A stochastic characterization of harmonic morphisms

Math. Ann. 287, 1-18 (1990) IlathmntmM AnnnR �9 Springer-Verlag 1990

A stochastic characterization of harmonic morphisms

Laszlo Csink 1, P. J. Fitzsimmons 2"*, Bernt Oksendal 3'** i Institute of Mathematics and Computer Seience, Kando KalmanCollege, Nagyszombat u. 19, H-1034 Budapest, Hungary 2 Department of Mathematics (C-012), University of California, San Diego, La Jolla, CA 92093, USA 3 Department of Mathematics, University of Oslo, P.O. Box 1053 Blindern, N-0316 Oslo 3, Norway

1. Introduction

The main purpose of this paper is to identify the harmonic morphisms between two U-harmonic spaces as the path-preserving functions between associated Markov processes. This identification paves the way for the use of stochastic methods in the study of harmonic morphisms.

To formulate our main result we need to recall the connection between U-harmonic spaces and Markov processes. Let (E, q/) be a U-harmonic space in the sense of Constantinescu and Cornea [CC2]. Thus E is a locally compact, second countable Hausdorff space, and q/:G~q/(G) (G open in E) is the sheaf of positive ql-hyperharmonicfunctions. See I-CC2, Chap. 2] for the relevant definitions. We assume that the constant function 1 is in q/(E), in which case there exists a bounded, continuous, strict potential p > 0 in q/(E); see I-CC2, Proposition 7.2.1]. If ue#l(E) is bounded and continuous then the abstract carrier S(u) is the complement in E of the largest open set on which u is harmonic. GivenfeC~(E) (the bounded, positive, continuous functions on E), we write f . p for the specific product induced by p and S('); see [CC2, Sect. 8.1]. By [CC2, Theorem 10.2.1] there is a unique subMarkov semigroup (P,),~_0, operating on the bounded Borel measurable functions on E, such that

(f*p)(x) = ~P,f(x)dt, V fEC~(E). {1.1) o

Results of Meyer, Boboc, Constantinescu and Cornea, and Hansen now yield the existence of a Hunt process X = (.(2, ~ , ~-,, 0,, X,, px) with state space E u {d} and semigroup (P,). (Here d is the cemetery point adjoined to E as the point at infinity if E is noncompact, and as an isolated point otherwise.) See ICC2, p. 271] for references and [BH1] or [BH2, Chapter IV] for detailed proofs of this result. The

* Research supported in part by NSF Grant DMS 87-21347 ** Research supported in part by NAVF (Norway) ref. D.93.10.000

2 L. Csink et al.

paths of X are continuous on [0, ~h where (:= inf{t > 0:X~ = A} is the lifetime of X. Note that P~'(~ < oo) = 1; indeed px(~) = p(x) < co.

If G c E is open, then the cone ql(G) of positive hyperharmonic functions on G is precisely the class of excessive functions of the subprocess X 6 obtained by killing X at the exit time zG:= inf{t > 0:X,r That is,

X ~ = ~ X t , 0 < t < ~ ; (1.2) [Lt, t>T~.

An alternative description of ql is provided by the harmonic measures Ho(x,') defined for open G c E by

H d x , f ) = HGf(x) = PX(f(X~,)). (1.3)

(Here and elsewhere a function f on E is extended to E~., {A} by setting f ( A ) = 0; thus the PX integral in (1.3) is really an integral over the set {t't~ < ~}.) A function u:G-~[0, oo] lies in 6~(G) if and only if it is lower semi-continuous (l.s.c.) and Hv u < u on G for all open V with P ~ G. Moreover heC~(G) is harmonic on G if and only if Hvh = h on G for all V as before.

Given two ~-harmonic spaces (E, q/) and (F, C), a continuous function ~o: E ~ F is a harmonic morphism provided uo~p~ql(~0- t(G)) whenever u ~ ( G ) and G ~ F is open (cf. Constantinescu and Cornea [CC1], Boboc [B], Fuglede IF1, F2a, b] and Laine [L]). Our main result is a stochastic characterization of the harmonic morphisms from (E, ql) to (F, C). This characterization involves a notion of path preservation which we now describe.

As before let X be the Hunt process associated with (E,q,') and the strict potential p. Let A = (A,) be a finite continuous additive functional (CAF) of X. In detail A is a real-valued process adapted to the filtration (~t) of X with paths t~-~At(o~) that are continuous, increasing and finite on [0, (I-, and

A,+,(co) = A,(co) + A,(O,o~), Vs, t >= O, co~s (1.4)

Here 0, is the usual shift operator on path space ~. Taking s = t = 0 in (1.4) we see that Ao = 0 and consequently At ~ 0 for all t > 0. See [BG] for details. The right continuous increasing inverse process B, = A- *(t):= inf {s: A, > t} is called the time change associated with A.

Now let (F, f ) be a second ~5-harmonic space such that 1 ~ ( F ) , with bounded continuous strict potential q, and let Y = (Yt,/3.) denote the associated Hunt process. (The "hat" will be used to signal objects defined relative to Y.) Let qJ:E-~F be continuous. Fix an open set G c E, put z = zo, and consider the stochastic process

~'q,(xB,(~)), 0 __< t < A,(~o); Zt(~o, ~b) (1 o5) , i

under the l a w / ~ defined on f2 • ~ by

/~"(do~, dd)) = P~'(dco)P'{x"~')}(do3). (1.6)

We call (Z,,/~') the zo-splicino of ~X~to) and Y.

A stochastic characterization of harmonic morphism 3

Definition. A continuous mapping ~o:E~F is X - - Y path preserving with time change B pr6vided the rG-splicing of <p(XBc,) ) and Y has the same law (under/Sx) as (Yt, P~'(x)), for all x e G and all relatively compact open sets G c E.

