Diffusion Processes on Nonholonomic Manifolds and Stochastic Solutions for Einstein Spaces

Diffusion Processes on Nonholonomic Manifoldsand Stochastic Solutions for Einstein Spaces

Sergiu I. Vacaru

Department of ScienceUniversity Al. I. Cuza (UAIC), Iasi, Romania

Multidisciplinary Research Seminarat

INSTITUTE OF MATHEMATICS "O. MAYER"ROMANIAN ACADEMY, IASI BRANCH

October 4, 2010

Sergiu I. Vacaru (UAIC, Iasi, Romania) Stochastic Einstein Spaces October 4, 2010 1 / 17

Outline

1 Aims and Motivation

2 Stochastic Processes & Nonholonomic ManifoldsGeometry of N–anholonomic manifoldsh– and v–adapted Euclidian diffusionDiffusion on nonholonomic manifolds

3 Nonholonomic Diffusion in General RelativityThe special relativistic nonholonomic diffusionNonholonomic diffusion and gravitational interactions

4 Exact Stochastic Solutions in GravityThe Einstein eqs on nonholonomic manifoldsNonholonomic separation of Einstein eqsStochastic solutions with h∗

3,4 6= 0 and Υ2,4 6= 0

5 Summary & Conclusions


Aims and Motivation

Aims and Motivation

Aims:Theory of stochastic processes onnonholonomic Eucliedean and Riemannan manifoldsRelativistic nonholonomic stochastic differential equations anddiffusionExact solutions for stochastic Einstein spacetimes

Definition:A nonholonomic manifold is a pair (V,N )

Review and new results:S. Vacaru, in: IJGMMP, JMP, JGP, CQG, IJTP)Details in: S. Vacaru, arXiv: 1010.0647Diffusion on Curved (Super) Manifolds and Bundle spaces


Aims and Motivation

Related directions

Stochastic GravityEinstein–Langeven eqs with additional noise source.Fluctuations of quantum fields in curved spacetime.Semi–classical approximation and renormalizedenergy–momentum tensors.

Diffusion on Curved SpacesRolling Wiener processes on curved manifolds and diffusion inRiemann–Cartan–Weyl spaces.Laplace–Beltrami operators and diffusion.Stochastic processes in Lagrange–Finsler spaces,supersymmetric and higher order generalzitations.


Stochastic Processes & Nonholonomic Manifolds Geometry of N–anholonomic manifolds

Stochastic Processes & Nonholonomic Manifolds

Geometry of N–anholonomic manifolds

N–connection splitting: T V = hV⊕vV, N = Nak , ex: V = TM

N–adapted d–frames:

eα + (ei = ∂i − Nai ∂a,eb = ∂b =

∂

∂yb ),

eβ + (ei = dx i ,ea = dya + Nai dx i).

Nonholonomic relations: [eα,eβ] = eαeβ − eβeα = wγαβ (u) eγ ,

d–metrics: g = gijdx i ⊗ dx j + hab(dya + Nak dxk )⊗(dyb + Nb

k dxk )

Signature (±,±,±,±), coordinates uα = (x i , ya), x i = (x1, x2)and ya =

(y3 = v , y4 = y

). Indices i , j , k , ... = 1,2 and

a,b, c, ... = 3,4 for (2 + 2)–splitting, when α, β, . . . = 1,2,3,4.


Stochastic Processes & Nonholonomic Manifolds Geometry of N–anholonomic manifolds

N–adapted d–connections

Canonical d–connection & Levi–Civita connectionA d–connection D = (hD, vD) preserves under parallelism the h–v–spitting. Metric compatible: Dg = 0

Canonic. d–con. D : Γγαβ = (Lijk , L

abk , C

ijc , C

abc); T i

jk = 0, T abc = 0.

Lijk =

12

g ir (ekgjr + ejgkr − er gjk

),

Labk = eb(Na

k ) +12

hac(

ekhbc − hdc ebNdk − hdb ecNd

k

),

C ijc =

12

g ikecgjk , Cabc =

12

had (echbd + echcd − edhbc) .

Levi–Civita con. ∇ = Γγαβ and distortion: Γγαβ = Γγαβ + Z γαβ ,

all components defined by metric and N–connection.Cartan d–connection is canonical almost symplectic.


Stochastic Processes & Nonholonomic Manifolds h– and v–adapted Euclidian diffusion

h– and v–adapted Euclidian diffusion

Wiener d–processes of dimension n + mLocally a couple of elementary (Wiener) h– and v–processesWα(τ) =

(W i(τ),Wa(τ)

), parameter τ (in particular, τ = t).

