1 Diffusions with periodic coefficients

7

Probabilistic approach to homogenization

1 Diffusions with periodic coefficients

We consider first one dimensional diffusions with periodic coefficients. Assume that aandU are 1–periodic functions belonging toC²(R) anda(x)≥c0>0 for all x∈R. We use the notationC^m(R) for the space ofm times continu- ously differentiable functions.C₀^m(R) is its subspace consisting of compactly supported functions. Suppose that (Zx(t))t≥0 is a diffusion that starts at x∈R, which corresponds to the generator

Lf(x) := 1

2e^U(x) d dx

e^−U(x)a(x)df dx

, f ∈C₀²(R). (1.1) It is defined as the unique solution of the Itˆo stochastic differential equation

dZx(t) =V(Zx(t))dt+a^1/2(Zx(t))dw(t) Zx(0) = x, (1.2) where V(x) := −(1/2)[U⁰(x)a(x) +a⁰(x)] and (w(t))t≥0 is a standard one dimensional Brownian motion over a probability space (Σ,W,Q). Under reg- ularity assumptions made about the coefficients the solution to (1.2) is unique and has the strong Markov property, see e.g. [?] Theorem 5.3.4, p. 112. Its probability transitions have densities with respect to the Lebesgue measure p(t, x, y), defined on (0,+∞)×R², which are strictly positive andC¹smooth in the t variable and of C² class in both x and y. As a consequence, the transition probability semigroup for the diffusion

P^tf(x) :=EQf(Zx(t)) = Z

R

p(t, x, y)f(y)dy, f∈B(R)

has the strong Feller property, i.e.P^t(B(R))⊂C_b²(R) for anyt >0. HereEQ

stands for the expectation with respect toQ.

It is clear from (1.2) that the position of a particle can be expressed as a sum of an additive functional of the process plus a martingale so in order to

analyze the asymptotic behavior of the first term we could, in principle, use the machinery developed in Chapter 2 of the book. It should be noted however that the invariant measure for the diffusion, which can easily be observed equals e^−U(x)dx, is of infinite mass. This fact prevents us from applying directly the general results obtained in Chapter 2. To remedy the situation we shall work with the diffusion that is induced by Zx(t) on the corresponding compact state space – the one dimensional torusT:=R/∼. Here∼is an equivalence relation defined by x ∼ y for x, y ∈ R iff x−y ∈ Z. Denote by ˙x ∈ T the equivalence class of a givenx ∈ R. Each 1 periodic functionf onR defines the corresponding function on the torusf( ˙x) :=f(x). For a given functionf and point ˙x∈Twe can define its derivative at the point of the torus as

Df( ˙x) := lim

h→0

f( ˙x⊕h)−f( ˙x)

h , (1.3)

where⊕denotes the addition modulo 1. We shall also denote byC^m(T^d) the space ofmtimes differentiable functions on addimensional torus.

The process ( ˙Zx(t))t≥0 is Markovian onTwhose transition of probability densities, with respect to the Lebesgue measure on the torus, are given by

p(t,x,˙ y) =˙ X

y∈y˙

p(t, x, y)>0. (1.4)

The generator of the process is given by Lf( ˙x) = 1

2e^{U( ˙x)}D

e^{−U( ˙x)}a( ˙x)Df( ˙x)

, f ∈C²(T). (1.5) We denote by ( ˙P^t)t≥0 the semigroup corresponding to ( ˙Zx(t))t≥0. It leaves C²(T) invariant, so it is a core of the generator. Letπbe a probability measure on T given by π(dx) :=˙ Z⁻¹e^{−U( ˙x)}dx, with˙ Z := R

Te^{−U( ˙x)}dx. Note that˙ R Lf dπ = 0 for all f ∈ C²(T), so we conclude that π is invariant for the semigroup ( ˙P^t)t≥0. It is clearly ergodic. Indeed, suppose that a Borel subset Asatisfies1{A}=P^t1{A}for somet >0. Then, for all ˙x∈T^d

0 =1{A^c}(x)P^t1{A}(x) = Z

A

1{A^c}( ˙x)p(t,x,˙ y)d˙ y,˙

which, thanks to strict positivity of the transition of probability density, im- plies that A is of trivial Lebesgue, thus also, π measure. As an immediate consequence of the ergodic theorem we conclude therefore that fort→+∞

Zx(t) t = 1

t



x˙ + Zt

0

V( ˙Zx˙(s))ds+ Zt 0

a^1/2( ˙Zx˙(s))dw(s)



 tends to

1 Diffusions with periodic coefficients 163 Z

T

V( ˙x)π(dx) =˙ 1 Z

Z

T

D

a( ˙x)e^{−U( ˙x)} dx˙ = 0 for a.e. ˙x andQa.s. realization of (Zx(t))t≥0.

To conclude the central limit theorem we look for the solution χof the equation

−Lχ=V (1.6)

calledthe cell problem,. The above equation reads

−1

2e^{U( ˙x)}D

e^{−U( ˙x)}a( ˙x)Dχ( ˙x)

= 1

2e^{U( ˙x)}D

e^{−U( ˙x)}a( ˙x)

(1.7) hence

Dχ( ˙x) =−1 +C1e^{U( ˙x)}a⁻¹( ˙x) (1.8) for some constant C1 that must be chosen so that R

TDχ( ˙x)dx˙ = 0. This requirement is satisfied iff

C1= Z

T

e^{U( ˙x)}a⁻¹( ˙x)dx˙ −1

. From (1.8) we obtain

χ( ˙x) =−x˙+C1

Z x˙ 0

e^{U( ˙y)}a⁻¹( ˙y)dy˙+C,

where C is a certain constant. It should be chosen in such a way that χ ∈ C²(T). We define χ:R→R, using the solution of (1.7), byχ(x) :=χ(˙)x. It is aC²regular, 1-periodic function that satisfies−Lχ(x) =V(x). Using Itˆo’s formula forχ(Zx(t)) we obtain then

χ(Zx(t)) =χ(x) + Zt 0

Lχ(Zx(s))ds+ Zt

0

a^1/2(Zx(s))χ⁰(Zx(s))dw(s).

Hence,

Zx(t) =x+χ(x)−χ(Zx(t)) + Zt 0

a^1/2(Zx(s))[χ⁰(Zx(s)) + 1]dw(s).

Denote by Mt the martingale term appearing on the right hand side above.

Using (1.8) we obtain that its quadratic variation satisfies 1

thMi^t= 1 t Zt

0

a( ˙Zx˙(s))[Dχ( ˙Zx˙(s)) + 1]²ds.