The above notion of "path preserving" is appropriate only under the following condition

the constant functions are ~-harmonic. (1.7)

The hypothesis will be in force throughout the paper. The relevant consequences of (1.7) are detailed at the beginning of Sect. 2, but it should be noted at this point that (1.7) implies that z~ < ~ almost surely for all relatively compact open G ~ E.

Before stating our main result we introduce the potential kernels U, U, and UA:

U f (x) = S Ptf(x)dt = px f (Xt)d t , 0

Of(x) = [. e , f (x )d t = e ~ f ( ~ ) d t , 0 0

) Theorem 1.1. Let (E, ~//) and (F, ~e-) be U-harmonic spaces, let p~all(E) and q~e'(F) be bounded continuous strict potentials, and let X and Y be the associated Hunt processes. Assume condition (1.7). I f ~p:E ~ F is continuous, then the followin# are equivalent:

(i) <p is a harmonic morphism (of (E, dll) into (F, ~r (ii) there is a CAF (At) of X such that

O f(~p(x))= PX~ ! f(<p(Xt))dA,) + H~(U f o~p)(x) (1.8)

for all f r x~G, and relatively compact open sets G c E; (iii) there is a CAF (At) of X with inverse B such that <p is X - Y path preserving

with time change B.

Remark. It should be noted that the CAF A in Theorem 1.1 need not be strictly increasing, hence B need not be continuous. Nonetheless t~q)(XB~o) is continuous since q~(X) is constant on each closed interval on which A is constant. See Proposition 3.1, and also Theorem 3.2 for a description of the fine support of A in terms of ~p.

The connection between harmonic morphisms and Brownian path-preserving functions (in the classical case where harmonic means Ah = 0) goes back to L6vy when E = F = R2; see McKean IMcK] for a proof using stochastic integrals. L6vy's result was extended to arbitrary dimensions by Bernard et al. [BCD]. The analogous result for finely (classical) harmonic morphisms and Brownian path-preserving functions was proved by Oksendal [01]. Our Theorem 1.1 in case E and F are Euclidean domains, and X and Y are diffusions (i.e., their generators

4 L. Csink et al.

operate on C 2 functions), was established by Csink and Oksendal [CO] using a method different from that used in this paper. The condition (ii) in Theorem t.1 is (at least formally) equivalent to a condition on generators used in [ c o ] to characterize harmonic morphisms; see Sect. 5 for more on this point.

The proof of Theorem 1.1 occupies Sect. 2. In Sect. 3 we characterize the fine support of A in terms of (p, and we show that ~p restricted to this fine support is a finely open mapping. Several further applications of Theorem 1.1 are gathered in Sect. 4. Section 5 contains some complementary results and open questions.

2. Proof of Theorem 1.1

The notation established in Sect. 1 is used in this and the following sections and will not be repeated. Before proceeding to the proof of Theorem 1.1 we record several consequences of the condition (1.7).

Clearly (1.7) implies

Hal = 1, u relatively compact, open G c E, (2.1)

which is equivalent to

P~(T~ < 0 = 1, V xeE, V relatively compact, open G c E. (2.2)

If {G,} is an increasing sequence of relatively compact open sets with union E, then ~,~ ' ( , so {z~} announces ( because of (2.2), and therefore ( is a predictable stopping time (see [DMI]). Since X is a quasi-left continuous on [0, ool- it must be continuous at (, i.e.,

l i m X , = d , a . s .P x, Vx~E. tT~

Consequently A cannot be isolated in E u {A) and E must be noncompact. The following consequence of the predictability of ( will be used several times in the sequel.

Lemma 2.1. Let ueall(E) be bounded and harmonic. I f T( < 0 is a stopping time of X, then

ex(u(X)r_ ) = u(x), u (2.3)

In particular, if P~(T < ()= 1, then

PX(u(X r)) = u(x). (2.4)

(The existence of the left limit u(X)r_ := lira u(Xt) follows since u(X) is a right fT1 r

continuous positive supermartingale.)

Proof. Since u is continuous on E, (2.4) is a special case of (2.3). To see (2.3) choose relatively compact open sets G, TE and let T,:= T ^ r(G,). Since u is harmonic, u = Ha.u = P'(u(X,(o,)). But T~ ___ ~ so the optional sampling theorem implies

u(x) ~ P(u(Xr) ) _. e~(utX,c~.))) = u(x).


Thus u(x)= P~(u(Xr.)). Letting n ~ oo we obtain (2.3) by bounded convergence since u(Xr) = u(X)r- on {T < ~}, while T, TT( on {T = ~}. []

Proof of Theorem 1.1. (i)~(ii). We require two lemmas.

Lemma 2.2. Let u ~ ( E) be a bounded continuous potential. Then there is a (unique) CAF, (A,) of X such that u(x) = P~(Ag) for all xeE.

Proof. The lemma will follow from [BG, IV (3.13)] once we verify that

lim PX(u(Xr.)) = PX(u(Xr)), VxeE, (2.5) n

whenever {Tn) is an increasing sequence of stopping times with limit T < (. (Since u is bounded and ~ < oo, P,u(x) ~ IlullooPtl(x)= II u II~oP~(~ > t)--*O as t ~ oo.) First note that

h(x) := ex(u(X)~_)

defines a bounded element of ql(E). Indeed ~~ = (~ - t) +, so

Pth = e'(u(X);_ o0,; t < ~) = P'(u(X)r t < ~)~h(x), tJ, O,

and h is X-excessive. Because of (2.2), a similar computation shows that Hoh = h for all relatively compact open G = E, so h is even harmonic. Let {Tn} be a sequence of stopping times announcing ~ (for example ~(G~) where G, TT E). By Fatou's lemma,

