .0/ NT50-76K5 / B Q X\+ / + ξn =(Bn∆t −B(n−1)∆t)/

√∆t .0/21L +"- Xn

: n = 0, . . . , N 3 + 1 + / + 13"> -I?S+ Y 8aL +"C . C 8a>D. N . N+<-%?@5 1FG6I-%? 9 -#+ V 9 6 b_+ ∆t = T/N .9e.D3 5f+ Q 8 CU-I?S+<CUN5C`+ .99U8 NT+B-%? . -∣

∣b(x, t) − b(y, t)∣

∣+∣

∣σ(x, t) − σ(y, t)∣

∣ ≤ K1|x− y|∣

∣b(x, t)∣

∣+∣

∣σ(x, t)∣

∣ ≤ K2

(

1 + |x|)

∣

∣b(x, s) − b(x, t)∣

∣+∣

∣σ(x, s) − σ(x, t)∣

∣ ≤ K3

(

1 + |x|)

|s− t|1/2

= 5C .LIL x, y ∈ d .0/21i.0LIL s, t ∈ [0;T ] : F^?S+<C`+h-%?@+45 /S9 - ./ - 9 K1: K2

:./21 K31 5 / 5- 1 + V + /215 / N Q R ?S+ / -I?S+<C`+h6 9T. 45 /S9 - .0/ - K4

: F^?a6K4`?6 9.0L 9 5P6 /H1 + V + /21 + / - 5 = N : 9U8 4`? -%? . -O-%?@+eY 8aL +"C.`VMV CE5 M6IN . -76K5 / X 9". -76 9 + 9E(

|XT − XN |)

≤ K4∆t1/2.

`ba]TNAEa]TUD/BCTuÁ^Ca]D,^ :ACa(V,AfAEa]Tds[GIXLTB ¸¹VBEG]k0VuVyuyT ACaIDjF¢/KLº/T^mV\IV,AEajtK<^CTdV\]\IBCD:K<uyVACK<DP?ACDACaITd^nD/XLG:AEKLD/?-9]`ba]TNTj\TH ACTFlTBCBEDPBbPDjTÂCD UTBCDytKLACa¢DPBWF:TUB0.59

½m¾ 5 ²±.1 &' '& «]DPBOAEa]TµmBE?I^nACTKL? ¡aIXLT?0oTHW³¢\]BED:H T^C^mtKAEa£\(VBWVuyT AETUB α T¼aIVºPT b(x, t) =−αx VP?IF σ(x, t) = 1

g$D/B+VPXLXx ∈

t ≥ 09Z`ba]THUDP?IF:KLACK<DP?(^mDPg%AEa]TvACa]TDPBETUu VBET¼ÊV,AEK<^ ITF¹g$D/B

K1 = αK2 = max(1, α)

VP?IFK3 = 0

I^CDvACa]TNs[GIXLTB ¸¹VBEG]k0VuVduyT AEa]D:FtKLX<XH D/?jºPTUBEPTm\IV,AEajtK<^CTP9`ba]T+BET^CG]XAtDPgVv?jG]uyTUBEK<HVX-^CKLu¼G]XwV,ACK<DP?tKLACa∆t = 0.005

Kw^^Ca]D,t?KL? IPGIBCTN_]9<; \IVPPTd8,r J9

&! # " =@uy\(D/BnAWV?IHUTvi:VPuy\]XLK<?]K<ÔVÂº,VBEKwV,ACK<DP?DPg%¸DP?0AET ·VBEXLDlÊVuy\]X<K<?](ta]TUBETÀTÀGI^CT¼^CDPuyT¼³0?ID,tXLTF:PTVoDPG]AfACaIT¼K<?0ACTU/BEVACTFg$G]?IHJAEKLD/?¹AEDBETF:G(H TdAEa]TÀº,VPBCKwV?IHUTÀDPg|ACa]TvT^nACK<uV,ACT/9`baIK<ÔKw^fGI^CT g$GIX oTHUVPGI^CT^CuyVPXLXº,VBEKwV?IHUTÀuyTVP?I^+^nuVX<XTUBEBCD/BE^fKL?£ACaITvT^xAEKLuVACTP9Z`ba]TyoIV/^nKwHU^mDg%ACa]Kw^+uyT ACaIDjF VBETÀT:\]X<VPKL?ITFK<? Pq©9

fÊ^CG]uyT+ACaIVAX,X1, X2, . . .

K<^mV?¢K©9 K©9 F-9^CT0G]T?IH TNDPgBWV?(F:DPu ºVPBCKwVoIXLT^V?IFfK<^mVyuyTVP^CG]BWVo]X<T

g$G]?IH ACK<DP?-9(V/^nKwH+¸DP?0AET ·VBEXLD¼KL?0ACTPBWV,AEKLD/?lGI^CTÂEa]TNX<VDgX<VPBC/Tm?jG]u¼o(TBE^bK<?ACa]T+g$D/BCu¶Dg1

n

n∑

k=1

f(Xk) −→ E(

f(X))

. :9 8M

`ba]Ty^nGIu Dg[AEa]TvXLTUgA+a(V?IF£^nKwF:TvK<ÔGI^CTF VPÔV? V\]\]BED:KLuVACK<DP?¢g$DPBOAEa]T¼T:\(THJAEVACK<DP?£DP?¹AEa]TyBCK<Pa0AaIVP?IF¢^nKwF:TP9]`baITd^n\TUTFlDPgHUDP?jºPTBC/TU?IHUTfKw^tF:T AETUBEuvK<?]TFojkAEa]TNº,VBEK<VP?IH T+DPgACaITNXLTUgA aIV?IF¢^CK<F]T

Var( 1

n

n∑

k=1

f(Xk))

=1

nVar(

f(X))

. :9 _d

D,XLTUA Y1, Y2, . . .o(TdVP?]DAEa]TUB^CT0G]T?IH T+DgK©9 K©9 F-9]BEVP?IF:D/u º,VBEK<VPo]XLT^:^CGIHWaACaIVAtACa]TÀF]K<^nACBEKLoIG:ACK<DP?

DgYkaIV/^tVyF:TU?I^CKLAxk

gtKLACa¢BCT^n\TH AbACDyACaITdF:K<^nACBEK<o]G:ACK<DP?Dg

X

g =dL(Yk)

dL(X)

g$D/BVX<Xk ∈ g 9

0

`ba]T?AEa]TNXwV DgX<VPBC/Tm?jG]u¼o(TBE^b/KLº/T^1

n

n∑

k=1

f(Yk)

g(Yk)−→ E

(f(Y1)

g(Y1)

)

:9 r

=

∫

f(y)

g(y)g(y)dL(X)(y)

= E(

f(X)) :9 M g$DPB

n → ∞ 9Z/VPKL? (AEa]T^nGIu Dg[AEa]TyXLTUgAdaIV?(FM^CK<F:THV?MoTyGI^CTFMV/^+V?£VP\]\]BED:KLuV,AEKLD/?g$DPBOAEa]TT:\TH AEV,AEKLD/?¹DP?MACa]T¼BCK<Pa0AOaIV?IF£^nKwF:T/9`ba]Tvº,VBEK<VP?IH TÀDg

f(Yk)/g(Yk)Kw^+^nuVPXLX (KLg

fV?(F

gVBETÀV\ \]BED:KLuV,AETUX<k\]BEDP\DPBCACK<DP?IVPX(AEDyTVPHWaDAEa]TUB9]`ba]TNoDPG]?(F]VBEkÂHUV/^nT+Kw^

g(y) =f(y)

E(

f(X)) .

`ba]T?¢VPXLXAEa]TNK<?:g$DPBEuV,ACK<DP?¢VoDPG]AE(

f(X)) Kw^tVX<BETVPF]kÂH DP?0AWVK<?]TFKL?

gVP?IF

f(Yk)

g(Yk)= E

(

f(X))

Kw^tH DP?(^xAWV?0Abg$DPBVX<Xk ∈ g 9µfg|H DPGIBE^CTOACa]Kw^tAEBCKwHW³F:DjT^t?]DAfHWa(V?]/TmAEa]TdD/BEF:TBtDgACaITNuvTUACa]D:FZ9`ba]Tdº,VBEKwV?IHUTODPgAEa]TdT^xAEKLuV,AET

Kw^Var( 1

n

n∑

k=1

f(Yk)

g(Yk)

)

=1

nVar(f(Y1)

g(Y1)

)

, :9 qd K©9 T/9IAEa]TduyT AEa]D:FK<^m^xAEKLX<XDg|DPBWF:TUB

1/√no]G:AO^CDPuyTUACK<uvT^D/?]TÀHUVP?¹HWa]DjD/^CTdVyg$G]?IH ACK<DP?

gACDÂD/o:AEVPKL?¹V

uÀG(HWaoT ACACTUBmH DP?(^xAWV?0AtKL?ACaITNºVPBCKwV?(H TP9 TvVPBCTvKL?0AETUBET^nACTFMKL? T^nACK<uV,AEKL?]ACaITv\IBCD/oIVo]K<X<KAxk¢DgACa]TvTUº/TU?0A X ∈ A g$D/B+VuyTV/^nG]BWVoIXLT^CT AAIK'9 TP9:K<?ACa]TdHVP^CT

f = 1A9 TUBET+ACaIT+¸¹DP?0ACT ·VPBCX<D¼uyTUACa]D:F :9 8M %oTHUDPuyT^

1

n

n∑

k=1

1A(Xk) −→ P (X ∈ A)

V?(FÂAEa]TNº,VBEK<VP?IH T :9 _d %g$DPBbACaIK<^tT^xAEKLuVACT+K<^

σ2 = Var( 1

n

n∑

k=1

1A(Xk))

=1

n

(

P (A) − P (A)2)

.

