Invariance principle for the random conductance model with dynamic bounded conductances

www.imstat.org/aihp 2014, Vol. 50, No. 2, 352–374

DOI:10.1214/12-AIHP527

© Association des Publications de l’Institut Henri Poincaré, 2014

Invariance principle for the random conductance model with dynamic bounded conductances

Sebastian Andres

Institut für Angewandte Mathematik, Rheinische Friedrich-Wilhelms-Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany.

E-mail:andres@iam.uni-bonn.de

Received 20 January 2012; revised 20 September 2012; accepted 28 September 2012

Abstract. We study a continuous time random walkXin an environment of dynamic random conductances inZ^d. We assume that the conductances are stationary ergodic, uniformly bounded and bounded away from zero and polynomially mixing in space and time. We prove a quenched invariance principle forX, and obtain Green’s functions bounds and a local limit theorem. We also discuss a connection to stochastic interface models.

Résumé. Nous étudions une chaîne de Markov en temps continuXdans un environnement dynamique de conductances aléatoires dansZ^d. Nous supposons que les conductances sont stationnaires ergodiques, uniformément positives et polynomialement mélan- geantes en espace et en temps. Nous montrons un principe d’invariance « quenched » pourX, et nous obtenons des bornes sur les fonctions de Green et un théorème limite local. Nous discutons aussi les liens avec les modèles d’interfaces aléatoires.

MSC:60K37; 60F17; 82C41

Keywords:Random conductance model; Dynamic environment; Invariance principle; Ergodic; Corrector; Point of view of the particle; Stochastic interface model

1. Introduction

We consider the Euclidean latticeZ^dequipped with the setE_d of non-oriented nearest neighbour bonds:E_d= {e= {x, y}: x, y∈Z^d,|x−y| =1}. We will also writex∼y when{x, y} ∈E_d. Denote byΩˆ = [0,∞)^Ed and byΩ the set of all measurable functions fromRtoΩ. We equipˆ Ω with aσ-algebraF and a probability measurePso that (Ω,F,P)becomes a probability space. The random environment is given by the coordinate mapsμ^ω_e(t)=ω_e(t), t∈R, e∈E_d. We will refer toμ_e(t)as theconductanceof the edgeeat timet. Further, writeμ^ω_xy(t)=μ_{x,y}(t)= μ_yx(t), andμ_xy(t)=0 if{x, y}∈/E_d, and set

μx(t)=

y∈Z^d

μxy(t)=

y∼x

μxy(t). (1.1)

We denote byD(R,Z^d)the space ofZ^d-valued càdlàg functions onR. For a givenω∈Ωand fors∈Randx∈Z^d, letP_s,x^ω be the probability measure onD(R,Z^d), under which the coordinate process(Xt)t∈Ris the continuous-time Markov chain onZ^d starting inxat timet=swith time-dependent generator given by:

L^ω_tf (x)=

y∼x

μ^ω_xy(t)

f (y)−f (x)

. (1.2)

1Research supported in part by NSERC (Canada).

That is,Xis the time-inhomogeneous random walk, whose time-dependent jump rates are given by the conductances.

Note that the counting measure, independent oft, is an invariant measure forX. Further, we denote byp^ω(s, x;t, y), x, y∈Z^d,s≤t, the transition densities of the time-inhomogeneous random walkX. This model of a random walk in a random environment is known in the literature – at least in the case of time-independent conductances – as the Random Conductance Modelor RCM. Note that the total jump rate out of any sitexis not normalized, in particular the sojourn time at sitexdepends onx. Therefore, the random walkXis sometimes called thevariable speed random walk (VSRW). However, for the purpose of this paper it would also be possible to consider theconstant speed random walk (CSRW)with total jump rates normalized to one (cf. Remark1.5below).

On(Ω,F,P)we define ad+1 parameter group of transformations(τ_t,x)_(t,x)∈R×Zd by τ_t,x:Ω→Ω,

μ_e(s)

s∈R,e∈Ed→

μ_x+e(t+s)

s∈R,e∈Ed

so that obviouslyτ_s+t,x+y=τ_s,x◦τ_t,y. Notice that

p^τh,zω(s, x;t, y)=p^ω(s+h, x+z;t+h, y+z), μ^τ_xy^h,zω(t)=μ^ω_x+z,y+z(t+h). (1.3) We are interested in the P almost sure or quenched long range behavior, in particular in obtaining a quenched functional limit theorem (QFCLT) or invariance principle for the processX starting in 0 at time 0. To that aim we need to state some assumptions on the environment measureP.