A random (stochastic) curve on V is lifted to a horizontal curve onthe frame of orthonormalized bundles O(V) related by transformseα′ = eα

α′(u)∂α = eαα′(u)eα, ∂α = ∂/∂uα = (∂i = ∂/∂x i , ∂a = ∂/∂ya).

Itô d–calculusA diffusion d–process by a couple of h- and v- SDE,dUα = σαα′(τ,U)δWα′

+ bα(τ,U)dτ, where U = (hU, vU) ∈ Rn+m

is a stochastic d–process with U(0) = u, for u = uβ = (x j , yc),Markovian process by Itô stochastic N-adapt. integral (equation)

Uατ = Uα

0 +τ∫0σαα′(ς,U)δWα′

ς +τ∫0

bα(ς,U)dς,

diffusion coeff. σαα′ , drift. coeff. bα.Sergiu I. Vacaru (UAIC, Iasi, Romania) Stochastic Einstein Spaces October 4, 2010 7 / 17


Itô d–processes and associated diffusion d–operator

Associated diffusion d–operatorStochastic N–adapted differential δf = Af with A = hA ⊕ vA,

A =ρ

2

n+m∑

α′=1

12σiα′(τ,U)σ

jα′(τ,U)

(eiej + ejei

)f

+σaα′(τ,U)σb

α′(τ,U)eaebf + bα(τ,U)eαf,

hA =ρ

2

n∑

α′=1

12σiα′(τ,U)σj

α′(τ,U)(eiej + ejei)f + bi(τ,U)ei f.

∀ stochastic N–adapted process Uατ ∈ Rn+m ∃ the probability

density function φ(τ,u), evolution on τ. ∀ f of U, expected valuef (τ,u) := uE [f (Uτ )] :=

∫U

f (u)φ(τ,u)δu1...δun+m.



Stratonovich d–calculus

Focker–Plank/forward Kolmogorov eq

∂τ f (τ,u) = Af (τ,u), f (0,u) = f (τ,u),

Stratonovich stochastic integral

More convenient for curved spaces, with ”∼ ” and ””,dUα = σαα′(τ,U) δWα′

+ bα(τ,U)dτ. Equivalence:

σαα′(τ,U) = σα

α′(τ,U), bα(τ,U) = bα(τ,U) − ρ2

n+m∑α′=1

σβα′(τ,U)eβσ

αα′(τ,U).

Diffus.oper.: A = ρ2

n+m∑α′=1

Lα′Lα′ + L0; Lα′ = σβα′(τ,u)eβ ,L0 = bα(τ,u)eβ

Associated N–adapted Focker–Plank eq

∂τφ(τ,u) = ρ2

n+m∑α′=1

eβσβα′(τ,u)eγ [σγ

α′(τ,u)φ(τ,u)] − eαbα(τ,u)φ(τ,u)


Stochastic Processes & Nonholonomic Manifolds Diffusion on nonholonomic manifolds

Diffusion on nonholonomic manifolds

N–adapted parallel transport r = (u,e) = (uα,eβ′

β) ∈ O(V)

δuα = eαα′(uβ)δγα′ and δeαα′(uµ) = −Γαβν(u

µ)eνα′(uµ)δuβ

Extending on O(V) the fundamental d–vector fieldsLα′ → OLα′ = eαα′e α − Γαβν(u

µ)eβα′eνβ′(∂/∂eββ′),

L0 → OL0 = Aα(τ,U)e α − ΓαβνAβ eνβ′(τ)(∂/∂eββ′)

Proj. f (r) = f (u,0), r = (uα,eβ′

β)),OA = ρ

2

n+m∑α′=1

OLα′OLα′ + OL0

Diffusion and Lapplace–Beltrami d–operators OAf (r) = VAf (u),

VA =ρ

2

∑

α′

eαα′e α(eβα′e β) + Aβ e β =

ρ

24 + Aβ e β

4 =12

gαβ[e αe β + e βe α +

(Γναβ + Γνβα

)e ν

]


Nonholonomic Diffusion in General Relativity The special relativistic nonholonomic diffusion

Nonholonomic Diffusion in General Relativity

The special relativistic nonholonomic diffusion

4–d Minkovski spacetime 31M , hyperbolic structure,

−(v1)2 + (v2)2 + (v3)2 + (v4)2 = −1,on T ( 3

1M), uα = (x i , ya = va); i , j , ... = 1, 2, 3, 4; a, b, c... = 5, 6, 7, 8.