`DÂuV³PT¼KLuy\DPBCAEVP?IH Tv^CVPuv\IXLK<?]ÂGI^CT g$GIXg$D/BP (X ∈ A) ≈ 0

(T¼HWaID0D0^nTÀBEVP?IF:DPuÁº,VBEK<VPo]X<T^YktKLACa

P (Yk ∈ A) P (X ∈ A)9(`baITOK<uy\(D/BnAWV?IHUTN^CVPuv\IXLK<?]yuyT ACaIDjF j9 rd K<^

1

n

n∑

k=1

1A(Yk)

g(Yk)−→ P (X ∈ A)

g$DPBn→ ∞

V?(FÂAEa]TdH D/BCBET^C\(D/?IF:K<?]vºVPBCKwV?(H Tmg$BCD/u :9 qd o(TH DPuyT^

σ2 = Var( 1

n

n∑

k=1

1A(Yk)

g(Yk)

)

=1

nVar(1A(Yk)

g(Yk)

)

=1

n

(

E(1A(X)

g(X)

)

− P (A)2)

.

`ba]TBWV,AEKLDDPgAEa]TÂº,VBEKwV?IHUTÔg$DPBdACa]TKLuy\DPBCAEV?(H TÂÊVuy\]X<KL?]uyTUACa]D:F²V?(F ACa]Tl¸¹DP?0ACT ·VPBCX<DuyT AEa]D:FKw^σ2

σ2=E( 1A(X)

g(X)

)

− P (A)2

P (A) − P (A)2.

= ggK<^bXwVBEPT+D/?

AACaIK<^tBWV,AEKLDyoTHUDPuyT^^CuVX<X :K©9 TP9]K<?ACa]T^nTdHVP^CTÂCa]TÀT^nACK<uyVACTOg$BEDPu AEa]TNK<uv\DPBCAEVP?IH T

ÊVuy\]X<KL?]yuyT AEa]D:FaIVP^tVvoT AnAETUBº,VPBCKwV?IHUTP9

P

plain Monte−Carlo sampling

0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040 0.00450

100

200

importance sampling

0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040 0.0045

040

010

00

«KL/G]BET£j9<; R ?6 9 A 8 CE+ 6 LIL]89 -7C . - + 9 -%?@+ f . CU6 ./ 4`+CE+ 18 4"-J6K5 /@: F^?a6K4`? 4 ./ 3 + . 4`?a6K+<f+ 1 3"> 1 + V2L 5 > 6 / A6IN V 50CU- .0/ 4+ 9<. N VHL 6 / A Q / -%?@+ 8_VDV +"C V 6K4"- 8 C`+ 9 F +T+ 9 -J6IN . -#+h-%?@+ V C`5 3.D3 6 L 6I- > -%? . - . CE5F / 6 ./ NT50-76K5 /+ d4++ 19 -%?@+ L +<f+ L 3 3 + = 5CE+h-76IN+ 3"> AM+ / +"C . -76 / A .9<. N VHL +T5 = CE50F / 6 .0/PVS. -%? 9: 9". N V2L + 1 FG6I-I?.i9 - + V 9 6 b_+5 = DT = 0.001 :./21 45 8a/ -J6 / A?@50F N ./S> 5 = -I?S+ 9 +CE+ . 4`? . f .0L]8 + ADCE+ . -#+<Ch-I? ./ QZR ?@+8_VDV +"C V 6K4"- 8 CE+AD6If+ 9 -I?S+?a6 9 -#5À0C . N = 50CP-%?@+ 1 6 9 -7CU6 3"8 -J6K5 / 5 = + 9 -J6IN . - + 9 AM+ / +"C . - + 1 6 / -I?a6 9 F .0> QR ?@+ L 5F +"C V 6K4"- 8 CE+hAD6If+ 9 -I?S+?a6 9 - 5À0C . N = 5C + 9 -J6IN . - + 9 5 3 - . 6 / + 13"> -I?S+P6IN V 50CU- .0/ 4+ 9". N VHL 6 / AN+<-%?@5 1Z= CE5N$+ . N V2L + _QdQ A . 6 / + . 4? + 9 -J6IN . - +6 9 4 .0L 4 8aL . -#+ 1Z= CE5N .9". N V2L +P5 = V@. -I? 9`: 3"8 -5 / + 4 ./ 9 +`+-%? . -e-I?S+ + 9 -J6IN . - + 9 5 3 - . 6 / + 1 3"> 6IN V 50CU- .0/ 4+ 9". N V2L 6 / A . CE+N 8 4? 3 +"-7-#+"C45 / 4+ / -JC . -#+ 1. C`5 8a/21 -%?@+e-I?S+`50CE+"-76K4 .L f .L]8 + 2.69 · 10−3 Q½m¾ 5 ²±.1 &w2 & `DÂAET^nAmAEa]TÀK<uy\(D/BnAWV?IHUT¼ÊVuy\]X<KL?]uyT ACaIDjF¹TÀACBEkAEDlGI^CTvKAmACDlT^xAEKLuVACT¼ACa]T\]BEDPoIVPo]K<XLKLAxkjAEaIV,AVBCD,t?IK<VP?ÂuyDACK<DP?¢T:HUTUTF]ÂCa]TNX<TUº/TUX

3oT g$DPBETOACK<uyT;P9 TdF:TI?IT

A =

ω : [0; 1] → ∣

∣ sup0≤t≤1

ωt > 3

.

«]BEDPu¶ACa]TNBET (THJAEKLD/?¢\]BCK<?IHUKL\]X<T+g$DPBmVBED,t?]KwV?¸DAEKLD/?XTN³0?ID, AEa]TNT]VPH Aº,VX<G]TNDgACa]Kw^\IBCD/oIV o]K<XLKLAxk

P (X ∈ A) = P(

sup0≤t≤1

Xt > 3)

= 2P (X1 > 3) = 0.00269. . . .

«]D/BfAEa]Tv\]BCD:HUTÊ^YTyHUV?MGI^nTyBCD,t?IK<VP?¹¸DAEKLD/?¹tKAEa£VH DP?(^xAWV?0AOF:BEKLgA K'9 TP9

Yt = Xt + bt9

«]BEDPu HWaIV\:AETUBN;+TO³j?]D, ACa]TdF]TU?I^CKAxk

g(ω) =dL(Y )

dL(X)(ω) = exp(b · ωt − b2/2).

`ba]TNBET^CG]XLAbDPgVy?jG]uyTUBEKwHUVXZ^CKLu¼G]XwV,ACK<DP?Kw^F:Kw^n\]XwVkPTFÂK<? IPG]BETÀj9<;P9

"# `ba]TvBCTnxTH ACK<DP?MuyTUACa]D:F¹Kw^+VAETHWa]?]Kw0G]Tta]KwHWa£HV?£o(TvGI^CTFAED/TU?]TBEVACTÀÊVuy\]X<T^+V/HUH D/BEF]KL?]ÂACD¢VF:Kw^xAEBCK<o]G:AEKLD/?ta]TUBETNAEa]T¼F:T?I^CKAxktKAEa¹BET^C\(THJAAEDl^nD/uyTdDPBEKL/KL?(VXZuyTV/^nG]BETNKw^f/KLº/TU?-9I«IG]BCACa]TBmF:TUAEVK<Xw^HUVP?oT+g$DPGI?IFKL? +?0G( ]; 9I`ba]TNoD0D/³ Y%` mz08H D/?0AEVK<?I^tVyÊVuy\]X<TOK<uy\]X<TUuyTU?0AEVACK<DP?-9

Pz

d.jI,.j1 &2 CE+ +4"-76K5 / N+<-%?@5 1 Q *^+"- f : g 3 + V CE5 3.M3 6 L 6I- >1 + /S9 6I-76K+ 9 5 / 9 50NT+hNT+ .98 C .D3"L +9JV@. 4+ (X ,F , µ) ./21 λ ≥ 1 3 + .P/S8 N 3 +<CZFG6I-%? λf ≥ g Q *^+<- (Xn)n∈ 3 + .0/ 6 Q 6 Q 1 Q 9 +`m 8 + / 4+h5 =C .0/21 50N f . CU6 .M3<L + 9 FG6I-%? f .0L]8 + 9 6 / X ./H1i1 + /@9 6I- > f :./21iL +"- (Un)n∈ 3 + ./ 6 Q 6 Q 1 Q 9 +m 8 + / 4+5 =C .0/21 50N f . CU6 .D3"L + 9: 8a/ 6 = 50CUN L >1 6 9 -JCU6 3<8 -#+ 1 5 / -%?@+\6 / - +"CUf .0L [0; 1] : 6 /21 + V + /H1 + / - L]> 5 = -%?@+ (Xn) QX+ / + N = minn ∈ g | λUnf(Xn) ≤ g(Xn) QkR ?@+ / -%?@+ 1 6 9 -7CU6 3"8 -76K5 / 5 = XN? .9h1 + /@9 6I- > g5 / (X ,F , µ) Q