Assumption A1 (Ergodicity). τ_t,x(A)∈Ffor allA∈F,and the measurePis invariant and ergodic w.r.t.(τ_t,x),i.e.

P[A] ∈ {0,1}for any eventAsuch thatτ_t,x(A)=Afor allt∈Randx∈Z^d. Assumption A2 (Stochastic Continuity). For anyδ >0andf ∈L²(P)we have

hlim→0Pf (τ_h,0ω)−f (ω)≥δ

=0.

Thanks to AssumptionA1 andA2 the family of operators(T_t)_t∈R acting onL²(P), defined byT_tf =f ◦τ_t,0, forms a strongly continuous group of unitary operators. ItsL²(P)-generator will be denoted byDt:D(Dt)→L²(P), defined by

Dtf (ω)= ∂

∂tTtf_|t=0(ω)= ∂

∂t_|t=0f (τt,0ω).

By Corollary 1.1.6 in [15] the generator is closed and densely defined. Note thatD_t is an anti-selfadjoint operator in L²(P), i.e.

D_tf, gP= −f, D_tgP, f, g∈D(D_t), in particular

D_tf, f_P=0, f ∈D(D_t). (1.4)

Assumption A3 (Ellipticity). There exist positive constantsCl andCusuch that P

C_l≤μ_e(t)≤C_u,∀e∈E_d, t∈R

=1. (1.5)

We recall that under AssumptionA3the following heat kernel estimates have been proven in [11] (see also [18], Appendix B, for similar bounds).

Proposition 1.1. There exist constantsc1, . . . , c5such that forP-a.e.ωand for everyt≥s≥0the following holds:

(i) Ifx, y∈Z^d andD= |x−y| ≤c₁(t−s),then p^ω(s, x;t, y)≤ c2

(t−s)^d/2exp

−c₃D²/(t−s)

(Gaussian regime).

(ii) Ifx, y∈Z^dandD= |x−y| ≥c₁(t−s),then p^ω(s, x;t, y)≤ c4

1∨(t−s)^d/2exp

−c₅D 1+log

D/(t−s)

(Poisson regime).

Our first result is the following averaged or annealed FCLT. LetP⊗P_s,x^ω be the joint law of the environment and the random walk, and the annealed law is defined to be the marginalP^∗_s,x=

ΩP_s,x^ω dP(ω). Further, let X^(ε)_t =εX_{t /ε}2, t≥0.

Theorem 1.2. Letd≥1and suppose that AssumptionsA1–A3hold.Then,the law ofX^(ε)converges underP^∗_0,0to the law of a Brownian motion onR^dwith a deterministic non-degenerate covariance matrixΣ.

To prove a QFCLT we will need some mixing assumptions on the environment. We denote byB(Ω) the set of bounded and measurable functions onΩ andC_b,loc¹ (Ω)ˆ the set of differentiable functions on Ωˆ = [0,∞)^Ed with bounded derivatives depending only on a finite number of variables.

Assumption A4 (Time-mixing of the environment). There existsp₁>1 such that for everym∈Nthe following holds:For eachϕ, ψ∈B(Ω)of the formϕ(ω)= ˜ϕ(ω(t₁))andψ (ω)= ˜ψ(ω(t₂))with|t₁−t₂| ≥1for someϕ,˜ ψ˜ ∈ C_b,loc¹ (Ω)ˆ depending onmvariables we have

E[ϕψ] −E[ϕ]E[ψ]≤c_m|t1−t2|^−p1ϕL^∞(P)ψL^∞(P).

Assumption A5 (Space-mixing of the environment). Letd ≥3.There existsp2>2d/(d−2)such that for every m∈Nand for everyx∈Z^d the following holds:For each ϕ, ψ∈B(Ω)of the form ϕ(ω)= ˜ϕ(ω(t₀))andψ (ω)= ψ (ω(t˜ ₀))for someϕ,˜ ψ˜ ∈C_b,loc¹ (Ω)ˆ depending onmvariables we have

E

ϕ(ω)ψ (τ_0,xω)

−E[ϕ]E[ψ]≤c_m|x|^−p2ϕL^∞(P)ψL^∞(P). We are now ready to state the following QFCLT as our main result.

