Quenched local central limit theorem for random walks in a time-dependent balanced random environment

We prove a quenched local central limit theorem for continuous-time random walks in Zd,d≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Z}}^d, d\ge 2$$\end{document}, in a uniformly-elliptic time-dependent balanced random environment which is ergodic under space-time shifts. We also obtain Gaussian upper and lower bounds for quenched and (positive and negative) moment estimates of the transition probabilities and asymptotics of the discrete Green’s function.


Introduction
In this article we consider a random walk in a balanced uniformly-elliptic timedependent random environment on Z d , d ≥ 2.
For x, y ∈ Z d , we write x ∼ y if |x − y| 2 = 1. Denote by P the set (of nearestneighbor transition rates on Z d ) Equip P with the the product topology and the corresponding Borel σ -field. We denote by ⊂ P R the set of all measurable functions ω : t → ω t from R to P and call every ω ∈ a time-dependent environment. For ω ∈ , we define the parabolic difference operator L ω u(x, t) = y:y∼x ω t (x, y)(u(y, t) − u(x, t)) + ∂ t u(x, t) for every bounded function u : Z d ×R → R which is differentiable in t. Let (X t ) t≥0 = (X t , T t ) t≥0 denote the continuous-time Markov chain on Z d × R with generator L ω . Note that almost surely, T t = T 0 + t. We say that (X t ) t≥0 is a continuous-time random walk in the environment ω and denote by P x,t ω its law (called the quenched law) with initial state (x, t) ∈ Z d × R.
We equip ⊂ P R with the induced product topology and let P be a probability measure on the Borel σ -field B( ) of . An environment ω ∈ is said to be balanced if ω t (x, x + e) = ω t (x, x − e) for all e ∈ Z d with |e| = 1. and uniformly elliptic if there is a constant κ ∈ (0, 1) such that κ < ω t (x, y) < 1 κ for all t ∈ R, x, y ∈ Z d with x ∼ y.
We assume that the law P of the environment is translation-invariant and ergodic under the space-time shifts {θ x,t : x ∈ Z d , t ≥ 0}. I.e., P(A) ∈ {0, 1} for any A ∈ B( ) such that P(A θ −1 x A) = 0 for allx ∈ Z d × [0, ∞). Given ω, the environmental process with initial stateω 0 = ω is a Markov process on . With abuse of notation, we use P 0,0 ω to denote the quenched law of (ω t ) t≥0 . Assumptions: throughout this paper, we assume that P is balanced, ergodic, and uniformly elliptic with ellipticity constant κ > 0.
Theorem 1 [14,Theorem 1.2] Under the above assumptions of P, (a) there exists a unique invariant measure Q for the process (ω t ) t≥0 such that Q P and (ω t ) t≥0 is an ergodic flow under Q × P 0,0 ω . Let ρ(ω) := dQ/dP.
In the special case where the environment is time-independent, i.e., P(ω t = ω s for all t, s ∈ R) = 1, we say that the environment is static. Remark 2 For balanced random walks in a static, uniformly-elliptic, ergodic random environment on Z d , the QCLT has been first shown by Lawler [24], which is a discrete version of the result of Papanicolaou and Varadhan [27]. It is then generalized to static random environments with weaker ellipticity assumptions in [9,19].
Remark 3 Write f L p (P) := (E P [| f | p ]) 1/ p for p ∈ R. At the end of the proof of [14,Theorem 1.2], it is shown that E Q [g] ≤ C g L d+1 (P) for any bounded continuous function g, which implies (2).
Since is equipped with a product σ -field, for any fixed ω ∈ , the map R → defined by t → θ 0,t ω is measurable. Hence for almost-all ω, the function ρ ω (x, t) is measurable in t. Moreover, ρ ω possesses the following properties. For P-almost all ω, (i) ρ ω (x, t)δ x dt is an invariant measure for the processX t under P ω ; (ii) ρ ω (x, t) > 0 is the unique density (with respect to δ x dt) for an invariant measure ofX that satisfies E P [ρ ω (0, 0)] = 1; (iii) ρ ω has a version which is absolutely continuous with respect to t with for almost every t, where ω t (x, x) := − y:y∼x ω t (x, y).
The proof of these properties, which is rather standard, is given in Sect. A.1 for the purpose of completeness.
As a main result of our paper, we will present the following local limit theorem (LLT), which is a finer characterization of the local behavior of the random walk than the QCLT. Let0 The proof of the LLT follows from Theorem 1 and a localization of the heat kernel q ω (0, ·), an argument already implemented in [7] and [4] in the context of random conductance models. For this purpose, the regularity ofx → q ω (0,x) is essential. We use an analytical tool from classical PDE theory: the parabolic Harnack inequality (PHI) which yields not only Hölder continuity of q ω (0, ·) but also very sharp heat kernel estimates.
To state the PHI, we need some notations. Forx = (x, t) ∈ Z d × R, let and define the the adjoint operator L * ω by We say that a function u is ω-caloric (resp. ω * -caloric) on D ⊂ Z d × R if L ω u = 0 (resp. L * ω u = 0) on D. Throughout this paper, unless otherwise specified, C, c denote generic positive constants that depend only on (d, κ), and which may differ from line to line. If two functions f and g satisfy cg ≤ f ≤ Cg, we write f g.
First, let us state the PHI for the operator L ω . For r > 0, x ∈ R d , let We remark that for dynamical environments in the discrete time setting, the PHI is obtained by Kuo and Trudinger for the so-called implicit form operators, see [23, (1.16)]. For discrete-time random walks in a static environment, the PHI is shown by Lawler [25] for uniformly elliptic operators, and by Berger and Criens [8] (see also [11]) for a genuinely d-dimensional i.i.d. environment which is not necessarily elliptic.
Observe that for fixedx = (x, t), the functionŷ → q ω (ŷ,x) is ω-caloric on Z d × (−∞, t). Whereas, the heat kernelx → q ω (0,x) is ω * -caloric on Z d × (0, ∞). Hence, to obtain the regularity of the heat kernel, we need to prove, instead of the PHI for L ω , the following PHI for L * ω which is our second main result. Theorem 6 (PHI for L * ω ) For P-almost all ω, any non-negative ω * -caloric function v on B 2R × (0, As a standard consequence of the PHI for L * ω , we get the following Hölder estimate for ω * -caloric functions. (See a proof in Sect. A.2.) The main challenge in proving Theorem 6 is that ω * is neither balanced nor uniformly elliptic, and so the PHI for L ω (Proposition 5) is not immediately applicable. This is the main difference with the random conductance model with symmetric jump rates where ω t (x, y) = ω t (y, x) = ω * t (x, y), in which case the PHI for L ω is the same as PHI for L * ω . See [1,2,12,13,20]. In PDE, the Harnack inequality for the adjoint of non-divergence form elliptic differential operators was first proved by Bauman [6], and was generalized to the parabolic setting by Escauriaza [15]. Our proof of Theorem 6 follows the main idea of [15].
Let us explain the main idea for the proof of Theorem 6. An important observation is that ω * -caloric functions can be expressed in terms of hitting probabilities of the time-reversed process. Thus to compare values of an ω * -caloric function at different points, one only needs to estimate hitting probabilities of the original process that starts from the boundary. To this end, we will use a "boundary Harnack inequality" (Proposition 30) which compares ω-caloric functions near the boundary. We will also need the following parabolic volume-doubling property (VDP) for the invariant measure to control the change of probabilities due to time-reversal.