`bajGIÔACaITyVPXL/DPBEKAEa]u DPBE³:^mV/Ôg$DPX<XLD,^9ZfÊ^nG]uyTyAEaIV,AdTyVP?/AOACD/TU?]TBEVACTvBEVP?IF:D/u ºVPXLGIT^VPHH D/BEF:K<?]AEDVl\]BEDPo(Vo]K<XLKLAxkF:TU?(^nKLAxk

g9 T (BE^nAOa(VºPT¼ACD I?IFMVP?]DAEa]TUBNF:T?I^CKAxk

fVP?IFMV?jG]uÀoTUB

λ ≥ 1tKAEa

λf ≥ g]ta]TUBET+TdHV?VX<BETVPF]kÂPT?]TUBWV,AETOBWV?(F:DPu º,VPXLG]T^tVPHH DPBWF:K<?]vACDyAEa]TN\]BEDPoIVPo]KLX<KLAxkF:T?I^nKLAxk

f9I`D^CVPuy\]XLTOg$D/BbACa]TdF:T?I^CKAxk

gD/?]T+aIVPÂEDy\(TBng$D/BCu AEa]T+g$DPX<X<D,tKL?]^nACT\IÛ9

^nACT\ ;bPTU?ITUBWV,ACT+VyBWV?IF:D/u º,VX<G]TXV/HUHUDPBWF:KL?IdACD

f^nACT\8bPTU?ITUBWV,ACT+VyBWV?IF:D/u º,VX<G]TU:G]?]KLg$DPBEuyXLkF:K<^nACBEK<o]G:ACTFDP?

[0; 1]^nACT\¢_bKgU · λf(X) > g(X)

/D¼oIV/HW³ACD^nACT\ ;^nACT\r bTUuyKA

XTHVGI^CTmKLAPK<ºPT^b^nD/uvT+K<?I^CKL/a/AW^b=BETU\IBCD:F:GIHUTfAEa]TN\]BEDjDgDPgAEa]T+ACaITUDPBETUu¶a]TUBETP9)+,-¨ & ROK<ºPT?AEa]Tdº,VX<G]TNDPg

Xn(KAfKw^V/HUH T\:ACTF¢tKAEa\]BEDPoIVPo]KLX<KLAxk

g(Xn)/λf(Xn)9ijDyAEa]TNACDPAEVPX

\]BEDPoIVPo]K<XLKLAxkyACaIVAXn

Kw^tVPHH TU\]ACTF ]K<^

P(

λUnf(Xn) ≤ g(Xn))

=

∫

X

g(x)

λf(x)f(x) dµ(x) = 1/λ.

`ba]Tvº,VX<G]TNK<Ô/TUDPuyTUACBEK<HVX<XLk¢F:Kw^xAEBCK<o]G:AETF£tKAEaM\IVPBEVPuvTUACTB

1/λV?IF¹g$DPBOTUºPTBCk^CT A

A ∈ F V?IFn ∈ g T+PT A

P (Xn ∈ A,N = n) =

∫

A

(

1 − 1

λ

)k−1 g(x)

λf(x)f(x) dµ(x)

=(

1 − 1

λ

)k−1 1

λ·∫

A

g(x) dµ(x).

ijG]uyuV,AEKLD/?¢D,ºPTUBnPK<ºPT^

P (XN ∈ A) =∑

n∈ P (Xn ∈ A,N = n) =

∫

A

g(x) dµ(x).

`ba]Kw^t\]BED,ºPT^[AEa]TdH XwVK<u¢9 0TF' = g|T¼VP?0AtACDlG(^nTNAEa]Kw^fuyT AEa]D:F¢ACD^nK<uÀG]XwV,AETÀVÂHUDP?IF:KLACK<DP?(VXF]K<^nACBEKLoIG:ACK<DP?

P ( · | A)Td\]BED:H TTF

VP^g$D/XLX<D,^9 £TdHWa]DjD0^nT+^nD/uyTdF:TU?I^CKLAxkϕta]TBCT

AaIV/^Vva]KL/a¢TU?]D/G]Pa\]BEDPoIVPo]KLX<KLAxktBnA9

ϕVP?IFta]TUBET

TvHV?M/TU?]TBEVACT¼BEVP?IF:DPuº,VX<G]T^mtaIK<HWa VBET¼F:Kw^xAEBCK<o]G:AETF VPHH DPBWF:K<?]AEDϕ9-«]D/BOACa]TyVX<PD/BCKLACa]uT

HWa]DjD/^CTmACa]TdF:T?I^CKAEKLT^fVP?IF

gtKLACa

f(x) =ϕ(x)1A(x)∫

A ϕdP

V?IFg(x) =

1A(x)

P (A).

THVGI^CTAa(VP^fVÂa]KL/a¹TU?IDPG]/a\]BCD/oIVoIKLX<KAxktBnA9

ϕTÀHUVP?¹PTUAm^CVPuy\]XLT^fV/HUH D/BEF]KL?]yAED

fg$BEDPu ACa]T

?IVPKLº/TyVPXL/DPBEKAEa]u¢9Z`ba]TÂF:TU?(^nKLAxkgKw^+ACa]TÂF]TU?I^CKAxkMDgACa]TÂHUDP?IF:KLACK<DP?(VX|F:Kw^xAEBCK<o]G:AEKLD/? taIK<HWa*TyVPBCT

K<?/AETUBET^nACTFlK<?-9I«IDPBλTNHUV?¢HWa]DjD/^CT

λ = ess supx∈A

g(x)/f(x) =

∫

AϕdP

P (A) ess infx∈A ϕ(x).

µm?]T¼PDjD:FACa]K<?]VPo(D/G:AOACa]TvVX<PDPBEKLACa]uÁK<^ (ACaIVAmT¼F:D?]DAO?]TUTF¢ACD³0?ID,AEa]Tv\]BCD/oIVoIKLX<KAEKLT^P (A)V?(F ∫

A ϕdPK<?¢DPBWF:TUBACDÂVP\]\]X<kÂKLA :AEa]TdH D/?IF:KLACK<DP?

λUf(X) ≤ g(X)

zP~

g$DPBV/HUHUTU\:AEKL?]Vvº,VX<G]TNoTH D/uyTÛϕ(X) ≤ ess inf

x∈Aϕ(x) :9QD

a]TBCT/9`DlPT?]TUBWV,AETdDP?IT¼BWV?IF]DPuº,VX<G]TÀta]KwHWaMKwÔF:K<^nACBEK<o]G:ACTF£V/HUHUDPBWF:KL?IACD

fTÀ?ITUTFMKL?MACa]TvuyTVP?

1/∫

AϕdP

º,VPXLG]T^tta]KwHWaVBET+F]K<^nACBEKLoIG:ACTFVPHUHUDPBWF:K<?]ÀAEDϕ9I«]BEDPu¶ACa]Td\IBCDjDgVoD,ºPT+TN³0?ID,ACa(V,AK<?

ACaITNACa]TDPBETUu ACa]TÀ?0GIuÀoTUBNDPg|?]THUTÊ^CVPBCkÂK<?]\]G]Am^CVPuy\]XLTÀºVPXLGIT^Kw^t/TUDPuyTUACBEK<HVX<XLkF:K<^nACBEK<o]G:ACTF¢tKAEa

\IVPBEVPuvTUACTB1/λ

9]`ba]TNuyTV?º,VX<G]TNKwÊ(N) = λ

9ijDKL?ACaITNuvTV?¢TOa(VºPTmACDPT?]TUBWV,AET

m =1

∫

A ϕdP· λ =

1

P (A) ess infx∈A ϕ(x) :9 d

º,VX<G]T^mF:Kw^xAEBCK<o]G:AETF¹V/HUH D/BEF]KL?]yAEDϕK<?¹DPBWF:TUBACDl/T AfD/?]Tdº,VX<G]TvF:K<^nACBEK<o]G:ACTF¹VPHH D/BEF:K<?]yAED

g9= g

ϕK<^

H D/?IH T?0ACBWV,ACTFÂ?]TVBAAEa]K<^fHUV?oTNuÀGIHWao(TUAnAETUBtAEaIV?ACaITNºVPXLGIT

1/P (A)g$BCD/u ACa]TN?(VK<ºPT+VX<PDPBEKLACa]u¢9

½m¾ 5 ²±.1 & & £TyHUVP?MGI^CTÀAEa]TyBCTnxTH ACK<DP?¹uyT AEa]D:F¹AED^CK<uÀG]XwV,AETvVBCD,t?]KwV?¹¸¹DACK<DP?£DP?MACa]TACK<uyTÂK<?/AETUBEº,VX

[0; t]HUDP?IF]KAEKLD/?]TF DP? AEa]TTUº/TU?0ANACaIVA ∫ t

0 B2s ds < ε

g$DPBdV^CuVX<X[º,VX<G]TDgεVP?IF

|Bt| ≤ cg$D/B^nD/uyT

c > 09

THVGI^CT+ACa]TÀKL?0ACTPBWVXHUDP?IF]KAEKLD/?K<^fDP?]X<kÊV,AEK<^ ITFg$DPB\IVACaI^fta]KwHWa¹^xAWVkuyD/^nAfDPgAEa]TNAEKLuyTÀ?ITVBACaITÀDPBEK<PK<? ]TdGI^CTÀV? µmBE?I^nACTKL? ¡aIXLT?0oTHW³\]BEDjHUTÊ^bACD^CVPuv\IXLTNAEa]TÀD/BCK<PK<?IVPX-BEVP?IF:D/u¶\IVACaI^9«IBCD/ug$DPBEuÀGIX<V _I9 _d fTy³j?]D,ÁCa(V,ANACaITyF]TU?I^CKAxkMDgtV?MD/?]T F:K<uyTU?I^CKLD/?IVX%µmBE?I^nACTUK<? ¡a]X<TU?joTHW³\]BED:H T^C^tKLACa¢\IVBWVuyTUACTUB

α > 0K<^

ϕt(ω) = exp(α

2(t− ω2

t ) − α2

2

∫ t

0

ω2s ds

)

g$DPBVPXLXω ∈ C([0; t], )

9µm?lAEa]Td^CT A

Aε =

ω ∈ C([0; t], )∣

∣

∣

∫ t

0

ω2s ds < ε, |ωt| ≤ c

T (?IFess infx∈Aε

ϕt(x) = exp(α

2(t− c2) − α2

2ε)

.