Theorem 1.3. Letd≥3and suppose that AssumptionsA1–A5hold.Then,P-a.s.X^(ε)converges(underP_0,0^ω )in law to a Brownian motion onR^dwith a deterministic non-degenerate covariance matrixΣ.

Notice that Theorem1.3only covers the transient lattice dimensionsd≥3. In order to get an invariance principle forXalso in dimensionsd≤2, we need to modify the mixing assumptions as follows.

Assumption A4. AssumptionA4holds withp₁> d+1ifd≥2andp₁>4ifd=1.

Assumption A5. There existsp₂>1such that for everym∈Nand for everyL >0the following holds:For each ϕ, ψ∈B(Ω)of the formϕ(ω)= ˜ϕ(ω(t₁))andψ (ω)= ˜ψ (ω(t₂)),where|t₁−t₂| ≤Landϕ,˜ ψ˜ ∈C_b,loc¹ (Ω)ˆ depend on variables contained in two subsetsA_ϕandA_ψ ofZ^dwith diameter at mostmanddist(Aϕ, A_ψ)≥L,

E[ϕψ] −E[ϕ]E[ψ]≤c_mL^−p2ϕL^∞(P)ψL^∞(P).

Theorem 1.4. Letd≥1and suppose that AssumptionsA1–A3,A4andA5hold.Then,P-a.s.X^(ε)converges(under P_0,0^ω )in law to a Brownian motion onR^dwith a deterministic non-degenerate covariance matrixΣ.

Remark 1.5. One can also consider the time-inhomogeneous constant speed random walk or CSRWY =(Yt, t ∈ R, P_s,x^ω , (s, x)∈R×Z^d)with generator given by:

L^Yt f (x)=

y∼x

μ_xy(t) μ_x(t)

f (y)−f (x) .

In contrast to the VSRW X,whose waiting time at any sitex∈Z^d depends onx,the CSRW waits at each site an exponential time with mean one.Since the CSRW is a time change of the VSRW,an invariance principle forY follows from an invariance principle forX by the same arguments as in[1],Section6.2.In this case the limiting object is a Brownian motion inR^d with covariance matrixΣ_C=(1/Eμ₀(0))ΣV,whereΣ_V denotes the covariance matrix of the limiting Brownian motion in the invariance principle forX.

Next we state some consequences of our results, which follow from arguments in [4] by combining the invariance principle forXand the Gaussian bound for the heat kernel. First, we have a local limit theorem for the heat kernel.

Write

k_t(x)=k_t^{(Σ )}(x)= 1

(2πt )^ddetΣexp

−x·Σ⁻¹x/2t

(1.6) for the Gaussian heat kernel with diffusion matrixΣ.

Theorem 1.6. LetT >0.Forx∈R^dwritex =(x₁, . . . ,x_d).

(i) Suppose that AssumptionsA1–A3hold.Then,

nlim→∞ sup

x∈R^d

sup

t≥T

n^d/2E p^ω

0,0;nt,

n^1/2x

−k_t(x)=0.

(ii) Under the assumptions of Theorem1.3or Theorem1.4we have

nlim→∞ sup

x∈R^d

sup

t≥T

n^d/2p^ω

0,0;nt, n^1/2x

−k_t(x)=0, P-a.s.

Proof. Given the annealed or quenched invariance principle and the heat kernel bounds in Proposition1.1this can be

proven as in Section 4 of [4].

Whend≥3 the calculations in Section 6 of [4] then give the following bound on the Green kernelg^ω(x, y)defined by

g^ω(x, y)= _∞

0

p^ω(0, x;t, y)dt.

Theorem 1.7. Letd≥3and suppose that the assumptions of Theorem1.3or Theorem1.4hold.

(i) There exist constantsc₁andc₂such that forx=y c₁

|x−y|^d−2≤g^ω(x, y)≤ c₂

|x−y|^d−2.

(ii) LetC=Γ (^d₂−1)/2π^d/2detΣ.For anyε >0there existsM=M(ε, ω)withP[M <∞] =1such that (1−ε)C

|x|^d−2 ≤g^ω(0, x)≤(1+ε)C

|x|^d−2 for|x|> M(ω).