By the ergodic theorem, the right hand side of the above equality converges forQa.s. realization ofZx(t) and for a.e.x∈Rto

σ²:=

Z

a( ˙x)[−1 +C1e^{U( ˙x)}a⁻¹( ˙x)]²π(dx) =˙ C1

Z , (1.9)

as t →+∞. By the martingale central limit theorem, see Lemma 2.2.1, the laws of t^−1/2Zx(t) conditioned on the initial condition x converge to a zero mean normal distribution. The limiting variance is given by expression (1.9) above.

We remark here that the multi-dimensional case does not differ very much from the one dimensional situation discussed above. In that caseais a periodic d×d–matrix valued function whose symmetric parta^s(x)≥c0I, whereI is the identity matrix andc0>0. The driftV = (V1, . . . , Vd) is given by

Vk(x) =1

2e^U(x)∇^x·h

e^−U(x)a(x)ek

i. (1.10)

Analogously to the one dimensional case we introduce ad–dimensional torus T^d := R^d/ ∼, where x ∼ y iff x−y ∈ Z^d. We maintain the convention of denoting by ˙x the equivalence class of x and identifying the notation of a given 1-periodic functionf :R^d→Rwith the one induced onT^d.

Letπ(dx) :=˙ Z⁻¹e^{−U( ˙x)}dx˙ and letL²₀(π) := [f ∈L²(π) :R

Tf dπ= 0]. For each`∈R<sup>d</sup> V ·`∈L²₀(π) and we can formulate the respective cell problem

−Lχ`( ˙x) =V( ˙x)·`. (1.11) HereLis the closure, in L²₀(π), of the operator

Lf( ˙x) = 1 2e^{U( ˙x)}

Xd k,l=1

Dk

e^{−U( ˙x)}akl( ˙x)Dlf( ˙x)

, f ∈C²(T^d)∩L²₀(π).

(1.12) The partial derivatives on thed–dimensional torusDk,k= 1, . . . , dare defined analogously to (1.3) for the corresponding directions ek, k = 1, . . . , d. We recall the Poincare inequality , see Theorem 5.8.1 of [?], which states that quote evans

there exists a constantλ0>0 such that for allf ∈C¹(T^d)

λ0



 Z

T^d

f²dπ−



Z

T^d

f dπ





2

≤ Z

T^d

|∇f|²dπ. (1.13)

The inequality implies, in particular, that the resolvent set of−L:D(L)→ L²₀(π) contains 0, so equation (1.11) can be uniquely solved. Standard theory of elliptic partial differential equations implies thatχ`∈C²(T^d). The remaining part of the proof that t^−1/2Zx(t) satisfies the central limit theorem is the same as in the one dimensional case. The quadratic form corresponding to the limiting covariance matrixDis given by the formula

`·D`= Z

T^d

a(`+∇χ`)·(`+∇χ`)dπ. (1.14)

2 Quasi-periodic case. 165 The only significant difference from the one dimensional situation lies in the fact the formula for χ`, hence also the expression (1.14) for the limiting co- variance, is not given explicitly. We have not discussed above the details of the homogenization argument for multidimensional periodic diffusions since this type of processes can be considered as a special case of diffusions whose coefficients are given by stationary random fields that shall be discussed in the following sections.

2 Quasi-periodic case.

Suppose that a, U :R×R→RareC² smooth functions, 1-periodic in both variables,a(x, y)≥c0>0,x, y∈R. We consider a diffusion (Zx(t))t≥0 that starts atx and whose generator equals

Lf(x) := 1

2e^U(x,λx) d dx

e^−U(x,λx)a(x, λx)df dx

, f ∈C₀²(R). (2.1) Here λ is a certain irrational number. Let Zx^(λ)(t) := λZx(t). We define η(t) := ( ˙Zx(t),Z˙x^(λ)(t)),t≥0. It is a diffusion onT×T. The generator of the corresponding transition of probability semigroup (P^t)t≥0 can be calculated with the help of Itˆo’s formula and it equals

Lf( ˙x,y) =˙ 1

2e^{U( ˙x,y)˙} D

e^{−U( ˙x,y)˙} a( ˙x,y)Df˙ ( ˙x,y)˙

, f ∈C²(T×T). (2.2) The derivation operator

Df( ˙x,y) :=˙ Dx˙f( ˙x,y) +˙ λDy˙f( ˙x,y),˙ f ∈C²(T×T), (2.3) whereDx˙,Dy˙are given by (1.1.3) on the respective variables. In factC²(T×T) is a core ofL. This can be concluded from its invariance under the semigroup and Theorem 8.2.5, p. 373 of [?]. Note that

Df( ˙x,y) =˙ i X

(k,l)∈Z²

(k+λl) ˆf(k, l)e^{2πi(kx+l˙ y)˙} , (2.4)

where

fˆ(k, l) :=

Z

T²

e^{−2πi(kx+l˙ y)˙} f( ˙x,y)d˙ xd˙ y˙

are the Fourier coefficients off. We can extent operatorD to a subspaceH¹ ofL²(T×T) consisting of all functionsf for which

X

(k,l)∈Z²

|k+λl|²|fˆ(k, l)|²<+∞.

Arguing in the same way as in (1.??) we can conclude that the measure π(dx, d˙ y) :=˙ Z⁻¹e^{−U( ˙x,y)˙} dxd˙ y, where˙ Z is a normalizing factor, is invariant probability measure for the semigroup (P^t)t≥0. Observe thatD(L)⊂H¹and the Dirichlet form corresponding to (η(t))t≥0 equals

hf,(−L)fiπ= Z 2

Z

a(Df)²dπ, f ∈D(L). (2.5) From the above formule we conclude thatπ is an ergodic, invariant measure for (η(t))t≥0. Indeed, for anyf ∈D(L) that satisfiesLf= 0 we haveDf = 0 and, sinceλis irrational, we obtain, see (2.4), that ˆf(k, l) = 0 for all (k, l)∈ Z²\ {(0,0)}. This implies thatf ≡const.