`bajGI^fg$BCD/uÁHUDP?IF:KLACK<DP? j9 0 tTvH DP?(H X<GIF:TIACaIVAmT¼^Ca]DPGIX<FV/HUH T\:AOVÂ\IV,AEa¹Dg|AEa]TyµmBE?I^nACTUK<? ¡a]X<TU? oTHW³Â\]BED:H T^C^ :KgKLAK<^bK<?Aε

V?IFVPFIF:KAEKLD/?IVX<X<kl^CVACKw^ (T^

U exp(α

2(t−X2

t ) − α2

2

∫ t

0

X2s ds

)

≤ exp(α

2(t− c2) − α2

2ε)

DPBtT0G]KLº,VPXLT?/AEXLkU ≤ exp

(

−α2

(c2 −X2t ) − α2

2

(

ε−∫ t

0

X2s ds

)

)

.

TdVP?0AtACDÂ³/TUTU\AEa]TduyTVP??0GIuÀoTUBDPg|ÊVuy\]X<T^GI^nTFg$BCD/u T0GIV,AEKLD/?¹:9 ^CuVX<X'9fÊ^nGIuvTc2 <

t]?]D,N9]THUVG(^nTODg

m =1

P (Aε) ess infx∈Aεϕ(x)

=1

P (Aε)exp(α2

2ε− α

2(t− c2)

)

=1

P (Aε)exp((

α√

ε/2 − (t− c2)/√

8ε)2

−(

t− c2)2/8ε)

TOtK<X<XZACa]T?HWa]DjD/^CTα =

t− c2√8ε

·√

2

ε=t− c2

2ε

z];

0 0.2 0.4 0.6 0.8 1

−1

−0.5

0

0.5

1

t

x

«KL/G]BETj9Q8 W . N V2L + VS. -%? 5 =P. CE5F / 6 ./ NT50-76K5 / 5 / -I?S+6 / - +"CUf .0L [0; 1] : 45 /H1 6I-J6K5 / + 1 5 / -I?S++"f+ / --%? . - ∫ 1

0 B2s ds ≤ 0.01 ./21 |B1| ≤ 0.5 QOR ?6 9 Â 8 CE+ZF .9 4"CE+ . -#+ 1 FG6I-I?j-I?S+ZCE+ +`4"-J6K5 / N+<-%?@5 1P1 + 9 4"CU6 3 + 16 / + . N VHL + Q Q

ACDPTUAfAEa]T¼D/\:ACK<uVXuyTV?£?jG]uÀoTUBODgÊVuy\]X<T^mGI^CTF¢AED\IBCD:F:GIHUTÀD/?]TÀ\IVACaMg$BCD/u AEa]TyH DP?(F:KAEKLD/?]TFBCD,t?IK<VP?luyDAEKLD/? :ta]KwHWaKw^

m∗(ε) =exp(

−(t− c2)2/8ε)

P (Aε).

¡f^CK<?]yXLTuvuVyr(9 _¼TdHV?¢H D/?IH X<GIF:T

limε↓0

ε logm∗(ε) = −(

t− c2)2/8− lim

ε↓0ε logP (Aε) =

t2 − (t− c2)2

8.

ijDACaIT?0GIuÀoTUBNDg?]THUTÊ^CVPBCk¹ÊVuy\]X<T^N^nACK<X<X|PBED,^mT:\(D/?]TU?0ACKwVX<X<kg$DPBε ↓ 0

-oIG:AdtKAEa V¢oT ACACTBT:\DP?]T?0ACKwVXZBWV,ACTOAEaIV?ACa]TND/BCK<PK<?IVPX

t2/89

`DK<XLX<GI^nACBWV,AET¼AEa]TyTTH A+THUVP?¹ACBEkAEa]K<^+tKLACat = 1

c = 0.5

ZVP?IFε = 0.01

9(=O^nK<uÀG]XwV,AETF;~/~P~/~P~P~y\(V,ACa(^mtKLACa*^nACTU\ ^nK UT

∆t = 10−5 TVPHWa-9Z`ba]TvBET^CG]XAW^+VBET¼^CG]uyuyVPBCKw^CTFK<?MAEa]T¼g$DPX<XLD,tK<?]AEVPo]X<TP9µm?]T+DPgACaITNBCT^nG]XLACK<?]y\IVACaI^tKw^t^Ca]D,t?K<? IPGIBCTdj9Q8:9?IVPKLº/TOuyTUACa]D:F BETxxTHJACK<DP?uyT AEa]D:F

K<?]\]G:AfÊVuy\]X<T^ ;~P~P~/~P~/~ ;~P~/~P~/~P~ÊVuy\]X<T^tV/HUH D/BEF]KL?]¼ACD

f

;_P~/ ];_V/HUH T\:ACTFÊVuy\]X<T^ ~ zI;Ur

ijDACaIK<^NKw^+DP?ITvDPg%AEa]THUVP^CTÔta]TBCTvAEa]TBCTnxTH ACK<DP? uvTUACa]D:F£D/BC³:^+0G]KLACTyTXLX Zo]G:ANAEa]T?IVPKLº/TvVP\ \]BED/V/HWaÂgVK<XwÛ9

" " ¡f^CK<?]ACa]Ts[GIXLTB ¸¹VBEG]k0VuVÂg$BEDPu ^nTHJACK<DP?j9<;yD/BC³:ÔTUX<Xg$DPBNDPBWF:K<?IVBEk¹^nACD:HWaIVP^nACKwHyF:K TBCT?/AEK<VPXT0GIVACK<DP?I^ IoIG:AmKLAmF:DjT^f?]DA+VX<XLD,ACD^CVPuv\IXLTÀVÂ\]BCD:HUTÊ^H D/?IF:KLACK<DP?]TF¢DP?¹V/KLº/TU?º,VX<G]TNg$D/BACaITÀTU?(F\DPK<?/A9I=@?D/BEF:TBbACDl^CK<uÀG]XwV,AET+ACa]Td\IBCD:H T^C^fH DP?(F:KAEKLD/?]TFD/?AEa]TdT?IF¢\(D/KL?0A Vo]BEK<F]PT_ ]TN?]TUTF¢uvD/BCT^CDP\]a]Kw^nACKwHUV,AETFuyT ACaIDjFIÛ9I`baIK<^t^CTH ACK<DP?¢F:TÊH BEKLoT^t^CGIHWaVvuyT ACaIDjF-9

`ba]TvoIVP^CKwHÀ\]BEK<?IH K<\]X<TvG(^nTF¹a]TBCTvK<ÔAEa]TcVP?]PTº0K<?MuyTUACa]D:F -m^C^CG]uyTÀAEaIV,ANTyVP?/AmACDÊVuy\]X<Tg$BEDPu Vy/KLº/TU?\]BEDPoIVPo]KLX<KLAxkF:Kw^xAEBCK<o]G:AEKLD/?

µtKLACaF:T?I^CKAxk

ϕD/? d 9(`ba]TU?TdH D/?I^nKwF:TBbACa]TÀ^nACD:HWaIV/^xAEK<HF:K TBCT?0ACKwVXZT0GIVACK<DP?

dZt = grad logϕ(Zt) dt+√

2 dBt. :9 zd «]BEDPuÁAEa]TUD/BCTu ;/9 rT¼³j?]D, IAEaIV,AmACa]KwÔ\]BEDjHUTÊâIV/^µVP^mKAW^m^nAEVACK<DP?IVPBCk¢F:Kw^xAEBCK<o]G:AEKLD/?-9ZfÊ^nG]uyK<?]

TUBEPD:F:KwH KLAxkÂg$D/BZTvHUVP?¹V\]\]BED:KLuVACT+ACaIT¼F:Kw^xAEBCK<o]G:AEKLD/?

µojk^CK<uÀG]XwV,AEKL?]Vl^CDPX<G:ACK<DP?DPg :9 zd tV?IF

z/8

AEVP³jKL?]Ztg$DPBOX<VPBC/T

tVP?IFTÀHUVP?MV\I\]BCD:K<uV,ACTNT:\(THJAEVACK<DP?I^ ∫

f dµojk?jG]uyTBCKwHUVPXLX<kTº,VX<GIV,AEKL?]

1T

∫ T

0 f(Zt) dtg$D/BtX<VPBC/T

T9]efTUAEVK<Xw^VoDPG:AbAEa]K<ûvTUACa]D:F¢HUVP?oT+g$DPGI?IFKL? ªt·z/z 9

«]D/BmHUDP?I^nAEVP?/AW^c ∈ TÀaIVº/T

grad log(cϕ) = grad(log c + logϕ) = grad logϕK©9 T/9T¼HV?