/ \ h(x)= P~' L lim u( X r,) ) < lim inf PX( u( X r,) ) < u( x ),

since u(X.) is a supermartingale. Thus h is a harmonic minorant of the potential u, hence h = 0. In other words,

PX(u(X)r = O, VxeE. (2.6)

Now let {T,} be an increasing sequence of stopping times with limit T __< r On the set A:= {T < ~} u {T, < T = C, Vn} we have u(X)r- = u(Xr) a.s. because of (2.6). The same is true on A" because on that set u(Xr) = u(A) = 0 = u(X)~_ = u(X)r- by a second application of (2.6). Formula (2.5) now follows from Lemma 2.1 and bounded convergence, and the lemma is proved. []

Lemma 2.3. Under condition (i) of Theorem 1.1 there is a (unique) CAF, A, of X such that for all positive bounded measurable functions f on F,

O f(~o(x)) = U A(fo ~o)(x) + h:(x), Vx6E, (2.7)

where h:~ql(E) is harmonic.

Proof. Let u~- UI ocp = qo~o~ql(E). Clearly u is bounded and continuous, hence so is its potential part p. By Lemma 2.2 there is a unique CAF, A, of X with p = P'(A~) = UAI. In proving (2.7) it suffices to consider f e C ; ( F ) and for such f the stated identity is equivalent to

(fo q~),p is the potential part of ( f ,q )o q~. (2.8)

6 L. Csink et al.

Indeed g~-~ Uag is a positive linear form on C~(E) such that S(UAg ) C supp(g). (To r(G)

see this note that if G is an open subset of (supp(g)) c, then ~ g(X~)dA s = 0, so o

HG(UA,) = Ua, by the s trong Markov property.) By [CC2, Theorem 8.1.1], we

therefore have

UAo=g*p, Vg~C~(E), (2.9)

and (2.8) implies (2.7) as claimed. As to (2.8) note that if w'V'(F) is bounded (by 1, say) and # - h a r m o n i c , then

voto is q/-harmonic. Indeed both v and 1 - v are in ~ (F ) ; since to is a ha rmonic morphism both voto and 1 - vo~0 are in ~ Using now (2.1)

voto > H~(voto) = H~I -- H~(1 - v~ = 1 - H~(1 -- voto) > 1 -- (1 -- voto) = voto,

and v oto is harmonic. This a rgument can be localized to show that if V c F is open and v is harmonic on V, then voto is harmonic on to-l(V).

N o w let v = (qi)i~t be a part i t ion of q (see [CC2, p. 188]). Since to is a harmonic morphism,/~ = (qio to)~, is a part i t ion of u = q o tO. In view of the discussion in the preceding paragraph

S ( q i ~ = tO-~(S(ql)), ViEI.

Let qiotO = Pl + hi be the Riesz decomposi t ion of qiotO into potential and harmonic parts. Clearly S(pl)= S(qi~ Define

_h = ~ inf (foto)'h i, i S(pO

= ~ sup (foq~)'h,, i S(p i )

where f e C ~ ( F ) is fixed. Then, in the no ta t ion [of CC2, p. 189],

/%.(foto) + h > ~ inf (fotp)(pi + h,) i S(pi)

= ~ inf (f).(q, oto) i o ( s ( m ) )

=> ~ !nf (f)'(qi~ i S(qO

= Vq,(f)~ to.

Similarly #*(foto) + h < v*(f)oto, and so

Vq,(f)o~o < lap,(f oto) + h <= la~(f oto) + h <= v*(f)oto.

Given ~ > O, a suitable choice of the part i t ion v yields sup (v'~(f)oto - Vq,(f)o (p) < e. E

It follows that there is a harmonic function hseql(E) such that ( f ,q)oto = ( f o to), p + h I , so (2.8) holds as desired. [ ]

A stochastic characterization of harmonic morphisrn 7

Similarly

It is now a simple matter to deduce (ii) from (i). Fix a relatively compact open set G ~ E and x~G. Apply H G to both sides of (2.7) and subtract the resulting identity from (2.7). since hj. is harmonic, we obtain (1.8) after an application of the strong Markov property. []

(ii) ~ (iii). Assume condition (ii) of Theorem 1.1. We will show that (Zt,P ~) solves a certain "martingale problem" of which (Y, fl~(~) is the unique solution (in distribution). This will prove (iii). More precisely we claim that for fixed xeG c E (G open and relatively compact), if Z is the z~-splicing of q(Xn(.~) and Y, then

P~(Z0 = q~(x))= 1; (2.10) [

Of(Z,) + ~ f(Z~)ds (2.11) 0

is a/3~-martingale for each f~C[(F) . This being so, {DM2, XV, TI8] implies that (Z,,/3.) has the same law as (Yt, flr

We show in Sect. 3 that (1.8) implies

q(Xt)= ~p(x), Yt~[0, Bo]c~[0,~], a.s .W. (2.12)

But Z o = q~(XBo ̂ t~)) and r(G) < ~ a.s. W, so (2.10) follows. Next, note that (2.11) is equivalent to

P~ f(Zs)dslfC, = Of(Zt), Vt>0, (2.13)

where c~, = a{Z~:s < t}. To prove (2.13) we write

f(Z~)ds = ll,<a,i (~p(X~))dA~ + ds + II,> A~I f(Y,)du. t t - A ~

Evidently fr {t < At} = a{~p(X~)): s < t} ~ {t < A,} ~ ~ n , ) n {t < At}. In other words if H z~, then there exists H z ~ - s , ~ such that H ~ {t < AtL=/4 ~ {t < At}. (Of course we identify a subset -Qo ~ .(2 with ~o • ~ ~ O x 03 Since B, is an (~)-stopping time, {t < A~} = {B~ < r}z~B<o, and t = B, + zoos, ~ on {B, < z}, we may compute

W f(q~(X~))dAs; Hc~ {t < t = P q~(X~))dAs

(2.14)

P*(~f(Ys)ds;HC~o {t <A,} ) = PX(Of(q(Xt));Hc~ {t < At} ). (2.15)

On the other hand both r A ~ and A,IIt>A are measurable over fr {t > A,}, * " ~ x = t = r , - -

and tt ts clear that under P (restricted to {t > At} ) the conditional distribution of (Ys:s >=t-At) given ~p(Xt)= y, A t = a is the same as the flY-law of (Ys:s >=t-a).