In analogy with the periodic case we formulate the cell problem (1.6) which leads to the equality

Dχ( ˙x,y) =˙ F( ˙x,y),˙ (2.6) cf. (1.8), where

F( ˙x,y) =˙ −1 + Z

e^Ua⁻¹dπ −1

e^{U( ˙x,y)˙} a⁻¹( ˙x,y).˙

From (2.6) we obtain that the Fourier coefficients of the correctorχ( ˙x,y) are˙ equal ˆχ(0,0) = 0 and

ˆ

χ(k, l) := Fˆ(k, l)

i(k+λl) for (k, l)∈Z²\ {(0,0)}. One cannot guarantee that P

|χ(k, l)ˆ |² < +∞ without making some addi- tional assumptions onaandU. In general therefore the cell problem (1.6) for quasi-periodic coefficients has no solution inL²(π). As a result, we cannot pro- ceed with the argument made in the periodic setting. In the following chapters we describe how the results of Chapter 2 can be invoked to remedy the situ- ation. The crucial observation is that for homogenization it suffices to work only with the correctorχwhose gradient belongsL²(π). This is in fact typical for diffusions with random, stationary coefficients. In fact, as we explain in the following chapter, both homogenization of diffusions with periodic and quasi- periodic coefficients can be viewed as a special case of a general procedure that allows to homogenize random diffusions with stationary coefficients.

The simple example presented above can be generalized both in dimension and the number of irrational period ratios. We can consider therefore diffusions with the generators of the form

Lf(x) := 1

2e^U(x)∂xk

e^−U(x)akl(x)∂lf(x)

, f ∈C₀²(R^d), (2.7) where U(x) = V(x, λ1x, . . . , λmx) and a(x) = b(x, λ1x, . . . , λmx) for some m ≥1. The functions V and b are correspondingly scalar and d×dmatrix valued C² smooth functions, 1 periodic in all the variables, b satisfies the uniform ellipticity condition. The multipliers λ1, . . . , λm are irrational and linearly independent over the field of the rational numbers.

3 General remarks on homogenization of random Markov processes. 167

3 General remarks on homogenization of random Markov processes.

We describe here a general framework that allows us to work with Markov processes whose transition of probability functions are random fields. Assume that X is one of the additive groupsZ^d, or R^d. Suppose that (Ω,F,P) is a certain probability space. Let (τx)x∈X be a group of transformations acting on Ω and preserving measureP. By the above we mean that τx : Ω → Ω, τx◦τy = τx+y for all x, y ∈ X and the group action in question has the following properties:

τx(A)∈ F for eachA∈ F andx∈X (measurability), (3.1)

P◦τx=Pfor allx∈X (homogeneity), (3.2)

if1{A} ◦τx=1{A}for allx∈X thenP[A] = 0, or 1 (ergodicity). (3.3) Example 7.1.Suppose that Ω := T^d. In this case we let P := md be the Lebesgue measure on the d-dimensional torus and F theσ-algebra of Borel sets. The additive groupR^dacts onT^dvia shifts modulo 1, i.e.τx( ˙y) := ˙y⊕x,˙ where ˙y ∈ T^d, x ∈ R^d and ⊕ is the addition modulo 1. One can easily check that all conditions (3.1)-(3.3) are satisfied in this case. Note that any function f : T^d → R gives rise to a 1-periodic, homogeneous random field f˜(x; ˙y) =f(τxy), (x,˙ y)˙ ∈R^d×T^d.

Example 7.2.Assume that we are given mreal numbers λ1, . . . , λmthat are independent over the field of rationals. LetΩ:= (T^d)^m. In this case we letP be the tensor product ofmcopies of the Lebesgue measuremd. The additive groupR^d acts on (T^d)^m by

τx( ˙y1, . . . ,y˙m) := ( ˙y1⊕x˙1, . . . ,y˙m⊕x˙m),

where ˙y∈T^d, ˙xiis a projection ofλixontoT^d,i= 1, . . . , d. We leave a reader as an exercise to show that conditions (3.1)-(3.3) are met in this case as well.

Also in this case a quasi-periodic function f gives rise to a quasi-periodic, homogeneous random field via

f˜(x; ˙y1, . . . ,y˙m) =f(τx( ˙y1, . . . ,y˙m)), (x,y˙1, . . . ,y˙m)∈R^d×(T^d)^m. Example 7.3.Finally, suppose that ˜f :R^d×Σ→Ris a homogeneous, random field over a certain probability space (Σ,W,Q), see Definition??.??. Consider Ω as the space of all functionsω:R^d →R. Let F be the smallestσ algebra generated by all the sets of the form [ω:ω(x1)∈A1, . . . , ω(xn)∈An], where n≥1 is an integer,x1, . . . , xn ∈R^d,A1, . . . , An ∈ B(R) - Borel measurable subsets ofR. We letPbe the unique measure onF determined by equality

P[ω:ω(x1)∈A1, . . . , ω(xn)∈An] =Q[ ˜f(x1)∈A1, . . . ,f˜(xn)∈An]

for all n ≥ 1, x1, . . . , xn ∈ R^d, A1, . . . , An ∈ B(R). Let further τx(ω)(·) :=

ω(x+·) be the function obtained fromω by translation of its argument. It is clear that the field ˜f is stationary iff P◦τx =P for allx ∈R^d. Note that f˜1(x;ω) := f(τxω), where f(ω) := ω(0) is a random field whose all finite dimensional distributions coincide with those of ˜f.

To describe a random family of Markov processes we suppose that for each ω ∈Ω we are given a family of transition of probabilities P^ω(t, x,·),t ≥ 0, x∈X,ω∈Ωthat are homogeneous w.r.t. the action of the groupG, i.e. they satisfy

P^τyω(t, x, B) =P^ω(t, x+y, B+y) for allx, y∈X andB∈ B(X). (3.4) Let (Σ,W,Q) be a certain probability space and let EQ denote the re- spective mathematical expectation. We suppose that (Z_x^ω(t))t≥0 is a Markov process defined over this probability space that starts at x and corresponds to the given family of transition of probabilities. By a random Markov pro- cess (Zx(t))t≥0 we understand a stochastic process defined over the product probability space (Ω×Σ,F ⊗ W,P⊗Q) and given byZx(t;ω, ζ) :=Z_x^ω(t;ζ).

Our main objective is to find conditions under which (Zx(t))t≥0 satisfies the Central Limit Theorem, i.e. the laws of t^−1/2Zx(t) converge, as t→+∞, to the law of a normal random vector.