F:TUACTUBEuyKL?IT+ACa]TdF:BEKLgAg$D/BtACaITNi]efs j9 zM TUº/TU?Kg|TOD/?]X<kÂ³j?]D,AEa]TdF:TU?(^nKLAxk ϕ GI\AEDVvHUDP?I^nAEVP?0A9 D, VPÊ^nGIuvTOTNVP?/AACDÂ^CK<uÀG]XwV,AET+^CDPX<G:AEKLD/?I^bDgdXt = f(Xt) dt+ dBt

g$DPB0 < t ≤ T

X0 = a ∈ , :9L;~M ta]TBCT

BKw^dV

1 F:K<uvT?I^CKLD/?IVX|BCD,t?IK<VP?£uyDAEKLD/?²V?IF f : → Kw^d^CDPuyTF:BCKLgAdg$G]?(HJACK<DP? o]G:AH D/?IF:KLACK<DP?]TF¢DP?

XT = b ∈ 9 T¼HV?PTUAOVP?¹V\]\]BED:KLuVACK<DP? (X0, X1, . . . , XN ) ∈ N+1 DgAEa]TG]?IHUDP?IF]KAEKLD/?]TF¹^CDPX<G:AEKLD/?º0KwVs[GIXLTBmuyT AEa]D:FZ9X = (X1, . . . , XN )

KwÔVBEVP?IF:D/u º/THJAEDPBK<? N tKAEaF:T?I^nKLAxk

ϕ(x1, . . . , xN ) =1

(2π∆t)N/2exp(

−N∑

n=1

(

xn − xn−1 − f(xn−1)∆t)2

2∆t

)

ta]TBCT+TOG(^nTx0 = a

V/^bV?Vo]oIBCTº0KwV,AEKLD/?-9:`ba]TNH D/?IF:KLACK<DP?IVPX-F:TU?I^CKLAxklDgACa]TNº/TH ACDPB(X1, . . . , XN−1)

H D/?IF:KLACK<DP?]TFlD/?

XN = bKw^bAEa]TU?

ϕ(x1, . . . , xN−1|xN = b) = c exp(

−N∑

n=1

(

xn − xn−1 − f(xn−1)∆t)2

2∆t

)

ta]TBCTD/?MAEa]TÂBEKL/a/ANaIVP?IF*^nKwF:Tx0 = a

xN = b

-VP?IFcK<^+AEa]T?]D/BCuVX<Kw^nK<?]H D/?I^xAWV?0A ZtaIK<HWa

uV³/TÂCaIT+g$G]?IHJAEKLD/?V¼\IBCD/oIVo]K<X<KAxklF]TU?I^CKAxkV/VPKL?9 £TNbV?0AbACDV\I\]XLkÂACa]TdcV?]/TUºjKL?uyT AEa]D:FlAEDyACa]Kw^\]BEDPoIVPo]K<XLKLAxkÂF:TU?(^nKLAxkP9

emTI?]TI(x1, . . . , xN−1) =

N∑

n=1

(

xn − xn−1 − f(xn−1)∆t)2

2∆t,

ACaITU?¹AEa]TyF:BCKLgAOg$DPBmACa]TycVP?]PTº0K<?¹T/G(V,ACK<DP?MK<^grad logϕ( · |xN = b) = − grad I

V?IFMT¼PTUAfACaITcVP?]PTºjKL?T0GIVACK<DP?

dZs = −∇I(Zs) ds+√

2dBs.«]D/BfV(N − 1) F:K<uyTU?I^CK<DP?IVPXBCD,t?]KwV?uyDAEKLD/? B 9(=G(^nT s g$DPBAEa]TNAEKLuyTÀK<?¢ACaITÀcVP?]PTº0K<?¢T0GIVACK<DP?a]TBCTOACDÂF]K<^nACK<?]PGIK<^CaKLAtg$BEDPu AEa]T+ACK<uyT

tK<?ACa]TÀi:ems :9L;~M W9THVGI^CT¼aITUBETvTVBET¼D/?]X<k¹KL?0AETUBET^nACTF£K<?£AEa]TÂ^nAEV,AEKLD/?IVBEk¹F:Kw^xAEBCK<o]G:AEKLD/?£DPgAEa]Kw^di:efs TyHV?

g$BETUTUX<kBCT^CHVX<TmAEKLuyTdK<?AEa]TÀcVP?]PTºjKL?¢T0GIVACK<DP?-9(tgACTBAEa]Ts ACK<uvTdACBWV?I^ng$DPBEuV,ACK<DP? Zs = Zs/∆t

V?IFBs = Bs/∆t

T+/T AbACa]TÀi]efs

dZs = − 1

∆t∇I(Zs) ds+

√

2

∆tdBs. :9L;/;_

ta]TBCTBKw^tV?]DPACa]TB

(N − 1) F]KLuyTU?(^nK<DP?IVPX-BCD,t?]KwV?uyDACK<DP?9]´F:KLBETH AHUVXwH GIX<VACK<DP?PK<ºPT^

− 1

∆t∂nI(x) =

xn+1 − 2xn + xn−1

∆t2

− f(xn)f ′(xn)

− f(xn) − f(xn−1)

∆t+ f ′(xn)

xn+1 − xn

∆t :9L;8D

ta]TBCTvTyV0VK<?¹GI^CTvACa]TVoIo]BCTºjK<VACK<DP?I^x0 = a

V?(FxN = b

9Z`ba]TyBCTVP^CDP?g$D/BOBETÊHUVPXLK<?]lACa]Ts ACK<uyTÀKw^(ACa(V,Am?ID,AEa]TIBE^nAAETUBEuÁD/?¢ACa]TvBEKL/a/AmaIV?IFM^nKwF:TdKw^mVlF:K<ÊH BET AEK TFº/TUBW^nK<DP?¢DPg|ACa]TvcV\IX<V/H T

DP\TUBWV,AEDPBVP?IF¢DP?]T¼H DPGIX<F¢a]D/\(TdACaIVAg$DPBN → ∞ AEa]T

(N − 1) F:K<uyTU?I^CK<DP?IVPXi:ems :9L;/;_ tHUDP?jºPTBC/TÂCDV^nACD:HWaIV/^xAEK<Hm\IVBCACKwVX-F]KTUBETU?0ACKwVXZT0GIV,AEKLD/?D/? +× [0;T ]9

`ba]T¼uvTUACa]D:F¢AED^CKLu¼G]XwV,ACTvVl^CDPX<G:ACK<DP?XDg|ACaITyi:ems j9 zM fH DP?(F:KAEKLD/?]TF¹DP?AEa]TÀT?IF¹\DPK<?0A XTDPBE³j^?]D,´VP^tg$D/XLX<D,^9(«KLBW^nAmHUVPX<HUG]X<VACTNAEa]T¼HUDPBEBCT^n\DP?(F:KL?IyF]BCKLgAmg$D/BAEa]T¼cV?]/TUºjKL?¢T/G(V,ACK<DP?G(^nK<?]

zP_

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

t

Xt

«KL/G]BETj9 _ R ?a6 9 Â 8 CE+ 9 ?S5F 9.VS. -%? 5 =i.0/ C /@9 -#+<6 / ? L + /H3 +4 \V CE5 4+ 99 FG6I-%? VS. C . N+<-#+<C α = 5./H1j9 - . CU-B6 / 2 : 45 /21 6I-J6K5 / + 1 5 / X1 = 5 Q R ?@+ 9 6IN 8aL . -76K5 / F .9i1 5 / + 89 6 / A -%?@+* ./ Ad+"f 6 / NT+"-I?S5 1= CE50N + . N VHL + Q Q R ?@+ 9 6IN 8aL . -J6K5 /VS. C . N+<-#+<C 9e. CE+ N = 1000 : ∆t = 0.001 :.0/21 ∆s = 5 · 10−7 Qg$DPBEuÀGIX<V :9L;8D W9`baITU? GI^nTvAEa]K<^NF]BCKLgANACD^CKLu¼G]X<VACTyV¢^nD/XLG:AEKLD/? Z DPg%ACaITli:efs j9<;P;_ J9`ba]TyKL?IKAEK<VPXº,VX<G]T

Z0K<^+VBEo]KLACBWVBEkP9Zµm?]TvHUDPG]XwF g$DPBOT]VPuv\IXLTG(^nTvXLK<?]TVBOKL?0ACTBC\DPXwV,AEKLD/?MoT AxTUT?¹ACaIT

aV?(F

b9

D,´/T A Zsg$D/BOVlX<VPBC/TNACK<uyT

s9Z`ba]TU?MACa]TyH D/uy\(D/?]TU?0AE^

Zs,1, . . . , Zs,N−1VBET¼VP?MV\I\]BCD:K<uV,ACK<DP?

g$DPBX1·∆t, . . . , X(N−1)·DT

9µfg|H DPGIBE^CTfAEa]TN\]BEDPo]X<TUu¶tKAEaAEa]Kw^tuyT ACaIDjFKwÂCD I?(F/DjDjFº,VX<G]T^bDPgs9