8 L. Csink et al.

Thus with He(~ as before

PX(t~Af(Ys)ds;Hc~{t> A , } )=P~(Gf (Y ,_a , ) ;Hc~{ t> A,} ). (2.16)

Adding (2.14), (2.15), and (2.16)and using (1.8) and the definition of Zt we obtain

(+ ) P~ I f(Zs)ds; H = P~(gf(Z,); H n {t < A~}) + Px(0f(Z,); H c~ {t __> A~}) t

= e ~ ( u f ( z , ) ; H).

Since Hsff , was arbitrary, (2.13) is proved. []

(iii)~(i). Fix v e ~ ( F ) and put u = vo~0. Fix a relatively compact, open set G c E, and x~G. It is trivial that condition (iii) of Theorem 1.1 implies condition (ii), hence (1.8) is valid. Now since 0 is a bounded kernel there is a sequence {f,} of bounded Borel functions on F with Uf~ T e- The RHS of(1.8) defines an element of 0~(G), as is easily checked. Thus Uf , o ~ol ae~ so if Vis open with P c G, then

H v ( O L o ~ o ) < O f , oqo on G, Vn. (2.17)

Letting n o oo in (2.17) we find that Hvu < u on G. As u = vo~0 is clearly l.s.c, on G, we have u I~eq/(G). But G was arbitrary so ue~ by the sheaf property of a//.

N ow let f 'be an arbitrary open subset of F and put V = ~p-*(f'). The formula A[ = A t ̂ ,tv) defines a CAF of the subprocess X r (see (1.2)). It follows immediately from condition (iii) that ~olv is X v - yv path-preserving with time change B e = (AV) - ~. The argument of the preceding paragraph now shows that vo ~0eq/(V) whenever v e : ( V ) . Thus ~o is a harmonic morphism. []

3. The fine support of A

Let tp be a (continuous) harmonic morphism between (E, ~') and (F, ~F), and let A be the CAF of X provided by Theorem 1.1. In this section we give an explicit description of the fine support of A in terms of ~o, and we deduce several corollaries of this description. The fine support, D, of A may be defined as

D = {x~E:eX(At > O, Vt > O) = 1 } = {xEE:PX(Bo = O) = 1 }.

(By the Blumenthal 0 - 1 law, the above probabilities are either 0 or 1.) The set D is the smallest finely closed subset of E such that

t

Sla(Xs)dAs=A~, Vt~O, a.s.P ~, VxEE. 0

(See [BG, V, (3.10)] and note that X has a reference measure, being a transient process with 1.s.c. excessive functions.) It is known [BG, V, (3.5), (3.6)] that

T o = B o a.s.P x, Yx~E, (3.1)

where TD:= inf{t > 0:XtED}. Various characterizations of D can be found in [BH2, VI. 8.2].


Evidently A is strictly increasing (and B is continuous) if and only if D = E. The following simple example shows that this need not be the case. As we shall see this example is typical, at least locally. An example, due to A. Cornea, of the same phenomenon in the more restrictive context of ~-Brelot spaces, can be found in EE3].

Example. X is uniform motion to the right on E = R, while Y is uniform motion to the right on F = [0, oo[. A function u : G ~ [ 0 , oo] lies in ad(G) if and only if it is right continuous and decreasing on each connected component of G. The sheaf

admits an analogous description. The mapping cp(x)= x +, xeR, is a harmonic morphism and the CAF of Theorem 1.1 is just

t

A, = S l[o,oo[(X*)ds" 0

Thus D = [0, ool- ~ E. As Theorem 3.2 below indicates, it is no coincidence that E\D is the set of points at which tp is finely locally constant.

The following preliminary to Theorem 3.2 was used in Sect. 2.

Proposition 3.1. If x~E\D, then

tp(X,)=q~(x), Vt~[0, To] c~ [0, ([, a.s .e x.

In particular the set Bx:= { yeE\D: , ( y) = cp(x) ) is a fine neighborhood of x on which ~o is constant.

Proof. Fix xeE\D, and let G c E be a relatively compact, open neighborhood of x. Let S be any stopping time of X such that S ___ T o ^ z6( < 0, and define a probability measure on E by

t~ = P~(Xse').

By the optional sampling theorem Mu) < u(x) for all u~ql(E). In particular,

q~(g)(v) =/l(vo~o) < v(q~(x)) = r Vve~lr(F), (3.2)

since q~ is a harmonic morphism. By (1.8) (or more precisely the argument used to deduce (1.8) from (2.7)),

Ol(~p(x))= P~( i dA,) + PX(Ol(tp(Xs))). (3.3)

But A s = 0 since S __< T a (recall (3.1)) so (3.3) implies that

0 l(tp(x)) = #( 01 o ~o) = ~(1~)( 01).

Since the potential q = U1 is strict, we deduce from ~o(~t) = tp(ex) = er That is,/~ is carried by B~, so

P~(~(Xs) -- q,(x))= 1.

An application of the optional section theorem I-DM1, IV T84] now shows that

(3.4)

(3.2) and (3.4) that

PX(#,(x,) = q,(x), Vtel'0, To ^ T~]) -- 1.

10 L. Csink et al.

Letting G swell to E through a countable sequence we obtain the proposition. []

We say that ~p is finely locally constant at x~E provided there is a fine neighborhood N of x such that cp(N) = {tp(c)}.