It should be stressed that although for each ω the process (Z_x^ω(t))t≥0

is Markovian the process (Zx(t))t≥0 in general does not have the Markov property. In many cases however we are able to find the representation of the process as a sum of a martingale and an additive functional of a certain Ω–valued Markov process (η(t))t≥0. Namely, using the Markov structure of the process for a”frozen” ω, we are able to write

Z_x,k^ω (t) =M_k^ω(t) + Z t

0 L^ωfk(Z^ω(s))ds, k= 1, . . . , d, (3.5) where L^ω is the generator of Z_x^ω(t) = (Z_x,1^ω (t), . . . , Z_x,d^ω (t)),t ≥0,M^ω(t) = (M₁^ω(t), . . . , M_d^ω(t)),t≥0 is anR^d–valued martingale and

fk(y) =yk, y∈X, k= 1, . . . , d. (3.6) Sometimes we can write L^ωfk(x) =Vk(τxω), where V = (V1, . . . , Vd) :Ω→ R^d. Then the decomposition (3.5) leads to

Zx(t) =M^ω(t) + Z t

0

V(η(s))ds. (3.7)

The process η(t) appearing in (3.5) is obtained by lifting of the trajectory Z_x^ω(t) to the”large” spaceΩvia the action of the group (τx)x∈X, that is

η(t) :=τZ^ω_x(t)ω. (3.8)

4 Random walks on random lattices 169 The property (3.4) implies (see the proof of Proposition 7.4 below) that (η(t))t≥0 is Markovian. This process shall be called the environment process (sometimes it is also called the Lagrangian, or quasi-Lagrangian process).

In some examples we are able to determine an ergodic, invariant measure π for the environment process that is mutually absolutely continuous with respect to the homogeneous measure P. We can also introduce the L²(π) – generatorLof the environment process. To decompose the bounded variation part of Zx(t), see (3.7), into a martingale part and a perturbation that is negligible on the time scale∼t^1/2 we need to solvethe random cell problem

−Lχ^(p)=Vp, p= 1, . . . , d. (3.9) As we have indicated in Section 2 in general this problem is not solvable in L²(π). We apply therefore the method developed in Chapter 2 and find the solution to (3.9) in an appropriate space H1 defined with the help of the symmetric part of the generator of (η(t))t≥0. In the following chapters we illustrate this approach with some concrete examples of random Markov processes, such as diffusions with stationary random coefficients. We start however with a simple model of a double stochastic random walk on an integer valued random lattice.

4 Random walks on random lattices

In this section we suppose that X =Z^d and the probability space (Ω,F,P) together with a group G are as described in the introduction. By B(Ω) we shall denote the space of all bounded and measurable functions onΩ.

Assume that we are given a family of functionspz:Ω→[0,+∞),z∈Z^d that satisfy the following conditions:

• pz∈B(Ω) for allz∈Z^d, (4.1)

• pz is offinite range, i.e. there exists a deterministicR >0 (4.2) such thatpz= 0, if|z| ≥R,

• P

zp−z(τzω) =P

zpz(ω) forP–a.s.ω, (4.3)

• the sub-lattice generated by thosez, for which (4.4) pz(ω)>0 coincides with Z^d forP–a.s.ω (irreducibility).

Let (Z_x^ω(t))t≥0 be a continuous time random walk onZ^d, defined over a certain probability space (Σ,W,Q), that starts atxand jumps from a givenytoy+z at the rate

p(y, y+z;ω) :=pz(τyω). (4.5) Condition 4 guarantees that the random walk in question is irreducible. Note that thanks to property (4.5) the transition of probability functions of the random walk satisfy

P^ω(t, x+z, y+z) =P^τzω(t, x, y) (4.6) for allx, y∈Z^d, t≥0 andω∈Ω. The generator of the random walk described above is given by

Lωf(x) =X

z

p(x, x+z;ω) [f(x+z)−f(x)]

for all f ∈ B(Z^d) – the space of bounded functions on the lattice Z^d. Note that condition (4.3) implies thatP

yp(y, x;ω) =P

yp(x, y;ω) for allx ∈Z^d and a.s. ω. The adjoint generator with respect the counting measure on Z^d equals therefore

L^∗ωg(x) =X

z

p(x+z, x;ω) [g(x+z)−g(x)]

for anyg that belongs toB(Z^d). SinceL^∗ω1= 0, it is clear that the counting measure is invariant for the random walk corresponding to a givenω.

The random walk on a random integer lattice starting atx is the process Zx(t;ω, ζ) :=Z_x^ω(t;ζ) considered over the product space (Ω×Σ,F⊗W,P⊗Q).

We shall omit the starting point in the notation if x= 0.

Letfk(z) :=zk,k= 1, . . . , d. We definethe local drift of the random walk as the random vectorV = (V1, . . . , Vd), where

Vk(ω) :=Lωfk(0) =X

z

zk pz(ω) and assume that its average

hVi^P=X

z

zp¯z= 0. (4.7)

Here ¯pz:=hpzi. With these definitions Z^ω(t) =M^ω(t) +

Z t 0

V(η(s))ds. (4.8)

Here M^ω(t) = (M₁^ω(t), . . . , M_d^ω(t)) is a vector valued mean zero martingale with respect to the natural filtration corresponding to (Z^ω(t))t≥0and (η(t))t≥0

is the environment process defined by (3.8).

Proposition 7.4.The environment process (η(t))t≥0 is Markovian. Its gen- erator is given by

Lf(ω) =X

z

pz(ω)Dzf(ω). (4.9)

Here f ∈ B(Ω) and Dzf(ω) := f(τzω)−f(ω). Moreover, the measure P is invariant and ergodic under the process.

4 Random walks on random lattices 171 Proof. Suppose that n ≥1 is an integer 0 ≤t1 ≤. . . ≤ tn ≤t, h≥ 0 and g1, . . . , gn, f ∈B(Ω). From the definition of η(t) we obtain

hg1(η(t1)). . . gn(η(tn))f(η(t+h))i (4.10)

=hEQ[g1(τZ^ω(t1)ω). . . gn(τZ^ω(tn)ω)f(τZ^ω(t+h)ω)]i.

Using the Markov property of the random walk for a fixed environmentω we can rewrite the right hand side of (4.10) as being equal to

* EQ

"

g1(τZ^ω(t1)ω). . . gn(τZ^ω(tn)ω) X

y

P^ω(h, Z^ω(t), y)f(τyω)

!#+

P

(4.11)

(4.6)

=

* EQ

"

g1(η(t1)). . . gn(η(tn)) X

y

P^η(t)(h,0, y−Z^ω(t))f(τyω)

!#+

P

. Substituting y := y−Z^ω(t) in the summation above and using again the definition of the environment process we obtain that the right hand side of (4.11) equals

hEQ[g1(η(t1)). . . gn(η(tn))P^hf(η(t))]i^P, where

P^hf(ω) :=X

y

P^ω(h,0, y)f(τyω), for allf ∈B(Ω). (4.12) This proves the Markov property for the environment process and also iden- tifies the transition of probability semigroup as given by the formula (4.12).