Remark 9
For time discrete random walks in a static environment, Theorem 6 was shown by Mustapha [26]. His argument follows basically [15], and uses the PHI [23,Theorem 4.4] of Kuo and Trudinger in the time discrete situation. Note that for static environments, the PHI for L * ω follows from the PHI (for L ω )and a representation formula (See Remark 33), and a VDP is not needed. However, in our dynamical setting, the parabolic VDP is crucially employed. To this end, we adapt ideas of Safonov-Yuan [28] and results in the references therein [6,16,18] into our discrete space setting.

Remark 10
For adjoint solutions of non-divergence form elliptic PDE, a VDP was first established by Fabes and Stroock [17]. It was then generalized by Safonov and Yuan [28] to the parabolic case.
Recall the heat kernel q ω in (4). For any A ⊂ R d and s ∈ R, let We write the 2 -norm of x ∈ R d as |x| = |x| 2 . For r ≥ 0, t > 0, define h(r , t) = r 2 t∨r + r log( r t ∨ 1).
Our third main results are the following heat kernel estimates (HKE). Note that for a general balanced environment, the density ρ ω does not have deterministic (positive) upper and lower bounds, thus one cannot expect deterministic Gaussian bounds for p ω (0,x). However, our HKE below shows that p ω (0,x) has both L (d+1)/d (P) and L − p (P) moment bounds.
Theorem 11 (HKE) For P-almost every ω and allx = ( for all s ∈ [0, t] and y with |y| ≤ |x| + c √ t. Moreover, recalling the definition of L p (P) in Remark 3, there exists p = p(d, κ) > 0 such that and Furthermore, we can characterize asymptotics of the Green's function of the RWRE. Recall the notations (in Theorem 1 (b)), p t , and x (in Theorem 4).