µm?]T¼uvTUACa]D:F¹ACDl?jG]uyTBCKwHUVPXLX<k^CDPX<ºPTÀACa]Ti:ems j9<;P;_ K<^fACDGI^CTdAEa]T¼s%G]XLTB ¸¹VPBCG]k0VuVyuyT AEa]D:FZ9 DPACTÀACaIVAfg$DPBfACa]KwÔuyT ACaIDjFM/GIKAET¼Vl^CuVX<X^nACT\M^nK UT¼KL? s F:K<BCTHJAEKLD/?K<^m?]THUTÊ^CVPBCk]K<?¹DPBWF:TUBfACDÂ³/TUT\ACaITÀuyT AEa]D:FM^nAEVoIXLT/9(= g|AEa]T¼BEG]?]?IKL?]lACK<uyTÀDg|AEa]Tv\]BCD/PBWVu K<ÔV?¹K<uy\(D/BnAWV?0AgV/HJACD/B (ACaITU?¹uyT AEa]D:F]^X<KL³/TNACa]TÀKLuy\]X<K<HUKAOs[G]X<TUB ¸MVBEG]k/VPuyV¼uyT AEa]D:FDPBtAEa]Tv·BWV?]³ @ KwHWa]DPXw^CDP?ÊHWa]TuvTÀta]K<HWa¹VPXLX<D, /BCTV,AETUB^nACTU\^CK T^bK<?

s F:K<BCTHJACK<DP?uyKL/a0Ao(TdV/F:º,V?0AEVPPTUD/GI^9½m¾ 5 ²±.¹1 &< µmBC?(^xAETUK<? ¡fa]X<TU?jo(THW³o]BEK<F:/T^ J9 THUVP?MGI^CTvACa]TyuyT AEa]D:F£F:T^CHUBCK<oTF£KL?£ACa]Kw^

^CTHJAEKLD/?MAED^nK<uÀG]XwV,AETy\IV,AEaIÔg$BCD/uACa]TµmBE?I^nACTKL? ¡aIXLT?0oTHW³\IBCD:H T^CÔtKLACa*PK<ºPT?MK<?]KLACKwVX%VP?IF (?IVXº,VX<G]T^9

·DP?(^nKwF:TUBAEa]TÀi:emsdXt = −αXt dt+ dBt

g$DPB0 < t < 1tKLACa

α > 0VP?IF¹oDPGI?IF]VBEk¢H DP?(F:KAEKLD/?I^

X0 = aV?(F

X1 = b9Z`ba]T?MTÀaIVº/T

f(x) = −αx V?(FT0GIVACK<DP? :9L;8D o(TH D/uvT^

− 1

∆t∂nI(x) =

xn+1 − 2xn + xn−1

∆t2

− α2xn

+ α−xn+1 + 2xn − xn−1

∆t

= (1 − α∆t)xn+1 − 2xn + xn−1

∆t2− α2xn.

«KL/G]BETd:9 _y^Ca]D,ÂCa]TNBET^CG]XLAtDg|V¼?jG]uyTBCKwHUVPX-^nK<uÀG]XwV,AEKLD/?G(^nK<?]vACa]Kw^tuyT AEa]D:FZ9

zr

t§

1AACa]TNK<?IF:KwHUVACD/Bbg$G]?IHJAEKLD/?¢DgAEa]Td^CT A

AIK'9 TP9

1A(x) = 1Kgx ∈ A

VP?IF1A(x) = 0

TX<^CT(A)

ACa]TNK<?IF:KwHUVACDPBbg$GI?IHJAEKLD/?DPgAEa]Td^nTUAA]G(^nTFlKLg

AKw^V?¢H D/uy\]XLKwHUVACTFlTj\IBCT^C^CKLD/? \IV/T+q08D

Bδ(x)ACa]TND/\(T?o(VX<XZtKAEaBWVPF]KLGI^

δVBEDPGI?IF

x

B(λ, t)ACa]TbVkPT^fBEKw^n³¢g$D/BmAEa]T^nT\IVBWV,AEKLD/?¹DgAxD\]BED:H TÊ^CT^ (D/oI^nTBCº/TFGI?/AEKLXAEKLuyT

tg$D/B+V\]BEKLD/BCk

F:Kw^xAEBCK<o]G:AEKLD/?λ \-9PD

B( d)ACa]TdD/BCTX σ VPXL/TUo]BWVND/? d

C0([0; t], d)ACaITd^n\IV/H T+DgVX<XH D/?0ACK<?0GIDPGI^g$G]?IH ACK<DP?I^tg$BCD/u ACa]TNK<?0ACTBCº,VX

[0; t]KL?0ACD d \IVPPTd8~d s :\

(λ)AEa]TNT:\DP?]T?/AEK<VPX-F:Kw^xAEBCK<o]G:AEKLD/?tKLACa¢\IVPBEVPuyT ACTB

λ

IdACa]T

d× dKwF:T?/AEKAxkluyVACBEK

KrACa]TdHUXLD0^nTFlo(VX<XZtKAEaBWVPF]KLGI^

rVBEDPG]?(FÂAEa]TNDPBEKL/KL?

Kr = x | |x| ≤ r 9λd ACa]T d F:KLuyT?I^nK<DP?(VX-c-To(T^n/G]T+uyTVP^CG]BET \-9-;qM g ACa]TN?(V,ACGIBEVPXZ?0GIuÀoTUBW^

1, 2, 3, . . .

g 0ACa]TN?(V,ACGIBEVPXZ?0GIuÀoTUBW^bKL?IHUXLG(F:KL?I¼AEa]T UTBCD

0, 1, 2, . . .

N (µ, σ2)AEa]TÀR+VG(^C^CK<VP?F]K<^nACBEKLoIG:ACK<DP?¢tKLACa¢T:\THJAWV,AEKLD/?

µV?IFº,VBEK<VP?IH T

σ2

Prob( d)ACa]Td^C\IV/H TODg|VX<XZ\]BCD/oIVoIKLX<KAxkluvTVP^CG]BET^DP? d

+

ACa]TN\D/^CKLACK<ºPT+BCTVXZ?jG]u¼o(TBE^ :K'9 TP9 + = [0;∞)

ρ(A)AEa]Td^C\(THJACBWVXZBWVPF]KLGI^DPgAEa]TNuV,ACBEK

A]K©9 T/9jACa]TNuV jK<uÀGIu¶DgACa]TdVPoI^CDPX<G:ACT+º,VX<G]T^bDPgAEa]TNTUK<PT? º,VX<G]T^bDPg

A

s ∧ t ACa]TNuyK<?]KLu¼G]u¶DPgAEa]TN?jG]uÀoTUBW^sVP?IF

t

s ∨ t ACa]TNuV :K<uÀG]u¶DgACa]TN?jG]u¼o(TBE^sVP?IF

t

V ba (f)

AEa]T+ACDPAEVXºVPBCKwV,AEKLD/?lDPgACaIT+g$G]?IHJAEKLD/?foT AxTUT?

aVP?IF

b \-9(q/zM ACa]T K<TU?ITUBtuyTV/^nG]BET+DP?ACa]TN\(V,ACa^C\IVPHUT ε

ACa]TNXwV DPgÊHUVPXLTFF:D,t?¢BCD,t?IK<VP?luyDAEKLD/? \-9(8~d bxc ACa]TNXwVBEPT^xAbKL?0AETUPTB^nuVPXLX<TUBbDPBtT0GIVXAED¼AEa]TNBETVXZ?jG]u¼o(TB xdxe ACa]Td^CuVX<XLT^xAtK<?/AETU/TUBtPBETVACTBD/BtT0GIVXAEDyACa]T+BETVPXZ?0GIuÀoTUB xx+, x−

AEa]TÀ\D/^CKAEKLº/Td\IVPBnAfBET^C\-9(?]TU0V,ACK<ºPTN\(VBCAfDgx ∈ 9`ba]Kw^fKw^fF:TI?]TF¹VP^ x+ = max(0, x)

V?(Fx− = max(0,−x) 9

z/

zPq

?(F:TUBW^nD/?- ^bKL?]T0GIVX<KAxk _];bVkPT^BEKw^n³ P

g$D/BH DP?(^xAWV?0AbF]BCKLgA ,q

g$D/B¸¹VBE³PD,ºHWaIVPKL?I^ 08

g$D/BmµO¡\IBCD:H T^C^CT^ oTVG]ACKLg$G]X-\]BEDjDg ;Uro]BEK<F]PT^ zI;·VuyTBCD/? ¸MVBCACK<? «IDPBEuÀG]XwV 8/H D/?0ACK<?0GIKAxkl^CT A";zH D/?0ACBWVPHJAEKLD/?l\]BEK<?IH K<\]X<TS;F:K G(^nK<DP?H DjT"!HUKLT?0A F:K G(^nK<DP?¢\]BED:H TÊ^

IF:TU?(^nKLAxkÂDg zF:BEKgAtg$GI?IHJAEKLD/?