The formula (4.9) is obtained by differentiating the expression (4.12) with respect to hath= 0.

Thanks to conditions (4.3) and (4.4) we can conclude that the probability measure Pis stationary and ergodic for this process. Indeed, condition (4.3) and homogeneity ofPtogether imply that

hLfi^P=X

z

hpz◦τ−zfi −X

z

hpzfi^P

=X

z

hp−z◦τzfi −X

z

hpzfi^P= 0

and the stationarity ofPfollows. To show ergodicity let us calculate the Dirich- let form of the process. Using again the condition (4.3) we obtain that the form equals

hf,(−L)fi^P =1 2

X

z

hpz(Dzf)²i^P. (4.13) Ergodicity of the stationary measure P can be now easily checked. Indeed, let f ∈ L² be such that Lf(ω) = 0. Then, by (4.13) and the irreducibility assumption, f is invariant under the action of the group of shifts (τx)x∈Z^d.

Since the action of this group is ergodic, it follows thatf is constant P-a.s.

We are going to rewrite further decomposition (4.8) representing the ad- ditive functional in (3.5), as a martingale plus a small remainder term as we have done in formula (2.4.7). Let us considerχ^(k)_λ ∈L²(P) – the solution of the resolvent equation

λχ^(k)_λ −Lχ^(k)_λ =Vk (4.14) for a givenk= 1, . . . , d. LetNλ(t) = (N1,λ(t), . . . , Nd,λ(t)) be anR^d –valued martingale given by

Nk,λ(t) :=χ^(k)_λ (η(t))−χ^(k)_λ (ω)− Z t

0

Lχ^(k)_λ (η(s))ds. (4.15) Then, from (3.5) we conclude that

Z(t) = ˜Mλ(t) +Rλ(t), (4.16) where

M˜λ(t) =M(t) +Nλ(t), (4.17) The remainder termRλ(t) = (R1,λ(t), . . . , Rd,λ(t)) is given by

Rk,λ(t) =χ^(k)_λ (ω)−χ^(k)_λ (η(t)) +λ Z t

0

χ^(k)_λ (η(s))ds, (4.18) k= 1, . . . , d. We calculate now the matrix of quadratic co-variation of the mar- tingale ˜Mλ(t). Let ˜χ^(k)_λ (x, ω) :=χ^(k)_λ (τxω) andyk(x, ω) :=fk(x) + ˜χ^(k)_λ (x, ω), fk(x) =xk, cf. (3.6). SinceLχ^(k)_λ (τxω) =Lωχ˜^(k)_λ (·, ω)(x) we can write

M˜k,λ(t) =yk(Z^ω(t))−yk(0)− Zt

0

Lωyk(Z^ω(s))ds.

The quadratic co-variation of the above martingale equals (see e.g. [?] Lemma 5.1., p. 330)

hM˜k,λ,M˜l,λit (4.19)

= Zt 0

[Lω(ykyl)(Z^ω(s))−(ykLωyl) (Z^ω(s))−(ylLωyk) (Z^ω(s))] ds The right hand side of the above equality can be computed explicitly and it equals

X

z

Zt 0

pz(η(s))h

zkzl+

Dzχ^(k)_λ Dzχ^(l)_λ (η(s))i

ds.

4 Random walks on random lattices 173 In light of the results of Section 2.4, see in particular the proofs of Lemmas 2.2.7 and 2.2.8, to claim the Central Limit Theorem for the process Z(t) it suffices only to show that bothλkχ^(k)_λ k²0→0 andχ^(k)_λ converges to someχ^(k) in H1, asλ→0+. These statements can be concluded however, see Section 2.6.2, if we admit the following hypotheses:

• V(·)∈ H−1, (4.20)

• Lsatisfies a sector condition. (4.21)

We have proved therefore the following.

Theorem 7.5.Suppose that the random rates pz(ω), z∈Z^d satisfy the con- ditions (4.1)−(4.4). In addition, we assume that the hypotheses (4.20) and (4.21) hold. Then, the laws of t^−1/2Z(t) converge in P-probability w.r.t. ω, as t → +∞, to a d-dimensional, mean zero Gaussian distribution with the covariance matrix D= [Dkl] given by

Dk,l=σ_k,l² + 2hχ^(k), χ^(l)i¹, (4.22) whereσ²_k,l:=P

zzkzlp¯z, k, l= 1, . . . , d.

Since D ≥σ² := [σ_kl²] it is clear from the irreducibility condition (4.4) that the limiting covariance matrix is non-trivial. Observe also that (4.20) requires that the local drift has zero mean, see (4.7).

In what follows we discuss some sufficient conditions that guarantee that (4.20) and (4.21) hold. Suppose that

pz(ω) =p−z(τzω), (i.e.p(x, y;ω) =p(y, x;ω)). (4.23) Then, measurePis reversible for (η(t))t≥0 and in this case (4.21) holds triv- ially. It is not difficult to verify (4.20) in this situation. Indeed,

V(ω) = 1 2

X

z

(zpz(ω) +zp−z(τzω)) =−1 2

X

z

zD^∗_zpz(ω),

where D^∗_z = D−z is the adjoint of Dz under the scalar product h·,·i^P. In consequence

|hVk, fi^P|= 1 2

X

z

zkhpz, Dzfi^P

(4.24)

≤1 2

X

z

z_k²hpzi^P

!1/2

X

z

hpz(Dzf)²i^P

!1/2

.

Conditions (4.3) and (4.4) guarantee that all the components of V belong to H−1. We have proved therefore the following.

Corollary 7.6.Suppose that the random ratespz(ω), z∈Z^d satisfy the con- ditions (4.1)–(4.4) and the reversibility assumption (4.23). Then, (Z(t))t≥0

satisfies the central limit theorem as asserted in Theorem 7.6.

5 Random walks admitting a cycle representation

In this section we give an example of a non-reversible random walk satisfying hypotheses (4.20) and (4.21). To construct it we start with a definition of a cycle.A cycle of length nis a sequence of points (0, z1, . . . , zn−1,0) inZ^d. We let

pC(x, y) :=

Xn j=1

1(zj−1,zj)(x, y), x, y ∈Z^d.

Here zn = z0 := 0. Given a cycleC, we consider all its translations C+z for allz∈Z^d. For a strictly positive ergodic random fieldW(x, ω) =W(τxω) such thatkWk∞<+∞, we define the family

pz(ω) =X

u

W(u, ω)pC+u(0, z)

and then subsequently we determinep(x, y;ω) using formula (4.5). More gen- erally, we can consider a finite number of cyclesC1, . . . , CM all of them con- taining 0 and let

pz(ω) = XM k=1

X

u

Wk(u, ω)pCk+u(0, z). (5.1) It is easy to see that the walk satisfies the conditions (4.1)– (4.3). We shall always assume thatpz(ω) defined by (5.1) satisfies the irreducibility condition (4.4).