Theorem 3.2. E\D = {xeE:tp is finely locally constant at x}.

Proof. The inclusion c follows immediately from Proposition 3.1. For the reverse inclusion let V be a fine neighborhood of x on which ~p is constant. Given a relatively compact, open set G c E, if we apply the operator Hcn v to both sides of (2.7) and then subtract, we obtain for x~G

U A( f o ~p)(x) = Hanv(U A(f ~ ~p))(x), (3.5)

since ~o(X,~,~v~)= cp(x) a.s. px by the continuity of cp. Clearly (3.5) implies that PX(A~ca,~v~) = 0, so that Z~nv < T~ a.s. PX. Letting G~E we get 0 < Zv < To a.s. px, hence xr since every point of D is regular for D [BG, V, (3.5)]. []

Corollary 3.3 Let ~p:E--* F be continuous and nowhere finely locally constant. Then the followino are equivalent:

(i) ~p is a harmonic morphism; (ii) cp is X - Y path-preserving with a continuous time chanoe.

Proof. This follows immediately from Theorems 1.1 and 3.2 since A is strictly increasing (and B is continuous) if and only if D = E. [ ]

In contrast to the situation on E\D, the restriction of q~ to D is a finely open mapping. Indeed consider the process Xn = (Xn~o, px). It is standard that Xn is a right continuous strong Markov process. However each point x 6 E \ D is a branch point for Xn:

P~(XB(o~e. ) ~= ~, x ~ E \ D

since XB(o~EDu {A} a.s. P~. The branch set is polar for XB:

P~(XBol~E\D for some t > 0) = 0, VxeE,

and if we restrict X n to D then we obtain a right Markov process X with state space D. (See Getoor [G] for details.)

It is easy to check that if ueal/(E) then ulo is excessive for X; conversely if Uo is ,~-excessive, then

u(x):= P~(uo(X B(oj))

defines an X-excessive extension of Uo to all of E. It follows that the ,~-fine topology on D is the relative X-fine topology.

Theorem 3.4. Suppose N c E is X-finely open. Then N n D is X-finely open and ~p(N c~ D) is Y-finely open.

Proof. The first assertion follows from the discussion preceding the theorem. As to the second assertion, if x EN n D then

1 = Px(X t~NnD, Vt~]O, to[ for some to >0)

= PX(~p(XB(o)e~p(NnD), Vte]O, to[ for some to > 0),


so Y t ~ ( N c~ D) in some initial interval of time a.s. p,tx) by Theorem l.l(iii). (Note that Bo = 0 a.s. PX since xeD.) []

Combining Theorems 3.2 and 3.4 we can now prove the main result of this section. Recall that p = Ual is the potential part of/~1 otp.

Theorem 3.5. The following are equivalent:

(i) p is finely locally harmonic at x; (ii) ~0 is finely locally constant at x;

(iii) there is a fine neighborhood N of x such that q~(x)q~q~(N)', where the "prime" denotes Y-fne interior;

(iv) xq~D.

Proof. (i)=~(ii): Let p be finely locally harmonic at x; thus there is a finely open neighborhood N ofx such that for each finely open, relatively compact set N O c N with x~No,

nNoP(X) = p(x),

which forces PX(A,tNo~) = 0, hence zNo < T o a.s. P~. But ZNo > 0 a.s. px so xq~D. Point (ii) now follows from Theorem 3.2.

(ii) =~ (iii): Trivial. (iii)=~(iv): This follows from Theorem 3.4: If x~D and N is finely open set

containing x, then q~(x)~ tp(N n D) = ~0(N n D)' c ~a(N)', which contradicts (iii). (iv)=~(i): This follows from Proposition 3.1. []

A fundamental result of Constantinescu and Cornea [CC1, Theorem 3.5] states that every nonconstant harmonic morphism between Brelot harmonic spaces E and F is finely open, provided E is connected and the points of F are polar. Subsequently Fuglede [F2, Cor. p. 190] proved that for general harmonic spaces, if we put

C = {x~E: x is not finely isolated in q~-l({~(x)})},

then tpr\c is finely open. Since E\D c C, Theorem 3.4 contains Fuglede's result when the constants are ~-harmonic (since by [CC2, Theorem 2.3.3] any harmonic space can be covered by open sets whose induced harmonic subspaces are ~-harmonic). If (E, q/) and (F, ~ ) are Brelot harmonic spaces and the points of F are polar, then ~0-1({y}) is polar in E for all yeF ([CC1, Theorem 3.2]) and therefore q~ is nowhere finely locally constant. Thus Theorem 3.4 contains the result of Constantinescu and Cornea as well.

The following dichotomy follows from Theorem 3.5, (ii)~(iii).

Corollary 3.6. For each x~E, either r is finely locally constant at x, or q~ maps each fine neighborhood of x onto a fine neighborhood of qg(x).

The process X is not uniquely determined by (E, ql) but depends upon the choice of the strict potential p. We end this section by proving that when a special choice of p is made, the additive functional A becomes the occupation time of D (which set does not depend on the choice of p, becauce of Theorem 3.2). We are grateful

12 L. Csink et ai.

to W. Hansen for pointing this out to us, thereby solving a problem stated in a previous version of the paper.