IH D/?I^xAWV?0A z(qTUuy\]K<BEK<HVX-F:Kw^xAEBCK<o]G:AEKLD/? ;zs[G]X<TUB cV/BEVP?]PTmT0GIVACK<DP?I^ rj8s[G]X<TUB ¸¹VBEG]k/VPuVduyT AEa]D:F /«]BETUKwF:X<KL? TU?0ATUX<X`ba]TDPBEk 8P8ROK<BEÊV?]D,º «]D/BCu¼G]X<V¬ K<uv\DPBCAEVP?IH TNÊVuy\]X<KL?I /qcVP?]PTºjKL?uyT ACaIDjF zI;cVP\]XwVPH TO\]BEKL?(H K<\]XLTS;qcVP\]XwVPH TmACBWV?(^xg$D/BCu 80cemY ;P;

]TV³ ;/;

g$D/BtTUuy\]K<BCKwHUVPX-F:K<^nACBEK<o]G:ACK<DP?(^ ;z

g$D/Bt\IV,AEaI^btKAEa^nuVX<XL2 ?IDPBEu _];

g$D/B^CVPuv\IXLT+\IVACaI^ 8P~]8P8

g$D/B^xAWV,ACK<DP?(VBEkÂF:K<^nACBEK<o]G:ACK<DP?(^ ;

g$D/B^xAEBCD/?]vF]BCKLgA 8P]qPz(:;I,_c-K<\IÊHWa]KLANH D/?IF:KLACK<DP? D:HUHUG]\IV,AEKLD/?uyTVP^CG]BCTjTUuy\]K<BCKwHUVPX;zµmBE?I^xAETUK<? ¡a]X<TU?joTHW³\]BED:H TÊ^ ;~I8_

I^nT\IVBWV,AEKLD/?lDPg P

HUDP?IF]KAEKLD/?]TFDP?ACaITNTU?IF:\DPK<?0A zP_

tKLACa¹^nACBEDP?IvF:BEKLgA 8,r\IBCDPxTHJAEKLº/TmX<K<uvKLA ;qBWV,AETOg$G]?IHJAEKLD/? ;P;

]/D0D:F ;P;BETxxTHJAEKLD/?uyT AEa]D:F PzBETUº/TUBW^nK<o]X<TOF]KGI^CKLD/? ^CuVX<XZ?]DPKw^CT>8P~I8/8^nAEVACK<DP?IVPBCkÂF]K<^nACBEKLoIG:ACK<DP? Z;q

DPgAEa]T¼µO¡ \]BEDjHUTÊ^ 8_^nACD:HWaIV/^xAEK<Hms[G]X<TUBfuvTUACa]D:F 0^nACBEDP?IvF:BEKLgA";qI8r]_jAEa]TUD/BCTu ]`VPG]o(TBCKwV? 8/

DPgeOV^CDP? R BCAC?]TB ;q

F:TdBCG]K x? 8/

fD/XLuyDP/DPBED,º4

i:HWa]K<X<F:TB 8~AEKL/a0AC?]T^C^ :T:\DP?]T?/AEK<VPX ;8AEBEV/H T+DgVv\IV,AEa r/rGIPX<klHUVPX<HUG]X<VACK<DP? ~[VPBEV/F:aIVP?c-TuyuyV ;q

z0

zP

b

?IFI/ `+9 ´9Z?IF:TBE^CDP?9¹`ba]TyKL?0ACTPBWVXDg^CkjuyuvTUACBEK<H¼G]?]K<uyD:F:G]XwVBmg$G]?IHJAEKLD/?I^+D,ºPTBmV^nkjuyuyT AEBCKwHH D/?0º/T^CT AVP?IF¢^nD/uvT+\]BEDPo(Vo]K<XLKLAxkK<?]T0GIVPXLKLACK<T^9 OCE5_4`++ 1 6 / A 9 5 = -%?@+ N+<CU6K4 .0/ . -I?S+<N . - 6K4 .L WH5_4"6K+<- > Iq<;~;,qI;z/P]9

BE?rD c-GIF]tKL*BE?]DPXwFZ9 W2- 5_4`? .9 -J6K4Xh6 +"CE+ / -J6 .L Y m 8S. -76K5 /@9"c R ?@+50C > .0/21 VDVHL 6K4 . -J6K5 /S9 9 D/a]? KLX<TUk ijD/?I^ -;zjrI9

bR+`t j_ 9 d9IK<?]PaIVPu ·m9I¸²9(ROD/X<F]KLTIV?IF 9Ic[9I`TUGIPTUXw^9 + A 8aL . C . CU6 . -76K5 / 9·Vu¼o]BCKwF:/T+¡f?]K ºPTBE^CKAxklY[BCT^C^ Z;z/ 0:9 ªf Pz R+VBEBCTUAnAmK<BC³ja]D VP?IF¹ROK<VP? ·VBEXLDªtDPAEV]9 C 1 6 /2. C > Xh6 +"CE+ / -J6 .L Y m 8S. -J6K5 /S9 9 D/a]? KLX<TUk

i:DP?I^ jg$DPG]BCACaTF]KAEKLD/? -;z/ PzI9 bi]zPq ?IF]BCTK 9(D/BCD:F:K<?¢V?IFY|VPVº/Dyi:VPXLuyK<?]TU?9 .0/21M3 5_5 5 = CE5F / 6 ./ 5-J6K5 / . 4"- 9e./H1

50CUN 8aL . + 9[Y[BEDPoIVPo]KLX<KLAxklV?IF= AE^f\]\]X<KwHUV,AEKLD/?I^9]K<BE³0a GI^CTUB :V/^nTX Z;zPz/q]9 eOi: /z 9 eÀ9IemTUG(^CHWa]TX-V?IF¢e¼9 ´9i0ACBEDjDjHW³9 * . C7Ad+ X+<f_6 . -J6K5 /S9 9tHUVPF]TUuyK<HOY[BETÊ^ Z;zP /z]9 ez/ uyK<BdefTuÀoDVP?IF µfg$TB&TKAEDPG]?IK'9 * . C7Ad+ZX\+"f_6 . -J6K5 /@9 R +4? / 6Km 8 + 9\./H1 VDVHL 6K4 . -J6K5 /@9 -º/DPX G]uyTd_P vDg VMV2L 6K4 . -76K5 /@9 5 = . -I?S+<N . -J6K4 9 9bij\]BEKL?IPTUB :^nTH D/?IFlTF:KLACK<DP? -;zPz/ ]9 «]TUX$j;" KLX<X<K<VPu «]TXLX<TUB9 / / -7CE5 18 4"-J6K5 / - 5 OC`5 3.D3 6 L 6I- > R ?S+5C >P./H1 6I- 9 VDVHL 6K4 . -J6K5 /S9 (ºPD/XLG]uyT+=n=J9

D/a]? KLX<TUkijD/?I^ ]^nTH DP?(FlTF:KAEKLD/? Z;zjj;/9 «cY%zPz0 emVP?]K<TUX<XLT «X<DPBETU?(^ cVP?IF]VKw^V?(F G]kU? Y[aIVPu¢9¬cVPBC/T¹F:TUºjKwV,AEKLD/?I^K<?T^nACK<uV,ACK<DP? DPgdV?

µmBC?(^xAETUK<? ¡fa]X<TU?jo(THW³ÂuyD:F:TUX©9 Q VMV2L Q OC`5 3 Q (_Pq ; qP~IP (;zPzPzI9QR+«q/_ =J9I¸²9(ROTXgVP?IF¢V?IF¢i9@¼9I«]D/uyKL?-9 .0L 4 8aL 89 5 = . CU6 . -J6K5 /S9 9|Y%BCT?/AEK<HUT fVPXLX Z;z/qP_]9 DPXw~P~0 «]BWV?]³²F:T? fDPX<X<VP?IF:TUB9 * . C7Ad+TX\+"f 6 . -76K5 /@9 |ºPD/XLGIuvT£;Ur£Dg 6K+ L 19 /@9 -76I- 8 -#+ 5 / 5EADC .V ? 9 9uyTUBEKwHUV?¸¹VACa]TuV,ACKwHUVPXijD:H K<T AxkI8~P~/~]9 = Pz 9-=@³/TF]V¢VP?IF iZ9 VAEVP?IVoTP9 W'-#5 4? .9 -J6K4 1 6 +<C`+ / -76 .0L +`m 8S. -J6K5 /S9T.0/21 1 6 89 6K5 /V CE5_4+ 99 + 9 ºPD/XLG]uyTd8,ryDPg 5CU-%? h5 LIL ./H1 . -%?@+"N . -76K4 .L *;6 3 C . C > 9 DPBCACa D/XLXwV?(F Z;z/ Pz]9 +?0G( ];< efD/?IVXwFs9S+?jG:AEa-9 WH+"N\6 /'8 NT+"CU6K4 .L L Ad5CU6I-%?N 9 9[tF]F:Kw^nD/? T^CX<TUk]^CTHUDP?IFTF]KAEKLD/? -;z/ ];/9 +Y%zPz0 YT ACTBÀs9;+X<D0TF:TU?VP?IF*s%HW³jaIVPBEF Y[XwV,AETU?-9 8 N+<CU6K4 .0LG9 5 L]8 -J6K5 / 5 =9 - 5_4`? .9 -J6K4 1 6 +<C`+ / -76 .0L+m 8S. -76K5 /@9 9 G]uÀoTUBÂ8_MKL?h\]\IXLKwHUVACK<DP?I^¼Dgm¸MV,ACaITUuV,AEK<HÛ9|ij\]BEKL?IPTUB ;zPz/z]9 ·DPBEBETHJAETF

`ba]K<BEFY[BEKL?0AEKL?](9 di:z];" =J9 NVBWV,AVP^mV?IF i9 s9ija]BETUº/TP9 CE5F / 6 ./ 5-J6K5 / .0/21 W'-#5 4? .9 -76K4 .L 4 8aL]89 Zº/DPX<G]uyT;/;_Dg C .D108S. -#+ R + M- 9 6 / . -I?S+<N . -J6K4 9 9ij\]BEKL?]/TUB ]^nTH D/?IFlTF:KLACK<DP? Z;zPz];/9 Pq¸¹VX<ºjKL? À9NVPXLD0^NV?IF*Y|VPG]X<VÀ9 aIKAEXLD:HW³9 5 / - + . C L 5 +"-%?@5 19: 5 L]8 NT+ c .9 6K4 9 9