We verify now the hypotheses (4.20) and (4.21). Obviously, it is enough to check them forM= 1. In fact, a simple calculation shows that

p(x, x+y;ω) = Xn j=1

X

z

W(x+z, ω)1z+zj−1,z+zj(0, y)

= Xn j=1

W(x−zj, ω)1zj−zj−1(y).

This permits to write the random generator of the random walk as L^ωf(x) =

Xn j=1

W(x−zj−1, ω)(f(x−zj−1+zj)−f(x)),

which implies in turn that the generator of the environment equals Lf(ω) =

Xn j=1

W(−zj−1, ω)(f(τzj−zj−1ω)−f(ω)).

SincePn

j=1(W(zj−1, ω)−W(zj, ω)) = 0, we have

5 Random walks admitting a cycle representation 175 hg, Lfi^P=

Xn j=1

hW g◦τzj−1, f◦τzj−f◦τzj−1i^P

= Xn j=1

hW(g◦τzj−1−g), f◦τzj−f◦τzj−1i^P

= Xn j=1

Xj−1 k=1

hW(g◦τzk−g◦τzk−1), f◦τzj−f◦τzj−1i^P

(5.2)

When f =g we get the formula for the Dirichlet form hf,(−L)fi^P

= 1 2

Pn j=1

W(f ◦τzj−f◦τzj−1)²

P−

WPn

j=1(f◦τzj −f◦τzj−1)2

P

=1 2

Pn

j=1hW(f ◦τzj−f◦τzj−1)²i^P.

(5.3) The sector condition (4.21) is then a consequence of the Schwarz inequality applied to the utmost right hand side of (5.2)

|hg, Lfi^P|²≤ hPn j=1

Pj−1

k=1hW(g◦τzk−g◦τzk−1)²i^Pi

×hPn j=1

Pj−1

k=1hW(f◦τzj −f◦τzj−1)²i^Pi

≤4n²hf,(−L)fi^Phg,(−L)gi^P. It remains yet to be checked that V ∈ H−1. Observe that

V(ω) = Xn j=1

(zj−zj−1)W(−zj−1, ω)

= Xn j=1

zj(W(−zj−1, ω)−W(−zj, ω)).

We can write therefore that

hV, fi^P =hW Xn j=1

zj(f◦τzj−f◦τzj−1)i^P

≤( Xn j=1

z_j²)^1/2hWi^1/2P h Xn j=1

W(f◦τzj −f◦τzj−1)²i^1/2P (7.2)

= ( Xn j=1

z_j²hWi^P)^1/2hf,(−L)fi^1/2P

As a result we conclude therefore the following.

Theorem 7.7.Suppose that the random ratespz, z∈Z^d are defined with the help of the cycles as in formula (5.1). Then, the random walk on the random lattice(Z(t))t≥0 satisfies the central limit theorem as asserted in Theorem7.6.

6 Random walks whose local drift belongs to H

7 Notes and references

8

Homogenization of random diffusions

1 Diffusions with stationary and ergodic coefficients

An important example of random Markov processes is provided by diffusions with stationary and ergodic random coefficients. Suppose that (Z_x^ω(t))t≥0 is a diffusion starting atx, whose generator is given by

L^ωf(x) := 1 2e^U(x;ω)˜

Xd k,l=1

∂xk

e^−U(x;ω)˜ a˜k,l(x;ω)∂xlf

, f ∈C_b²(R^d), (1.1) where ˜a(x;ω) = [˜ak,l(x;ω)], x ∈R^d and ˜U(x;ω),x ∈R^d are stationary and ergodic random fields over a probability space (Ω,F,P). The method outlined in Section 7.3 can be used to prove the central limit theorem fort^−1/2Z_x^ω(t), as t→+∞. In Section 3 we construct the environment process corresponding to a random diffusion and identify its Dirichlet form. The methods of Chapter 2 can be used then to obtain the central limit theorem, see Theorem 8.15 below. We start this chapter with a section that lists the hypotheses made of the coefficients of diffusions and introduces some basic notions that allow us to describe such objects as the domain of the generator of the environment process, its Dirichlet form etc.

2 Stationary, random environments

Throughout this chapter we shall deal with d–dimensional diffusions whose coefficients are stationary random fields. As before we let (Ω,F,P) be a proba- bility space and (τx)x∈R^dbe an additive group of transformations acting onΩ.

The group satisfies the hypotheses (7.3.1)–(7.3.3) withX =R^d. In addition, we suppose that:

• for eachA∈ F, the map (x, ω)7→1{A}(τxω) is jointly (2.1) measurable with respect toσ–algebraB(R^d)⊗ F,

and

•for any δ >0 andf ∈L²(P) we have

h→0limP(|f(τhω)−f(ω)| ≥δ) = 0 (stochastic continuity). (2.2) For anyx ∈R^d and f ∈L²(P) we letT^xf = f ◦τx. Thanks to (7.3.2) and stochastic continuity condition (2.2) the family of operators (T^x)_x∈R^d forms ad–parameter, strongly continuous group of unitary operators onL²(P).

Ergodicity assumption (7.3.3) is equivalent with the fact that anyf ∈L²(P) that satifiesT^xf =f for allx∈R^d is constantP-a.s.

The L²–generators of the group correspond to the differentiation in di- rections ek, k = 1, . . . , d and shall be denoted by Dk : D(Dk) → L²(P), k= 1, . . . , d. They are anti-selfadjoint operators, i.e.

hDkf, gi^P =−hDkf, gi^P, ∀f, g∈ D(Dk) andk= 1, . . . , d. (2.3) For anyf ∈Td

k=1D(Dk) we define its gradient as∇f := (D1f, . . . , Ddf).

2.1 Some spaces of functions

Let k be a positive integer. Denote by H₀^k(Ω) the space consisting of those elements f ∈ L²(P) that belong to the domain of each operator D^m :=

D^m₁¹. . . D^m_d^d corresponding to an integer multi-indexm= (m1, . . . , md) such that |m|=k. On this space we introduce the semi-norm

kfkH^k:=



 X

|m|=k

kD^mfk²0





1/2

(2.4) and consider the identification f ∼ g, if f −g is a multiplicity of 1. The quotientH₀^k(Ω)/∼, which we still denote by H₀^k(Ω), is a pre-Hilbert space.