(iii) We have,P-a.s.,

|xlim|→∞|x|^2−dg^ω(0, x)= lim

|x|→∞|x|^2−dE

g^ω(0, x)

=C.

In the case of static conductances, quenched invariance principles for the random conductance model have been proven by a number of different authors under various restrictions on the law of the conductances, see [3,7,22,29].

Recently, these results have been unified in [1], where a QFCLT has been obtained for the RCM with general non- negative i.i.d. conductances. We also refer the reader to [6] for a recent survey on this topic.

On the other hand, to our knowledge the present paper is the first one proving an invariance principle for the RCM with a time-dynamic environment. However, quenched invariance principles have been proven for several other discrete-time random walks in a dynamic random environment. In [9] a QFCLT is obtained for random walks in space–

time product environments by using Fourier-analytic methods. This result has been improved in [10] to environments satisfying an exponential spatial mixing assumption and in [2] to Markovian environments by using more probabilistic techniques. Another very successful approach is the well-established Kipnis–Varadhan technique based on the process of the environment as seen from the particle. In [25] this approach has been used to get a QFCLT for the random walk in space–time product environments. Moreover, it has been applied in [13] to random walks in a dynamic enviroment, which forms a Gibbsian Markov chain in time with spatial mixing, and in [20] to random walks onR^d, where the environment is i.i.d. in time and polynomially mixing in space. Recently, a general class of random walks in an ergodic Markovian environment satisfying some coupling conditions has been studied in [27].

Also in this paper we will follow the approach in [25], so we use the process of the environment as seen from the particle and the method of the “corrector,” that is we decompose the random walkX into a martingale and a time- dependent corrector function. Due to the time-inhomogeneity and the resulting lack of reversibility we need to apply the adaptions of the Kipnis–Varadhan method to non-reversible situations in [23] and [21]. In particular, in order to construct the corrector we show that the generator of the environment seen from the particle is a perturbation of a normal operator in the sense of [21], Section 2.7.5. This is done in Section2. As a byproduct this will already imply the annealed FCLT in Theorem1.2.

Once the corrector is constructed, the QFCLT for the martingale part is standard, so it remains to control the corrector. To that aim we still follow [25] and apply the theory of “fractional coboundaries” of Derriennic and Lin in [12]. The main step in this approach is to establish a subdiffusive bound on the corrector (see Proposition 3.1 below), which is done in Section3. To obtain this bound we establish so-called two-walk estimates, i.e. we consider the difference of two independent copies ofXevolving in the same fixed environmentω(cf. e.g. [20] or Appendix A in [26]). In d≥3, following [24] we show that the variance decay of the environment viewed from the particle is strong enough for our purposes by using the mixing AssumptionsA4 andA5 (see Lemma 3.3). In the recurrent lattice dimensionsd≤2 the estimate for the variance decay is not good enough, so we give a different argument here involving the modified mixing AsumptionsA4andA5.

In Section4we prove the main result, i.e. we state a tightness result, which is a direct consequence from the heat kernel bounds in Proposition1.1, and show the QFCLT for the martingale part. To control the corrector we apply the results in [12], which are stated in the discrete-time setting. Since it is not clear to us, how to apply them directly in the continuous-time setting, we first prove the QFCLT for the discretized process as in [3]. More precisely, we define Xn=Xn,n∈N, and consider the process

X^(ε)_t =εX_{t /ε}2.

We can control sup_t≤T |X_t^(ε)−X^(ε)_t |– see Lemma4.2– so an invariance principle forX^(ε)will follow from one for X^(ε).

Finally, in Section5we point out a link to stochastic interface models (see [16]). Namely, a local limit theorem for the RCM with dynamic conductances can be used to obtain scaling limits for the space–time covariation of the Ginzburg–Landau interface model via Helffer–Sjöstrand representation.

Throughout the paper we writecto denote a positive constant which may change on each appearance. Constants denotedc_iwill be the same through each argument.

2. Construction of the corrector

Throughout this section we suppose that AssumptionsA1–A3hold. We define the process of the environment seen from the particle by

η_t(ω)=τ_t,Xtω, ω∈Ω, t≥0.

Proposition 2.1.

(i) The process(η_t)_t≥0is Markovian with transition semigroup Ptf (ω)=

y∈Z^d

p^ω(0,0;t, y)f (τt,yω) for allf ∈B(Ω).