hyperboloid metric and Christoffel connectionhab(vc) = δab − v av b/(v1)2, γa

bc(ve) = v ahbc

N–connection and d-metric, N = N ak (uα), u = uα = (x i , v a),

g = ηijdx i ⊗ dx j + hab(dv a + N ak dxk )⊗(dv b + N b

k dxk )

can. d–con. Γγαβ, Eα′ = Eαα′(u)∂α = Eαα′(u)Eα;

gα′β′ = [gi ′j ′ = ηi ′j ′ ,ha′b′ = δa′b′ ] = Eαα′Eβ

β′gαβ, r = uα,Eαα′

N–adapted relativistic stochastic eqs, frame bundle spaceδuα = Eαα′(τ) δW α′

+ Aα(τ)dτ, δEαα′(τ) = −Γαβν

(τ)Eνα′(τ) δuβ


Nonholonomic Diffusion in General Relativity Nonholonomic diffusion and gravitational interactions

Nonholonomic diffusion and gravitational interactions

N–adapt. relativ. diffusion & gravit. fields

Local hyperbolic massive particles: gµν(uα)vµvν = −1

N–adapted frames: V, ηα′β′ = gαβeαα′(u) eββ′(u),

Paral. transp. δvα′= −Γα

′

βγ′(uβ)vγ

′δuβ, δuβ = eββ′(uα)vβ

′δτ

N–ad. SDE diffusion δuα = eαα′(uβ)vα′δτ, force exBα = F a/m0,

δv α = E αα′(τ) δW α′ − Γαβγ′(u

γ)eβα′(uγ)vγ′vα′

δτ + ex Bαδτ,

δE αα′(τ) = −γα

βγ(v ϕ)E γ

α′ δv β ,

(A,L)–diff., coord. E r = uα = (x i , ya), v β ,E αα′, diff. F(V)A and

Laplace–Beltrami from F La′ = E αα′

∂∂v α − γαεγ(v

ϕ)E γα′E ε

β′

∂∂E α

β′

,

F L0 = eαα′(uβ)vα′eα − Γαβγ′(u

γ)eβα′(uγ)vγ′vα′ ∂

∂v α + exBα ∂∂v α −

γαεγ(vϕ)E γ

β′Bε ∂

∂E α

β′

; Bε = −Γεβγ′(uγ)eβα′(uγ)vγ

′vα

′+ ex Bε.


Exact Stochastic Solutions in Gravity The Einstein eqs on nonholonomic manifolds

The Einstein eqs on nonholonomic manifolds

Two equivalent representations of Einstein eqs

Levi–Civita ∇, R βδ − 12gβδR = κTβδ,

canonic d–connect D, R βδ −12

gβδ sR = Υβδ,

Lcaj = ea(Nc

j ), C ijb = 0, Ωa

ji = 0.

Ansatz for solutions:

ηg = ηi (xk , v) gi(xk , v)dx i ⊗ dx i + ηa(xk , v) ha(xk , v)ea⊗ea,

e3 = dv + η3i (xk , v) wi(xk , v)dx i , e4 = dy4 + η4

i (xk , v) ni (xk , v)dx i

gij = diag[gi = ηigi ] and hab = diag[ha = ηa

ha] andN3

k = wi = η3i

wi and N4k = ni = η4

ini ; Gravit.’polarizations’ ηα

and ηai ,

g = [ gi ,ha,

Nak ] → ηg = [ gi ,ha,Na

k ], functions ofnecessary smooth class and/or any random (stochastic) variables.


Exact Stochastic Solutions in Gravity Nonholonomic separation of Einstein eqs

Nonholonomic separation of Einstein eqs

MAGIC d–connection SPLITTING for one Killing ansatz:ηg = gi(xk )dx i ⊗ dx i + h3(xk , t)e3⊗e3 + h4(xk , t)e4⊗e4,

e3 = dt + wi(xk , t)dx i ,e4 = dy4 + ni(xk , t)dx i

a• = ∂a/∂x1, a′ = ∂a/∂x2, a∗ = ∂a/∂v ; v = t ; τ via δτ =√|h3|δt .