Corollary 12
The following statements are true for P-almost every ω.
where h is as in (5). As a consequence, the RWRE is recurrent when d = 2 and transient when d ≥ 3.
Similar results as Corollary 12(ii)(iii) are also obtained recently for the conductance model [5].
Our major technical novelties and main results can be summarized as follows. (a) Using probability estimates, we solve the difficult analytic problem of obtaining a parabolic VDP (for the density of the invariant measure) in a time-dependent balanced environment. (b) Using the parabolic VDP, we established the A p bounds, and as a consequence proved the PHI for ω-caloric functions. The latter proof, which is of interest on its own, can be viewed as the parabolic version of Fabes and Stroock's [17] proof in the elliptic static setting. (c) Interpreting ω * -caloric functions in terms of a time reversed RWRE, and using the parabolic VDP and boundary PHI estimates, we prove the PHI for the adjoint operator. (d) As applications, we obtain LLT, quenched HKE, positive and negative L p bounds for the heat kernel, and Green's function asymptotics for the RWRE.
The organization of this paper is as follows. Section 2 contains probability estimates that are used in the later sections. In Sect. 3, we obtain the parabolic VDP and an A p inequality for ρ ω , and prove the PHI for L ω . In Sect. 4, we establish estimates of ω-caloric functions near the boundary, showing both the interior elliptic-type and boundary PHI's. We prove the PHI for the adjoint operator (Theorem 6) in Sect. 5. Finally, with the adjoint PHI, we prove Theorems 4, 11, and Corollary 12 in Sect. 6. Some classical estimates and standard arguments can be found in the Appendix.

Auxiliary probability estimates
This section contains probability estimates that are crucial in the rest of the paper. For a finite subgraph D ⊂ Z d , let For D ⊂ Z d × R, define the parabolic boundary of D as In the special case D = D × [0, T ) for some finite D ⊂ Z d , it is easily seen that Fig. 1.
Recall the definition of the function h(r , t) in (5).

Lemma 13
Assume ω ∈ κ . Then for t > 0, r > 0, Proof Let x(i), i = 1, . . . , d, denotes the i-th coordinate of x ∈ R d . It suffices to show that for i = 1, . . . , d, (r , t)) . We will prove the statement for i = 1. LetÑ t := #{0 ≤ s ≤ t : X s (1) = X s − (1)} be the number of jumps in the e 1 direction before time t. Let (S n ) be the discrete time simple random walk on Z, then X t (1) d = SÑ t . Note thatÑ t is stochastically dominated by a Poisson process N t with rate c 0 := 2d/κ, and so P0 ω (sup 0≤s≤t where we used Hoeffding's inequality in the second inequality. On the other hand, since the random walk is in a discrete set Z, we have, for any θ > 0, When r ≥ 9c 2 0 t, taking θ = log( r c 0 t ), we get an upper bound exp[− r 2 log( r t )]. Hence, letting f (r , t) = r 2 t∨r 1 r <9c 2 0 t + r log( r t )1 r ≥9c 2 0 t , we obtain ).
Since f (r , t) r 2 t∨r + r log( r t )1 r ≥9c 2 0 t h(r , t), our proof is complete.
There exists a constant α = α(κ, d, θ 1 , θ 2 ) ≥ 1 such that for any s ∈ (θ 1 R 2 , θ 2 R 2 ) and σ > 0 Proof It suffices to consider the case σ ∈ (0, 1) and is a large constant to be determined. Indeed, if R < K 1 , then by uniform ellipticity, for any x ∈ B R , Further, for R ≥ K 1 , if suffices to consider the case σ R ≥ √ K 1 . Indeed, assume the lemma is proved for R ≥ K 1 and σ R ≥ √ K 1 . Then, when σ R < √ K 1 and x ∈ B R , by uniform ellipticity, Hence in what follows we only consider the case R ≥ K 1 and σ R ≥ √ and, for some large constant q ≥ 2 to be chosen, Recall the parabolic boundary ∂ P in (10). We first show that w satisfies The first two properties in (12) are obvious. For the third property, note that For any unit vector e ∈ Z d , let ∈ U , then x · e ≥ 1 and ∃δ ∈ (0, 1) such that ψ 1 (x + δe, t) = 0. In both cases, there exists δ ∈ (0, 1] such that where in the last inequality we used the fact 1 ≤ x · e ≤ |x| ≤ 2Rψ 1/2 0 . Thus, letting Proof By uniform ellipticity, it suffices to consider r ≥ 10. Let (11). Then, by Lemma 15, The corollary is proved.
Proof It suffices to prove the lemma for R > α 2 , where α = α(κ, d, β, θ) is a large constant to be determined. We only need to consider y with dist(y, Using the inequalities e a + e −a ≥ 2 + a 2 and e a ≥ 1 + a, we get for Recall the definition of the stopping time in (11)
Next, set (Recall the stopping time in (11).) Then Our proof of Lemma 18 is complete.