The semigroup(P_t)extends uniquely to a strongly continuous semigroup of contractions onL²(P),whose gener- atorL:D(L)→L²(P)is given by

Lf (ω)=D_tf (ω)+

y∼0

μ^ω_0y(0)

f (τ0,yω)−f (ω)

with domainD(L)=D(D_t).

(ii) The measurePis invariant and ergodic forη.

Proof. (i) The Markov property as well as the representation of the semigroup follow from (1.3) by similar arguments as in Lemma 3.1 in [21]. For every boundedf ∈D(D_t)we have

P_tf (ω)−f (ω)

t =

y∈Z^d

p^ω(0,0;t, y) t

f (τ0,yω)−f (ω) +

y∈Z^d

p^ω(0,0;t, y)f (τ_t,yω)−f (τ_0,yω)

t .

Taking limits fort↓0, using the fact thatp^ω(0,0;t, y)→δ_0y, we obtain the formula forLf. Obviously, the operators LandD_t have the same domain.

(ii) Letf ∈D(L). Since the operatorD_t is anti-selfadjoint we haveD_tf_P=0. Hence, LfP=

y∈Z^d

μ^ω_0y(0)f (τ0,yω)

P−

μ^ω_0y(0)f (ω)

P=

y∈Z^d

μ^τ_0y^0,−yω(0)f (ω)

P−

μ^ω_0y(0)f (ω)

P

=

y∈Z^d

μ^ω_0,−y(0)f (ω)

P−

μ^ω_0y(0)f (ω)

P=0,

where we have used the invariance ofPw.r.t.τt,xand (1.3). Thus,Pis an invariant measure forη. To prove thatPis also ergodic, let nowA∈FwithP_t1A=1A. Then,

0=1A^c(ω)·P_t1A(ω)=

y∈Z^d

1A^c(ω)p^ω(0,0;t, y)1A(τ_t,yω).

Since for allt >0 andy∈Z^dthere is a stricly positive lower bound forp^ω(0,0;t, y)independent ofω(see Proposi- tion 4.3 in [11]) we get

1A^c(ω)·1A(τt,yω)=0.

Thus, the setAis invariant under τ_t,x. SincePis ergodic w.r.t. τ_t,x we conclude thatA isP-trivial and the claim

follows.

Lemma 2.2. Forf∈D(L), f, (−L)f

P=1 2

y∈Z^d

E μ^ω_0y(0)

f (τ_0,yω)−f (ω)2 .

Proof. Recall thatf, D_tf_P=0. Therefore, f, (−L)f

P= −

y∈Z^d

E

f (ω)μ^ω_0y(0)

f (τ_0,yω)−f (ω)

= −1 2

y∈Z^d

E

f (ω)μ^ω_0y(0)

f (τ_0,yω)−f (ω)

−1 2

y∈Z^d

E

f (ω)μ^ω_0,−y(0)

f (τ_0,−yω)−f (ω)

= −1 2

y∈Z^d

E

f (ω)μ^ω_0y(0)

f (τ0,yω)−f (ω)

−1 2

y∈Z^d

E

f (τ0,yω)μ^τ_0,^0,y₋^ω_y(0)

f (ω)−f (τ0,yω)

=1 2

y∈Z^d

E μ^ω_0y(0)

f (τ0,yω)−f (ω)2 ,

where we have used again the invariance ofPw.r.t.τt,xand (1.3).

LetP_t^∗andL^∗denote theL²(P)-adjoint operators ofPt andL, respectively.

Proposition 2.3. We have P_t^∗f (ω)=

y∈Z^d

ˆ

p^ω(0,0;t, y)f (τ_−t,yω), f ∈L²(P),

withpˆ^ω(s, x;t, y):=p^ω(−t, y; −s, x)and forf ∈D(L) L^∗f (ω)= −D_tf (ω)+

y∼0

μ^ω_0y(0)

f (τ_0,yω)−f (ω) .