R11 = R2

2 =−1

2g1g2[g••

2 − g•

1 g•

2

2g1− (g•

2 )2

2g2+ g′′

1 − g′

1g′

2

2g2− (g′

1)2

2g1] = −Υ4(xk ),

R33 = R4

4 = − 12h3h4

[h∗∗

4 − (h∗

4)2

2h4− h∗

3h∗

4

2h3] = −Υ2(xk , v),

R3k =wk

2h4[h∗∗

4 − (h∗

4)2

2h4− h∗

3h∗

4

2h3] +

h∗

4

4h4

(∂k h3

h3+∂k h4

h4

)− ∂k h∗

4

2h4= 0,

R4k =h4

2h3n∗∗

k +

(h4

h3h∗

3 − 32

h∗

4

)n∗

k2h3

= 0,

w∗

i = ei ln |h4|, ek wi = eiwk , n∗

i = 0, ∂ink = ∂k ni


Exact Stochastic Solutions in Gravity Stochastic solutions with h∗

3,4 6= 0 and Υ2,4 6= 0

Stochastic solutions with h∗3,4 6= 0 and Υ2,4 6= 0

Formal integration, sure solutions

h–metric: g1 = g2 = eψ(xk ), ψ + ψ′′ = 2Υ4(xk )

coefficients:φ = ln | h∗

4√|h3h4|

|, αi = h∗4∂iφ, β = h∗

4 φ∗, γ =

(ln |h4|3/2/|h3|

)∗

N–connection eqs: βwi + αi = 0, n∗∗i + γn∗

i = 0v–metric, if h∗

4 6= 0; Υ2 6= 0, we get φ∗ 6= 0. ∀φ = φ(x i , t) 6= const isa solution generating function, h∗

4 = 2h3h4Υ2(x i , t)/φ∗.

solution: h3 = ± |φ∗(x i ,t)|Υ2

, h4 = 0h4(xk ) ± 2∫ (exp[2 φ(xk ,t)])∗

Υ2dt ,

wi = −∂iφ/φ∗, ni = 1nk

(x i) + 2nk

(x i) ∫

[h3/(√|h4|)3]dt ,

integration functions 0h4(xk ), 1nk(x i) and 2nk

(x i)

Υi = λ, λ→ hλ(xk ) = Υ4(xk ) and λ→ vλ(xk , t) = Υ2(xk , t).


Exact Stochastic Solutions in Gravity Stochastic solutions with h∗

3,4 6= 0 and Υ2,4 6= 0

Stochastic solutions with h∗3,4 6= 0 and Υ2,4 6= 0

Metrics and sources with h–diffusion

Random gener. funct. φ(x k , t) → φ(xk , t) = φ(xk , t) +$φ(xk , t),where φ(xk , t) ∼ f (τ,u) = f (τ, x i) diffusion process from h–spaceto v–space; only h–operators and fix Aβ = 0 in VA, τ → t ,v = v0 = const and 4 is computed 4 = h4 for gij = δijeψ(xk ).

The generalized Kolmogorov backward equation∂τ f (τ, x i) = h4f ((τ, x i ) = h4f ((τ, x i), f (0, x i) = f (x i).

The generalized Fokker–Planck eq ∂τz = ρ2h4z,

z = z(τ, 1x i ; 0, 2x i) is the transition probability with the initialcondition z(0, 1x i ; 0, 2x i) = δ( 1x i − 2x i) for any two points1x i , 2x i∈ V and adequate boundary conditions at infinity.Solutions induced by random sources Υ2; similar to stochasticgravity (Einstein–Langeven eq with additional sources due to thenoise kernel).


Summary & Conclusions

Summary & Conclusions

We developed the theory of stochastic processed onnonholonomic Euclidean and Riemannian manifolds.Relativistic models of nonholonomic diffusion.Off–diagonal solutions for stochastic Einstein spacetimes.Multidisciplinary character of research: 1) Geometric Methods(nonholonomic, Riemann–Finsler, almost Kähler); 2) NonlinearEvolution/Field Equations (PDE, SDE, evolution eqs); 3)Probability and Stochastics (diffusion, anisotropic kinetics,statistics, thermodynamics); 4) Geometric quantization.

Outlook (recently developed, under elaboration):Gravity and quantum physics, geometric mechanics; variousapplications in modern cosmology and astrophysics, geometricmechanics etc. Generic nonlinear solutions, stochastic evolution,fractional derivatives, solitonics, singularities, memory etc.Diffusion and Porous Media with Self–Organized Criticality in RicciFlow Evolution of Einstein and Finsler Spaces.


Diffusion Processes on Nonholonomic Manifolds and Stochastic Solutions for Einstein Spaces

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