Its completion with respect to the norm (2.4) shall be denoted by H^k(Ω).

The differentiation operators D^m, |m| = k can be uniquely extended, via continuity, fromH₀^k(Ω) to the entireH^k(Ω). We identifyH⁰(Ω) withL²(P).

For any random element f, such as e.g. a random variable, vector, or matrix we shall denote by ˜f(x;ω) :=f(τxω) the corresponding random field.

Let C^k(Ω) be the subspace of H^k(Ω) consisting of those random variables f :Ω→R, for which the corresponding random field ˜f(x;ω), (x, ω)∈R^d×Ω has trajectories that belong toC^k(R^d) forP–a.s.ω. ByC_b^k(Ω) we denote the subspace ofC^k(Ω) consisting of thosef for which

kfkC_b^k(Ω):= X

|m|=k

kD^mfk^L∞ <+∞.

In the special casek= 0 we shall simply writeC(Ω) andCb(Ω). LetC^∞(Ω) :=

T

k≥1C^k(Ω) andC_b^∞(Ω) :=T

k≥1C_b^k(Ω).

2 Stationary, random environments 179 Proposition 8.1.The space C_b^∞(Ω)is dense in H^k(Ω)for any k≥0.

Proof. Suppose thatg∈H^k(Ω). Letφ∈C₀^∞(R^d) be such thatR

R^dφ(x)dx= 1. Letδ >0 and

gδ(ω) :=δ^−d Z

R^d

φx δ

g(x;˜ ω)dx.

It is straightforward to verify that

δ→0+lim kD^mgδ−D^mgk⁰= lim

δ→0+k(D^mg)δ−D^mgk⁰= 0

for any multi-index|m| ≤k. The conclusion of the lemma then easily follows.

2.2 Ito’s equations with stationary coefficients

We assume that we are given a random variable U and a random matrix a = [ak,l], where ak,l : Ω → R, k, l = 1, . . . , d that satisfy the following conditions:

•U and entriesak,l,k, l= 1, . . . , dbelong toC_b²(Ω), (2.5)

•(ellipticity assumption) the matrix a is a.s. positive definite, i.e. there exists a (deterministic) positive constantc0>0 such that

Xd k,l=1

ak,l(ω)ξkξl≥c0|ξ|², ∀ξ ∈R^d. (2.6)

Let ˜U(x;ω), a(x;ω) be the random fields that correspond to U and a.

For a given ω we define by (Z_x^ω(t))t≥0 a diffusion, defined over a certain probability space (Σ,W,Q), that starts at x ∈ R^d and whose generator is given by (1.1). This process can be defined with the help of Itˆo’s stochastic differential equation. We define a random vectorV = (V1, . . . , Vd), by

Vk(ω) =1 2

Xd l=1

[−ak,l(ω)DlU(ω) +Dlak,l(ω)] (2.7)

=1 2e^U∇ ·

e^−Ua(ω)ek

, k= 1, . . . , d

and a random matrix c(ω) = (a^s)^1/2(ω), where a^s := 1/2(a+a^t) is the symmetric part of a. Suppose also that w(t) = (w1(t), . . . , wd(t)), t≥0 is a d–dimensional standard Brownian motions over (Σ,W,Q). We shall denote by EQ the expectation with respect to the measure Q. For each ω ∈ Ω the

processZ_x^ω(t) is defined as the unique solution of the Itˆo stochastic differential equation

dZ_x^ω(t) = ˜V(Z_x^ω(t);ω)dt+ ˜c(Z_x^ω(t);ω)dw(t) Z_x^ω(0) = x (2.8) that is measurable with respect to the natural filtration (Wt)_t≥0of the Brow- nian motion (w(t))t≥0. We denote byP^ω(t, x,·) andp^ω(t, x,·) the transition of probability functions and their densities corresponding to the process de- scribed by (2.8). Their existence is guaranteed by the classical results con- cerning finite dimensional, non-degenerate diffusions, see e.g. [?], p. ???.

The random diffusionstarting atxand corresponding to the random gen- erator given by (1.1) is a stochastic process given by Zx(t;ω, ζ) :=Z_x^ω(t;ζ), (ω, ζ)∈Ω×Σ considered over the product space (Ω×Σ,F ⊗ W,P⊗Q). As in the previous section we shall omit the subscript for the starting point if it equals to 0.

Remark 8.2.Note that our framework incorporates the diffusions with peri- odic as well as quasi-periodic coefficients considered in Sections 7.1 and 7.2.

In the first case we assume that we are given C² smooth 1-periodic matrix a= [ak,l] and 1-periodic functionU such that there existsc0>0 for which

Xd k,l=1

ak,l(x)ξkξl≥c0|ξ|², ∀(ξ, x)∈R^d×R^d. Let

Vk(x) :=1

2e^U(x)∇x·h

e^−U(x)a(x)ek

i, k= 1, . . . , d (2.9) and letc(x) be the square root of the symmetric part ofa(x). We define then a diffusion (Zx(t))t≥0 as the solution of the Itrˆo stochastic differential equation dZx(t) =V(Zx(t))dt+c(Zx(t))dw(t) Zx(0) =x. (2.10) In this case, we letΩ:=T^d, where T^d is thed–dimensional torus introduced in Section 7.1. We define, the probability measureP and the group of shifts as in Example 7.1, i.e. Pis the Lebesgue measure on the torus,τx:Ω→Ω, τx(ω) := ˙x⊕ω, ω ∈Ω. The spaces H^k(Ω) coincide with the usual Sobolev spaces of periodic fields that are weaklyktimes differentiable, in the Sobolev sense, with square integrable derivatives. Note that Z_x^ω(t), the solution of (2.8), satisfies in this caseZ_x^ω(t) =Zx+ω(t)−ω.

In the quasi-periodic situation discussed in Section 7.2. we can admitΩ:=

T²withPthe product Lebesgue measure as in Example 7.7.2. Recall that for ω = ( ˙y1,y˙2) andx ∈ Rwe let τxω := ( ˙x⊕y˙1,x˙1⊕y˙2), where x1 =λx for some irrationalλ. The generator of the corresponding unitary group (T^x)x∈R

is given by (7.2.4).

3 The environment process 181

3 The environment process

The environment process corresponding to the random diffusion is defined as anΩ–valued process over (Σ,W,Q) by

η(t) :=τZ^ω(t)ω, t≥0. (3.1) Proposition 8.3.The process (η(t))_t≥0 is Markovian. Its transition of prob- ability semigroup is given by

P^tf(ω) = Z

p^ω(t,0, x) ˜f(x;ω)dx, for allf ∈B(Ω). (3.2) Proof. The proof is almost the same as in the discrete case presented in Chap- ter 4. It relies on the use of the Kolmogorov-Chapman property for transition of probability densities and the following lemma.