Proof. Using (1.3) we compute the adjoint ofP_t as Ptf, gP=

y∈Z^d

E

p^ω(0,0;t, y)f (τt,yω)g(ω)

=

y∈Z^d

E

p^{τ−t,−yω}(0,0;t, y)f (ω)g(τ₋t,−yω)

=

y∈Z^d

E

p^ω(−t,−y;0,0)f (ω)g(τ₋t,−yω)

=

y∈Z^d

E ˆ

p^ω(0,0;t, y)g(τ_−t,yω)f (ω) ,

and the representation forP_t^∗follows. To computeL^∗we use a similar procedure as in Lemma2.2and get Lf, gP= Dtf, gP+

y∈Z^d

E μ^ω_0y(0)

f (τ0,yω)−f (ω) g(ω)

= −f, Dtg_P−1 2

y∈Z^d

E μ^ω_0y(0)

f (τ0,yω)−f (ω)

g(τ0,yω)−g(ω)

= −f, D_tg_P+

y∈Z^d

E μ^ω_0y(0)

g(τ_0,yω)−g(ω) f (ω)

,

which gives the claim.

Next we introduce the Hilbert spacesH1andH−1. LetCbe a common core of the operatorsLandL^∗. OnCwe define the seminorm

f²_H1=

f, (−L)f

P, f ∈C.

LetH1be the completion ofC(or more precisely the completion of equivalence classes of elements inCw.r.t. the equivalence relationf ∼giff −g_H1 =0) w.r.t. · _H1. Then,H1is a Hilbert space with inner product·,·_H1

given by polarization:

f, g_H1=1 4

f +g²_H1− f−g²_H1 .

Associated withH1we define the dual spaceH−1as follows. Forf ∈L²(P)let f²_H−1 =sup

g∈C

2f, g_P− g²_H1 .

The Hilbert spaceH₋1is then defined as the · _H−1-completion of (equivalence classes of) elements inCwith finite · _H−1-norm. As before the inner product·,·_H−1 is defined through polarization. We refer to Section 2.2 in [21]

for more details.

Next we define the local drift V_j(ω)=

y∼0

μ^ω_0y(0)y^j=L^ω0f_j(0), j=1, . . . , d,

where fj(x)=x^j, x^j andy^j denoting the jth component ofx andy. Since μ^ω_0y(0)=0 unlessy∼0, we have V_j(ω)=μ^ω_0,e

j(0)−μ^ω_0,−e

j(0).

Lemma 2.4. For everyj =1, . . . , d,V_j∈L²(P)∩H₋1. Proof. It suffices to show that

V_j, fP²≤c

f, (−L)f

P for allf ∈H1 (2.1)

(cf. equation (2.12) in [21]). By definition ofV_j we have V_j, f_P=E

μ^ω_0,e

j(0)f (ω)

−E μ^ω_0,−e

j(0)f (ω)

=E μ^ω_0,e

j(0)f (ω)

−E μ^ω_0,e

j(0)f (τ0,ejω)

= −E μ^ω_0,e

j(0)

f (τ_0,ejω)−f (ω) . Hence, using Cauchy–Schwarz and Lemma2.2

V_j, f_P²≤E μ^ω_0,e

j(0) E

μ^ω_0,e

j(0)

f (τ_0,ejω)−f (ω)2

≤C_u

y∈Z^d

E μ^ω_0y(0)

f (τ_0,yω)−f (ω)2

=2Cu

f, (−L)f

P,

and we obtain (2.1).

Forλ >0, we consider for eachj the solutionu^j_λof the resolvent equation

(λ−L)u^j_λ=V_j. (2.2)

Proposition 2.5. For everyj=1, . . . , d,there existsu^j∈H1such that

λlim→0λu^j_λ²

L²(P)=0 and lim

λ→0u^j_λ=u^j strongly inH1.

The proof of Proposition2.5will be based on the following statement proven in [21].

Proposition 2.6. Suppose we have the decompositionL=L⁰+Bof the operatorLsuch that:

(i) The operatorL⁰is normal,i.e.L⁰(L⁰)^∗=(L⁰)^∗L⁰.

(ii) The Dirichlet forms ofLandL⁰are equivalent,i.e.there exist positive constantsc₁andc₂such that:

c₁

f, (−L)f

P≤ f,

−L⁰ f

P≤c₂

f, (−L)f

P for allf ∈D(L).

(iii) Bsatisfies a sector condition w.r.t.L⁰,i.e.there exists a positive constantcsuch that f, Bg²_P≤c

f,

−L⁰ f

P

g,

−L⁰ g

P f, g∈D(L).