Theorem 3.7. Let D denote the fine support of the CAF A, where U A1 is the potential part of U1 orp. Then there is a bounded continuous strict potential p* in all(E) with associated Hunt process (X*, Q'O such that if

t

o~t:= S lo(X*) ds, fit = ~-1 = inf{s:~s> t}, (3.6) 0

then (Xn~o,P 0 and * ~ taw for atI xeE. (X al o, Q ) have the same

Proof. Define p* = Ual + IE\D*p. (Recall that p = U1.) By Lemma 2.2 both UA1 and l~\D*p are potentials of CAF's of X with fine supports D and the fine closure of E\D respectively. Thus the fine support of the CAF associated with p* is D u (E\D) = E; i.e., p* is strict. Clearly p* is bounded and continuous. Let (X*, QX) be the associated Hunt process, and let % and fl, be defined as in (3.6). If C c E is a Borel set then by [CC2, Theorem 8.2.1],

= (lc~o)*p*(x) = (lc* UalXx)

When restricted to D, both X~ o and X~*(t~ are transient fight Markov processes, and their potential kernels coincide by (3.7). Thus (XB, ~, P~) and (Xp*~o, QX) have the same law if xsD. But the "branching laws"

P~(X.~o~'), Q~(X~o~r

both coincide with the P~-law of X at time To = B0 (a.s. P=) if xr It follows

that (XB~o,P 0 a=(X~,~,Q~) for all x~E. []

See Sect. 5 for another consequence of a judicious choice of the "time scale" of X (and Y).

4. Other applications

As in previous sections ~0 is a (continuous) harmonic morphism between ~-harmonic spaces (E,~') and (F, ~), and (1.7) is in force. Recall that D is the fine support of the CAF, A, of Theorem 1.1.

Theorem 4.1. Let C ~ F be Y-polar. Then ~ - I(C) r~D is X-polar.

Proof. It is clear from Theorem 1.1 0ii) that ~-t(C) is Xs-polar. Consider the

A stochastic characterization of harmonic morphism ! 3

(random) time set

M = M(co) = {t > 0:XtED}.

Since D is finely closed, M is right closed in [0, ~ [ a.s. px for all xeE. Also, it follows from Proposition 3.1, the continuity of r and the strong Markov property of X, that each of the connected components of ]0, ~ [ \ M is an interval of the form [a, b[, almost surely. Thus for PX-a.e. e~, t~--~At(o9 ) is a bijection of M(og) onto [0,A~(~o)[. The X-polarity of ~o-~(C)nD now follows from the Xn-polarity of ~p-1(C) noted earlier. []

For the rest of this section we assume, in addition to (1.7), that the constants are C-harmonic. The proof of Theorem 2 in [ c o ] can be adapted to prove the following boundary value result.

Theorem 4.2. The limit

tp(X)~_:= limtp(Xt) exists in F w { A } a.s. px VxeE. (4.1)

In particular, if V c E is open and ~p( V) is relatively compact in F, then

lim tp(Xt) exists in F a.s. Px, Vx~V. fiftY)

Proof. The second assertion is an immediate consequence of the first, and the fact (Sect. 2) that Y~ ---, z~ as t]'(. Here zl is the point at infinity for F. Since Yis continuous on [0, ~1- with values in the compact metric space F u {z~}, the tightness argument used in the proof the Theorem 2 in [CO-] shows that condition (iii) of Theorem 1.1 implies

lim~p(XB~t~ ) exists in Fu{,4} a.s. px, u (4.2) t~a~

But ~p(Xm,)) = ~o(Xmt _)), because ~p(X ) is constant on [B(t), B(t - )] by Theorem 3.1, the strong Markov property, and the continuity of ~p. Thus (4.2) implies (4.1). []

The following "boundary" version of Theorem 1.1 now follows from Theorem 4.2 by taking relatively compact open sets G, T E and considering z(G,)T]'(. We omit the details.

Corollary 4.3. Let ~p:E ~ F be continuous. Then the following are equivalent under condition (1.7):

(i) q~ is a harmonic morphism between (E, qt) and (F, ~e'); (ii) there is a CAF, A, of X such that for all f ~C~(F), x~E,

O f(q~(x)) = U ~( f o q~)(x) + W (O f(~o(X)r (4.3)

(iii) there is a CAF, A, of X such that the r of q~(Xnl.)) and Y, under P*(deg, d~b) = P*(&o)p*(x)~-(~')(do3), has the same law as Y under p,~,o, for all x~E.

Note that if ~p is proper (~p(K) is compact in F if K r E is compact) or only "stochastically proper" in the sense that iim q~Xt) = z~, then the harmonic term in

14 L. Csink et al.

x d (4.3) vanishes, and (tp(Xaco),P)=(y,,f~,tx~) with no "splicing" necessary. In particular if E = F and the identity mapping on E is a harmonic morphism, then X and Yare (continuous) time changes of one another. This sharpens a result of Shih [Sh] which would yield the same conclusion under the apparently stronger hypothesis that the harmonic measures of Y dominate those of X. (However Shih's result is valid in a much wider context than that considered here.)

If G c E is open and f : G -+ F, we say that f has an asymptotic value w at y ~ G provided there is a curve ~ in G terminating at y such that

lira f (z) = w. Z ~ y ZEy

Our final result is an immediate consequence of Theorem 4.2.

Theorem 4.4. (Asymptotic value theorem). Let G ~ E be open and suppose that ~o(G) is relatively compact in F. Then r has an asymptotic value at a.a. points y~aG w.r.t. the harmonic measure He(x, .),for all x~G.

5. Complements

Most of the results of the paper hold under somewhat weaker hypotheses. For example, it would suffice for X and Y to be path-continuous transient Hunt processes with reference measures, and for the constants to be X-harmonic (and Y-harmonic in the latter part of Sect. 4). The only substantial change that must be made is in the statement and proof of Lemma 2.2, where "regularity" [BG, IV (32)] must be substituted for continuity (of the potential u). Moreover, results of a local character stated in Sect. 3 and 4 are valid for general harmonic spaces (in which the constants are harmonic) since these spaces can be covered by ~3-sets [CC2, Theorem 2.3.3], as noted already.