Lemma 8.4.We have

p^τzω(t, x, y) =p^ω(t, x+z, y+z), for allt >0, x, y, z∈R^d. (3.3) Proof. Since ˜V(·;τyω) = ˜V(·+y;ω) and ˜c(·;τyω) = ˜c(·+y;ω) we conclude from uniqueness of solutions to (2.8) thatZx^τyω(t) +y=Z_x+y^ω (t),Q–a.s. The respective transition of probability functions satisfy therefore

P^τyω(t, x, A) =P^ω(t, x+y, A+y) for allA∈ B(R^d). (3.4) The desired equality of densities follows from (3.4) and the continuity of the transition of probability density function, see Theorem 6.5.4 of [?].

Next, we wish to identify invariant probability measures for the environ- ment process that are absolutely continuous with respect toP. We can easily find an invariant measure (non-probabilistic) for the diffusion process corre- sponding to a fixed ω. Denote by (P_ω^t)_t≥0 the transition of probability semi- group for such a diffusion. A direct calculation, (1.1), shows that

d dt

Z

P_ω^tf(x)mω(dx) = Z

LωP_ω^tf(x)mω(dx) = 0 for allf ∈C_b²(R^d), where mω(dx) := e^−U(x;ω)˜ dx. Hence, mω is an invariant measure for the semigroup. We have the following.

Proposition 8.5.LetZ:=he^−Ui. Then,

π(dω) :=Z⁻¹e^−U(ω)P(dω) (3.5) is an invariant and ergodic probability measure for the semigroup (P^t)t≥0. In addition, the semigroup extends to a strongly continuous semigroup of con- tractions on L²(π).

Proof. Note that for anyf ∈C_b²(Ω) andt≥0 we have hP^tfi=Z⁻¹

Z

P_ω^tf˜(0)e^−U(ω)P(dω).

For anyg∈C_b²(R^d), t≥0 we haveP_ω^tg∈C_b²(R^d). Forx ∈R^d we can write therefore

P_ω^tg(x) =g(x) + Zt 0

LωP_ω^sg(x)ds, (3.6) whereLωis given by (1.1). Differentiating with respect totwe get

d

dthP^tfi=Z⁻¹ Z

LωP_ω^tf˜(0)e^−U(ω)dP(dω)

=Z⁻¹ Xd k,l=1

Z

Dk(akle^−UDlP^tf)dP= 0.

The last equality follows from the integration by parts. This proves the in- variance ofπ. Suppose thatA∈ F satisfiesP^t1{A}=1{A}for somet >0.

From (3.2) we have

0 =h1{A^c}, P^t1{A}i^P= Z

h1{A^c}(ω)p^ω(t,0, x)1{A}(τxω)i^Pdx.

Sincep^ω(t,0, x)>0 for allx∈R^d andP-a.s.ω we conclude that h1{A^c}(ω)1{A}(τxω)i^P = 0, ∀x∈R^d.

This however means thatAis invariant under (τx)_x∈R^d. It must be thereforeP- trivial thanks to ergodicity condition G 3). The conclusion of the proposition follows from the fact thatπ is equivalent with respect toP.

The extension of the semigroup to a semigroup of positive contractions on L²(π), denoted also by (P^t)t≥0, is standard. To show that (P^t)t≥0is a strongly continuous semigroup it suffices only to prove that for any f ∈ C^∞(Ω) we have limh→0+kP^hf −fkL²(π)= 0. Observe here that since C^∞(Ω) is dense in L²(P) it is also dense inL²(π). Using (3.6) we conclude that

|P^hf(ω)−f(ω)| ≤hmax{kσk^∞,kbk^∞}kfkC²_b(Ω)

for anyh >0 and the desired continuity property follows from the dominated convergence theorem.

Let L : D(L) → L²(π) be the generator of (P^t)_t≥0. Suppose that f ∈ C_b²(Ω). Using (3.6) we conclude, thanks to the dominated convergence theorem, that theL²(π) limit

3 The environment process 183 Lf(ω) = lim

h→0+

P^hf(ω)−f(ω)

h =Lωf˜(0).

Thus,C_b²(Ω)⊂D(L) and Lf=1

2e^U(ω) Xd k,l=1

Dk e^−Uak,lDlf

, f ∈C_b²(Ω). (3.7)

Proposition 8.6.The space C_b²(Ω)is a core of the generator L.

Proof. According to Proposition 1.3.3 of [?] it suffices only to check that C_b²(Ω) is dense in L²(π) and P^t(C_b²(Ω)) ⊂ C_b²(Ω) for all t ≥ 0. The first part holds, thanks to Proposition 8.1. The second is a conclusion from the following.

Lemma 8.7.For any f ∈B(Ω)andt >0we have P^tf ∈C_b²(Ω)and D^mP^tf(ω) =

Z

R^d

∂_|x=0^m p^ω(t, x, y)f(τyω)dy (3.8)

for |m| ≤2.

Proof. According to (3.2) we have P^tf(τxω) =

Z

R^d

p^τxω(t,0, y)f(τy+xω)dy= Z

R^d

p^ω(t, x, y)f(τyω)dy.

The last equality follows from homogeneity of transition of probability densi- ties, see (3.3), and the substitutiony :=y+x. The fact thatP^tf ∈C_b²(Ω) and formula (3.8) are easy consequences of differentiability ofp^ω(t, x, y) in bothx andyvariables and gaussian bounds of the respective derivatives, i.e. for any T >0 there existsCT, cT >0 such that for m1= 0,1,|m2|+|m3| ≤2

|∂ⁱ_t¹∂_xⁱ²∂_yⁱ³p(t, x, y)| ≤ C1

t^{(d+i2+i3)/2+i1} exp

−cT|x−y|²/t for allt∈[0, T],x, y∈R^d, see e.g. Theorems 6.4.5 and 6.4.7 of [?].

Our next task is to describe theL²(π)–adjoint of the generator. Suppose that ( ˆZ_x^ω(t))t≥0 is a random diffusion that starts atx ∈R^d and corresponds to a random generator

Lˆωf(x) := 1 2e^U˜(x;ω)

Xd k,l=1

∂xk

e^−U(x;ω)˜ ˜al,k(x;ω)∂xlf

, (3.9)

forf ∈C_b²(R^d).