Then,for any fixedV ∈L²(P)∩H₋1the solutionf_λof the resolvent equation(λ−L)f_λ=V satisfies

λlim→0λf_λ²_L2(P)=0 and lim

λ→0f_λ=f strongly inH1

for somef∈H1.

Proof. By Proposition 2.25 in [21] the assumptions imply that sup

0<λ≤1

Lf_λH−1<∞.

The claim follows then from Lemma 2.16 in [21].

Proof of Proposition2.5. We decompose the operatorL=L⁰+Bwith L⁰f :=D_tf+

y∼0

C_l

f (τ_0,yω)−f (ω)

, f ∈D(L),

and

Bf :=

y∼0

μ^ω_0y(0)−Cl

f (τ0,yω)−f (ω)

, f∈D(L).

A similar calculation as in the proof of Lemma2.2shows that f,

−L⁰ f

P=1 2

y∼0

E C_l

f (τ_0,yω)−f (ω)2

, (2.3)

f, (−B)g

P=1 2

y∼0

E

μ^ω_0y(0)−C_l

f (τ_0,yω)−f (ω)

g(τ_0,yω)−g(ω)

. (2.4)

The claim will follow from Proposition2.6and Lemma2.4once we have verified conditions (i)–(iii) in Proposi- tion2.6. To show (i), note that the closure ofL⁰is the generator of a semigroup(P_t⁰)that corresponds to a process seen from the particle associated with a simple random walk onZ^dwith constant jump ratesC_l. In particular, the associated process is time-homogeneous, i.e. the corresponding transition probabilities satisfyp₀^ω(s, x;t, y)=p^ω₀(t−s, x, y) andpˆ^ω₀(s, x;t, y)= ˆp^ω₀(t−s, x, y), wherep^ω₀(t, x, y)=p^ω₀(0, x;t, y) andpˆ₀^ω(t, x, y)= ˆp^ω₀(0, x;t, y). Since this random walk is obviously reversible w.r.t. the counting measure, we havep^ω₀(t, x, y)= ˆp₀^ω(t, x, y). Then, since we have similar representations forP_t⁰and(P_t⁰)^∗as for the semigroups in Proposition2.1and Proposition2.3, we get

P_t⁰_∗

P_t⁰=P_t⁰ P_t⁰_∗

, t≥0,

which implies that the closure ofL⁰is normal (see Theorem 13.37 in [28]).

Condition (ii) is immediate from Lemma2.2, (2.3) and the ellipticity condition (1.5). To prove (iii) we use (2.4), Cauchy–Schwarz and the ellipticity condition (1.5), which gives

f, Bg²_P≤ 1

2C_u²dE

y∼0

f (τ_0,yω)−f (ω)2

×E

y∼0

g(τ_0,yω)−g(ω)2

≤ C_u²d 2C_l²

f,

−L⁰ f

P

g,

−L⁰ g

P,

and the claim follows.

For abbreviation we writeu_λ=(u¹_λ, . . . , u^d_λ)and χ_λ(t, x, ω):=u_λ◦τ_t,x−u_λ.

Proposition 2.7. For all non-negativet∈Qandx∈Z^dthe limit lim

λ→0χ_λ(t, x, ω)=:χ (t, x, ω) (2.5)

exists along a subfamily (λ) for P-a.e. ω. Moreover, the mapping t →χ (t, X_t, ω) can be extended to a right- continuous function on[0,∞)such that

M_t=X_t+χ (t, X_t, ω), t≥0, (2.6)

is aP_0,0^ω -martingale.

Proof. For everyj=1, . . . , dand everyλ >0 we have that forP-a.e.ωthe processes N_t^j,λ=u^j_λ(η_t)−u^j_λ(ω)−

t 0

Lu^j_λ(η_s)ds (2.7)

and

M˜_t^j=X_t^j− _t

0 L^ωsf_j(X_s)ds (2.8)

are bothP_0,0^ω -martingales, where as beforef_j(x)=x^j. Then, using the definition ofV_j and the fact thatu^j_λsolves the resolvent equation (2.2) we get

X^j_t = ˜M_t^j+ _t

0

L^ω_sf_j(X_s)ds= ˜M_t^j+ _t

0

V_j(η_s)ds= ˜M_t^j+ _t

0

(λ−L)u^j_λ(η_s)ds

= ˜M_t^j+N_t^j,λ−

u^j_λ(η_t)−u^j_λ(ω) +λ

_t

0

u^j_λ(η_s)ds. (2.9)