The notion of harmonic morphism (under the name harmonic map) was introduced by Constantinescu and Cornea [CC 1] in the context of Brelot harmonic spaces. In [CC1]a continuous map c p : E ~ F is a harmonic map provided hoop is harmonic on r whenever h is harmonic on an open set V c F. The stronger definition of harmonic morphism used in this paper appears to be due to Boboc [B], who shows that the two notions coincide provided F has a base of regular sets (e.g. ff (F, ~ ) is a Bauer space). This fact has also been observed by Fuglede IF2]. See Sibony [S] for a related notion.

In previous characterizations of harmonic morphisms (e.g. [ F l l and [CO]) an identity relating the infinitesimal generators of X and Y has been used instead of a potential kernel identity like (1.8) or (4.3). For such an identity to hold one must have

t

A t = Sa(Xs)ds, Vt > O, (5.1) 0

for some Betel function a >-- 0 on E. We shall see that under a mild nondegeneracy condition on Y there is an a priori choice of the "speed" of X and Y (equivalently of the strict potentials p and q) ensuring (5.1).

A s tochas t ic charac te r iza t ion of h a r m o n i c m o r p h i s m 15

According to a result of Kunita [K] there is a CAF K of X such that each martingale of the form

t

M{:= f(X,) - f ( X o ) - Sg(X,)ds (geCb(E),f = -- Ug) (5.2) 0

has quadratic variation of the form

t

< M J" >t = ~ F(f)(X~)dK~, 0

for some Borel function F ( f ) > 0 on E. In the present circumstances one can choose K so that the potential Uxl is bounded and continuous (cf. the proof of Kunita's result in [DM2, XV, T30]). If we now use the continuous inverse of t + K t to time change X (equivalently, replace p = U1 by U1 + UK1) then every local martingale of the time changed process has absolutely continuous quadratic variation. (See [DM2, XV, T26], and note that every martingale over X has continuous paths, since X is a continuous Hunt process.) The time changed process induces the same harmonic structure on E, so we can assume without loss of generality that p = U1 has been chosen so that

t

<MI)t = SF(f)CXs)ds, Vt > 0 (5.3) 0

for all f as in (5.2). Incidentally, a formal application of It6's formula shows that F (the "gradient squared" operator) is given by

F ( f ) = c~(f2) _ 2ff~(f),

where f~ is the infinitesimal generator of X. In the sequel we assume that p and q have been chosen so that (5.3) and its analog for Yare valid, and we assume that constants are harmonic for both X and Y.

Now assume that ~0: E ~ F is a harmonic morphism and that A is the CAF of X provided by Theorem 1.1 (or Corollary 4.3). Fix a = - Ug where g~C~(F). Then using the obvious notation we have

t t

S P(a)(q~(Xs))da s = $ F(ao~o)(Xs)ds, Yt > 0 (5.4) 0 0

a.s. px for all xcE. To see (5.4) note that

t

/~,:= a(Yt) - d(Yo) - ~g(Y,)ds 0

is a/~Y-martingale for all yeF, while by (4.3),

t

M,:= a(~o(x,)) - a(+(Xo)) - ~ O(+(X~))dAs + 1. ~_~a(~o(x k_) 0

is a PX-martingale for all x~E. A straightforward calculation leads to

O ( P ( a ) X q , ( x ) ) = P ' ~ ) ( < / ~ > 3 = 2 O ( a o ) ( q , ( x ) ) - a(c,(x)) 2, (5.5)

V(F(u ) ) ( x ) = P X ( < M > c ) = 2U a(aoq~.oo~)(x) - a(q~(x)) 2 + P:'(a(,(x)r (5.6)

16_ L. Csink el al,

for all xeE . Note that the RHS's in (5.5) and (5.6) are bounded in x. Using (4.3) to eliminate the / ) terms we can combine (5.5) and (5.6) to obtain (after taking potential parts)

A

u,,(r(u)o~o)(x) = U(F(aoq,) ) (x) < ~ , VxEE. (5.7)

Now (5.7) amounts to the statement that the two CAF's in (5.4) have the same finite potential, hence (5.4) follows from the uniqueness theorem [BG, IV (2.13)] for potentials of CAF's.

To deduce (5.1) from (5.4) one must impose a nondegeneracy condition on F. For example suppose there is a countable cover {G,} of F comprised of relatively compact open sets, and functions g,~Cb(F) such that with t~, = - 0y, ,

/~(~l.)(Ys)>0 for a.e. s~[_O,e(G,,)] (5.8)

a.s. P for all yzG.ncp(E) . Using Corollary 4.3 it is not hard to conclude that A A

F(u,,)(q~X~)) > 0 for dAs-a.e, sz[0, ~(r I(G,))] (5.9)

a.s. px for all xe~0-I(G,). Evidently (5.4) and (5.9) yield t

A t = J a,tX~)ds, Vt~[0, r(cp-I(G,))] (5.10) 0

a .s . /~ for all x6q~-~(G,), where

a,(x) = F(a , ocp)(x) xE~o- X(G,), (0/0 = 0). f'(a.)(,p(x))'

The local result (5.10) is enough to ensure the global result (5.1); one can use, e.g. the "pasting" argument of [BG, V, Sect. 5]. Condition (5.8) certainly holds if Y is an elliptic diffusion in a Euclidean domain. We conjecture that (5.8) holds whenever (F, 3w) is a ~-Berlot space.

Let us now reformulate (4.3) in terms of infinitesimal generators. Define

~ ( ~ ) = { U o + h: U Ig[, h~ Cb(E), h harmonic};

c ~ f = _ 0 if f = U g + h ~ ( f g ) ,

and note that i f f e~(Cr then ~r only determined a.e. (i.e., up to a set of potential zero for X), but regardless of the choice of CO f ,

I

Mr:= f ( X t ) - f ( X o ) - S fc f ( x~)as + 1~, >=g~f(x);_ 0

is a continuous P~-martingale for all xeE . We define ~(~) and c~ analogously for Y. Assume that (5.8) holds, in addition to (5.3) and its analog for Y. Then by Corollary 4.3, if ~0 is a harmonic morphism between (E, q/) and (F, ~ ) we must have

f e ~ ( ~ ) ~ f o q ~ e ~ ( f r and fC(foq~)=a'(f~f)oq~ a.e., (5.11)

where a is as in (5.1). (Note that the extreme right hand term in (4.3) is harmonic.)