D/a]? KLX<TUkijD/?I^ Z;zP /q]9 z0_ 9H+BEDPoV?IF À9 º9£TK ^ /HW³PTB9Àµm?AEa]TdBWV,ACTÀDg[K<?:g$DPBEuV,ACK<DP?¢0VK<?KL?¹T:\(TBCK<uyTU?0AE^ftKAEaMV

I?]KLACTd\(VBWVuyT AETUBt^nTUA9 W'- . -J6 9 -76K4 9Z.0/21 X+4<6 9 6K5 /S9 -; 8P ];]8zPrI;zPzjj9 cVz/ ROBCTPD/BCkÂ«9IcVtXLTB9 / -7CE5 18 4"-J6K5 / - 5\W2- 5_4? .9 -J6K4 OCE5_4+ 99 + 9 9·a(V\]uV? fVX<X Z;zPz0:9

zPz

;~/~

c-Kw~];< £T?jo(DjÀ9-cK'9²ijuVX<X|oIVPXLX|\]BEDPo(Vo]K<XLKLACK<T^fg$DPBdR+VPGI^C^CKwV?£¸MVBE³PD,º\]BCD:HUTÊ^nT^mGI?IF:TUBOAEa]TLp?]DPBEu¢9 W'-#5 4? .9 -76K4 CE5 4+ 99 + 9Z.0/21 R ?@+"6IC VMV2L 6K4 . -J6K5 /S9 z08 0 ;~08 /8P~P~];/9

Y%` mz08 KLX<X<K<VPu d9ZY[BETÊ^ Zi]VG]X[À9`TUGI³PDPXw^C³0kZV?IF KLX<XLKwVu `+92|TUAnAETUBEXLK<?]I9 8 N+"CU6K4 .0L +`4"6 V + 96 / 9·Vu¼o]BCKwF:/TO¡f?]KLº/TUBW^nKLAxkY[BETÊ^]^CTHUDP?IFTF]KAEKLD/? -;z/z/8]9 ªt·z/z ·m9]Y9IªtD/o(TBnAfVP?IFRv9·VP^CTUX<X<VI9 5 / -#+ . C L 5 9 - . -J6 9 -76K4 .L N+<-%?@5 19 9%ij\IBCK<?]PTB Z;zPz/z]9 ªfiI8 ¸KwHWaIVTXªtTUTF¢V?IF¢bVBEBCkli:KLuyDP?9 +"-I?S5 19 5 = 5 1 +"C /. -%?@+"N . -76K4 .0L ? >9 6K4 9 (º/DPX<G]uyTN=J9tHUV/F:TUuyKwH+Y[BCT^C^ T [D/BC³ Z;zj8]9 ª +zPz0 emVP?]K<TUX[ªtTº0GvVP?IFM¸¹VPBEH [DPB9 5 / -76 /'8 5 89. CU-J6 / A .L + 9./21 CE5F / 6 ./ 5-J6K5 / 9 GIu o(TBv8PzP_KL?§ROBEG]?IF:X<TUaIBCT?*F]TUBd¸¹VACa]TuyVACKw^CHWaITU? K<Ê^CTU?IÊHWaIV,gAETU?-9ij\]BEKL?]/TUB -TUBEXLK<? ZAEa]KLBWF

TF:KLACK<DP? -;z/zPz]9Qi:HWaIz0"YT ACTBdi:HWa]TTUX©9 Y V 5 / + / -J6 .L 6 9 . -#+ 9 6 / Xh6 9 4<C`+<-#+ . C 50f 5 1 + L 9 9Y[aIe ACa]T^nKw^ ¡f?]K ºPTBE^CKA ,A NVKw^CTUBW^nXwVG:AETUBE? ;zPzjj9Qi+z eÀ9 ´9ijACBED0D:HW³MVP?IF*i9 ªd9 i9,[VBWVPF:aIVP?-9 8aL -J6 1 6IN+ /@9 6K5 /H.L1 6 89 6K5 /jV CE5 4+ 99 + 9 9 G]u¼o(TB8_/_¼K<?ROBCG]?(F:XLTa]BETU?F:TUB¸MV,ACaITUuV,AEK<ÊHWa]TU? Kw^C^CTU?IÊHWaIVgACTU?9Iij\]BEKL?]/TUB Z;zj,z]9 |D /zj_ D:HWa]T? |D (9 oTUB+F:K<T¼f^Ckjuy\:ACDPACK<³F:T^mbVk/T^CBCKw^nK<³PD0ô(TKemKGI^CKLD/?I^C\]BCDUT^C^CTU?-9demK<\]XLD/u VBEo(TKA I¡?IKLº/TUBW^nKLAA NVPK<^CTUBW^CX<VPG:ACTBC? ;z/z0:9 zP~ d9 º9 TUK ^PHW³/TUBÀVP?IF§Rv9 KL?I³0X<TUB9 W'-#5 4? .9 -J6K4 / - + ADC .L 9: / / -JCE5 18 4"-76K5 / 9ÁtF:º,V?IHUTF

c-THJACGIBCT^bKL?¸¹VACa]TuyVACKwHU^9afK<TUTU ]BWVGI?I^CHWajTUK< KLT^no(VPF:T? (;zPz/~]9

¯ %+. ¯ (

?IVPuvT D:HWa]T?|DoDPBE? I8zI9IijTU\:AETUu¼o(TBd;zj,~TUuVPKLX :ºPD0^CÔ^nTTUajG]a]?9 F:T

½O 5 5 ( ½m¾Z±%.0,.j/.µOHJA9I8~/~P_ G]?ITÀ8~P~Prv¡f?]K<ºPTUBW^CKAxkDPg *VBEtK<HW³

¸¹VPBCK<T ·G]BEKLT+ÊHWa]DPXwVBW^naIKL\ijT\:A9-;zPz/ fG]I9(8P~P~/_À¡?IKLº/TUBW^nKLAxkÂDg;NVKw^nTBE^CXwVG:AETUBE?s[uy\]X<D,kPTFMojkAEa]TÂemTU\IVPBnAEuvT?0A+DPgb¸MV,ACaITUuV,AEK<H^-¡f?]KLº/TUBW^nKLAxkDg NVKw^CTUBW^nXwVG:AETUBE?MVP^+VBET ^CTVPBEHWaVPÊ^nKw^xAWV?0A tKw^C^CTU?(^CHWaIVgACX<K<HWaITUBb¸KAWVBEo(TKAETUB k 9«]To-9-;zPzP ¸¹VPB9Z;zPzP s[uy\]X<D,kPTF¢ojklAEa]T¼emTU\IVPBnAEuyTU?0AfDg[¸¹VACa]TuV,ACKwHU^ (¡f?]K<ºPTUBW^CKAxkDg NVKw^CTUBW^nXwVG:AETUBE?V/^fVvAETVPHWa K<?]ÂVPÊ^CK<^nAEVP?/A tKwÊ^nT?I^CHWa(V,gACX<KwHWa]T fKLXLg^C³0BWV,gA k 9;z/z]; ;z/z0À¡?]K<ºPTBE^CKAxkDPg;NVPK<^CTUBW^CX<VPG:ACTBC?

CefK<\]X<DPu OK<?¸¹V,AEa]TUuVACKwHU^btKLACa^nTH D/?IF]VBEkyACD/\]KwH+Y[ajkj^CKwHU^`ba]T^nKw^ o(TBfF:K<TÀf^Ckjuv\]ACDAEKL³¢F:T^bVkPT^CBEK<^CKL³/D/ô(TKemKGI^CKLD/?I^C\]BCDUT^C^CTU? µm?¢ACaITÀf^Ckjuv\ AEDACKwHUVPX-TUaIVºjK<DPG]BbDgACaITdVk/T^ªtKw^n³lg$DPBemKGI^CKLD/?¢¸¹DjF]TUXw^`

;z/ ]; ;z/zP~vX<o(TBnAmi:HWajTUKLATUBROkjuv?(VP^CKLG]u SNVPK<^CTUBW^CX<VPG:ACTBC?

)N' 5 & DjHWaITU? |D (9 oTUB+F:K<T¼m^nkjuy\:ACDPACK<³F:TÔbVkPT^CBEK<^CKL³/D/^boTUKefK GI^nK<DP?(^n\]BEDTÊ^nT?-9 emKL\]X<DPuVPBCoTUKLA ¡?IKLº/TUBW^nKLAA NVPK<^CTUBW^CX<VPG:ACTBC? ;z/z0:9?(F:BCTVP^ |D H+XwVGI^mªtDAEa]TUBEuÀG]?(F D:HWa]TU?|D I9 =@?0AETUBE\]BCTUACK<?]ÂAEa]TÀ\(VBWVuyT AETUBW^Dg|AEa]T¼F]KGI^CKLD/?uyD:F:TUX I?¢Tuv\IKLBEK<HVXZº,VX<KwF]V,AEKLD/?-9

/ACDÂVP\]\(TVB9

kluedo.ub.uni-kl.de · /01 # , 243 57698;:=

Documents