In a first step we show that the martingaleN_t^j,λconverges inL²(P⊗P_0,0^ω )asλ↓0 to a martingaleN_t^j. To that aim it is enough to prove thatN_t^j,λ is a Cauchy sequence inL²(P⊗P_0,0^ω ). SincePis an invariant measure forηwe use Lemma2.2to obtain

EE_0,0^ω

N^j,λ−N^j,λ

t = _t

0 EE_0,0^ω L

u^j_λ−u^j_λ2

−2

u^j_λ−u^j_λ L

u^j_λ−u^j_λ (ηs)ds

=2t

u^j_λ−u^j_λ , (−L)

u^j_λ−u^j_λ

P

=2tu^j

λ−u^j_λ²

H1,

which implies thatN_t^j,λ is a Cauchy sequence in L²(P⊗P_0,0^ω ) by Proposition2.5. Thus, the martingale M_t^j,λ = M˜_t^j +N_t^j,λ converges to a martingale, whose right-continuous modification we denote by M_t^j. We define M_t = (M_t¹, . . . , M_t^d).

The next step is to show that the last term in (2.9) converges to zero inL²(P⊗P_0,0^ω )asλ↓0. SinceV_j∈H₋1we have that limλλu^j_λ=0 inL²(P)(cf. equation (2.15) in [21]). Thus, for everyj =1, . . . , d,

λ t

0

u^j_λ(ηs)ds

L²(P⊗P_0,0^ω)

≤λ t

0

u^j_λ(ηs)

L²(P⊗P_0,0^ω)ds=t λu^j_λ

L²(P)→0.

Thus, by takingL²(P⊗P_0,0^ω )-limits in (2.9) we get thatχ_λ(t, X_t,·)converges inL²(P⊗P_0,0^ω )asλ↓0 for every t≥0. By a diagonal procedure we can extract a suitable subsequenceλsuch that forP-a.e.ωwe have thatχλ(t, Xt, ω) has a limit inL²(P_0,0^ω )andP_0,0^ω -a.s. alongλfor all non-negativet∈Q. In particular, the limit isσ (Xt)-measurable and will therefore be denoted byχ (t, Xt, ω). Hence,

Xt=Mt−χ (t, Xt, ω). (2.10)

Moreover, forP-a.e.ω,

y∈Z^d

p^ω(0,0;t, y)χλ(t, y, ω)−χ (t, y, ω)²=E_ω⁰χλ(t, Xt, ω)

−χ (t, Xt, ω)²→0

alongλ. Sincep^ω(0,0;t, y) >0 for allt >0 andx∈Z^d, we conclude that forP-a.e.ωthe limit in (2.5) exists for every non-negativet∈Qand everyy∈Z^d. Finally, using (2.10) and the fact thatX_t andM_t have right-continuous trajectories, we can extendχ (t, X_t, ω)to a right-continuous function on[0,∞)and (2.6) follows.

Remark 2.8. Note that for all non-negatives, t∈Qandx, y∈Z^d, u_λ(τ_t,yω)−u_λ(τ_s,xω) =

u_λ(τ_t,yω)−u_λ(ω)

−

u_λ(τ_s,xω)−u_λ(ω)

→χ (t, y, ω)−χ (s, x, ω)

along the chosen subsequence forP-a.e.ω.The functionhλ(ω0, ω1):=uλ(ω1)−uλ(ω0)onΩ×Ω converges in L²(Ω×Ω,P◦(τs,x, τt,y)⁻¹)to a functionh.In particular,forP-a.e.ω,

h(τ_s,xω, τ_t,yω)=χ (t, y, ω)−χ (s, x, ω).

Corollary 2.9. ForP-a.e.ωthe corrector satisfies the cocycle property χ (s+t, x+y, ω)=χ (s, x, ω)+χ (t, y, τ_s,xω)

for non-negatives, t∈Q.

Proof. We have

u_λ◦τ_s+t,x+y−u_λ=(u_λ◦τ_s,x−u_λ)+(u_λ◦τ_t,y−u_λ)◦τ_s,x,

and the claim follows by taking theL²(P)-limit alongλon both sides.

In the following, for anyG:Z^d×Ω→Rwe shall write G²_ω:=

y∼0

μ^ω_0y(0)G(y, ω)².