Of course (5.11) implies condition (ii) of Corollary 4.3 with A t = .[ a(X,)ds , hence


that ~p is a harmonic morphism. An identity like (5.11) was used in [F1] to characterize harmonic morphisms between Riemannian manifolds, and to the same end in [CO] when X and Y are diffusions on Euclidean domains.

Our final remark concerns the removal of condition (1.7). One way to proceed if (1.7) fails is as follows: X is the subprocess of a second Hunt process .~ (with hyperharmonic sheaf qT, say) for which the constants are harmonic. Indeed if 1 = k + h is the decomposition of 1 into potential and harmonic parts, then k is the potential of a CAF K, and the semigroup of X is given by

P , f ( x ) = PX(f(X,)elC'). (5.12)

In effect, X is obtained by killing X at the first instant K, surpasses an independent exponential random variable of mean one. It is easy to check that a bounded l.s.c. function u -> 0 is X-excessive if and only if u - u . k is X-excessive. This observation can be localized to obtain a complete description of ql in terms of q/. Likewise Y is the subprocess of a Hunt process ~" on F for which constants are harmonic. One can show that

if~p is a harmonic morphis between (E, ~') and (F, ~) , then it is also

a harmonic morphism between (E, ~ ) and (F, C). (5.13)

Using (5.13) one can easily reduce the general case to that in which (1.7) holds, and thereby obtain the analog of Theorem 1.1 without hypothesis (1.7). We hope to return to this point in a future publication.

Acknowledgements. We wish to thank A. Cornea, A.M. Davie, and G. Michaletzky for their comments on a previous version of this paper. And we are very grateful to B. Fuglede, W. Hansen, and R. K. Getoor for useful communications throughout this work.

References

[BCD]

[BG]

[BGM]

[BHI]

[BH2]

[B]

[CCl]

[CC2]

[co]

[DMI] [DM2]

[F1]

Bernard, A., Campbell, E.A., Davie, A.M.: Brownian motion and generalized analytic and inner functions. Ann. Inst. Fourier 29, 207-228 (1979) Blumenthal, R.M., Getoor, R.K.: Markov processes and potential theory. New York: Academic Press 1968 Blumenthal, R.M., Getoor, R.K., McKean, H.P. Jr.: Markov processes with identical hitting distributions. IU. J. Math. 6, 402-420 (1962); 7, 540-542 (1963) Bliedtner, J., Hansen, W.: Markov processes and harmonic spaces. Z. Wahrschein- lichkeitstheorie verw. Geb. 42, 309-325 (1978) Bliedtner, J., Hansen, W.: Potential Theory. Universitext, Berlin Heidelberg New York: Springer 1986 Boboc, N.: Asupra prelungiri aplicatiilor armoniee. Ann. Univ. Craiova, Ser. Mat. Fiz. Chim. 6, 31-36 (1978) Constantineseu, C., Cornea, A.: Compactifications of harmonic spaces. Nagoya Math. J., 25, 1-57 (1965) Constantineseu, C., Cornea, A.: Potential Theory on Harmonic Spaces. Berlin Heidelberg New York: Springer 1972 Csink, L., Oksendal, B.: Stochastic harmonic morphisms: Functions mapping the paths of one diffusion into the paths of another. Ann. Inst. Fourier 33, 219-240 (1983) Dellacherie, C., Meyer, P.-A.: Probabilit6s et potentiel I (Chap. I fi IV). Paris: Hermann 1975 Dellacherie, C., Meyer, P.-A.: Probabilit6s et potentiel VI (Chap. XII fi XV). Paris: Hermann 1987 Fuglede, B.: Harmonic morphisms between Riemannian manifolds. Ann. Inst. Fourier 28, 107-144 (1978)

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Fuglede, B.: lnvariant characterizations of the fine topology in potential theory. Math. Ann. 241, 187-192 (1979) Fuglede, B.: Harnack sets and openness of harmonic morphisms. Math. Ann. 241, 181-186 0979) Getoor, R.K.: Markov processes: Ray processes and right processes. (Leer. Notes Math., vol. 444) Berlin Heidelberg New York: Springer 1975 Getoor, R.K. Sharpe, M.J.: Conformal martingales, Invent. Math. 16 271-308 (1972) Kunita, H.: Absolute continuity of Markov processes and generators. Nagoya Math. J. 36, 1-26 0969) Laine, I.: Coveting properties of harmonic Bl-mapping. Ann. Acad. Sci. Fenn., Ser. AI, 1, 309-325 (1975) McKean, H.P. Jr.: Stochastic integrals. New York: Academic Press 1969 Oksendal, B.: Finely harmonic morphisms, Brov, nian path preserving functions and conformal martingales. Invent. Math. 75, 179-187 (1984) Oksendal, B.: Stochastic processes, infinitesimal generators and function theory. In Operators and function theory. S.C. Power {ed.) Dordrecht: Reidel 1985 Shih, C.T.: Markov processes whose hitting distributions are dominated by those of a given process. Trans. Am. Math. Soc. 129, 157-179 (1967) Sibony, D.: Allure ~ la fronti6re minimale d'une classe de transformation, Th6or~me de Doob g6n6ralis6, Ann. Inst. Fourier 18, 91-120 (1968)

Received May 18, 1989

A stochastic characterization of harmonic morphisms

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