A local volume-doubling property and its consequences
The purpose of this section is to obtain the parabolic VDP (Theorem 8) and a negative moment estimate (Theorem 26) for the density ρ ω . The former is an essential part for the proof of the PHI for L * ω , while the latter will imply the negative moment bound (8) for the heat kernel. Their proofs rely crucially on a VDP for hitting probabilities restricted in a finite ball (Lemma 19), which is an improved version of [28, Theorem 1.1] by Safonov and Yuan in the PDE setting.
As a by-product, we obtain a new proof of the classical PHI of Krylov and Safonov [22] in the lattice (Proposition 5). Our proof can be viewed as the parabolic version of Fabes and Stroock's [17] proof of the elliptic HI in the static PDE setting.
In the course of our proof, we also use the maximum principle (Theorem A.3.1) and mean-value inequality (Theorem A.4.1) both of which are standard results for ω-caloric functions. Their statements and proofs are included in Sects. A.3 and A.4 for completeness.

Volume-doubling properties
By the optional stopping theorem, for any (x, t) ∈ D ⊂ Z d × R and any bounded integrable function u on D ∪ ∂ P D, where τ = inf{r ≥ 0 : (X r , T r ) / ∈ D}. To prove Theorem 8, a crucial estimate is a VDP (Lemma 19) for the hitting measure of the random walk, which we will obtain by adapting some ideas of Safonov and Yuan [28] in the PDE setting. In contrast to [28, Theorem 1.1], our proof relies on a probabilistic estimate (Lemma 15) rather than the PHI (Proposition 5).
Here C k depends only on (k, d, κ). In particular, for any k ≥ 1, |y| ≤ k √ s,

Proof of Lemma 19:
Since B 1 = {0}, we only consider r ≥ 1/2. Fix s, r , and let k 0 ≥ 1 be a large constant to be determined. For ρ ≥ 0, k ≥ k 0 , define L k,ρ = B kρ × {s − ρ 2 } and (See Fig. 3.) For any R ≤ ρ, by Lemma 15, there exists α k > 0 depending on (k, κ, d) such that Let β k > 1 be a large constant to be determined later. Then, letting Clearly, R ρ ≥ (4r ) ∧ ρ. We will prove that Assuming (15) fails, then R ρ < ρ for some ρ > 4r . We will show that this is impossible via contradiction. First, for such ρ > 4r , we claim that there exists a constant γ = γ (d, κ) > 0 such that Further, by the Markov property and that R ρ < ρ, for R ∈ [2r , R ρ ), n ≥ 1, where Inequality (17) then follows from the fact where A k depends on (k, κ, d). Taking k 0 > c 1 to be big enough such that −c/ log q k > γ for k ≥ k 0 and choosing β k > A −1 k , the above inequality then implies inf D k,2Rρ v ≥ 0, which contradicts our definition of R ρ . Display (15) is proved, and therefore, min x∈B k √ s v √ s (x, 0) ≥ 0. The lemma follows.
Proof It suffices to consider r < √ s, because otherwise, by Lemma 15, the right side is bigger than a constant. When t ∈ [0 ∨ (s − r 2 ), s], By Lemma 19, we can replace 4r in the above inequality by r . When t ∈ [s, s + r 2 ], for any y ∈ B k √ s , Observing that (cf. Lemma 19) our proof is complete.
We define, forx = (x, t) ∈ R d × R, the parabolic balls Proof of Theorem 8: Let k 0 ≥ 2 be as in Lemma 19. Recallω t , , Q r in (1), (11), (18). For fixed ξ ∈ κ , define a probability measure where C R is a normalization constant such that Q R is a probability.
First, we claim that C R R 2 . Clearly, C R ≤ 2R 2 . On the other hand, which implies E x,0 ξ [τ ] ≥ cR 2 . Thus C R ≥ C R 2 and so C R R 2 . Next, since is pre-compact, by Prohorov's theorem, there is a subsequence of Q R that converges weakly, as R → ∞, to a probability measureQ on . We will show thatQ is an invariant measure of the process (ω t ). Indeed, let p R = p R,ξ denote the kernel p R (x; y, s) for f ∈ dom(L ), where dom(L ) denotes the domain of the generator L . Note that similar to ρ ω , the function v(x) = p R (0, −R 2 ;x) satisfies the equality (3): t). Hence, using integration by parts, for all f ∈ dom(L ), where and u| ∂ P D = 0. By the parabolic maximum principle (Theorem A.3.1), we get u(0, −R 2 ) ≤ C R (2d+1)/(d+1) . Hence, by (19), (20), and C R R 2 , and so EQ[L f ] = 0, which implies thatQ is an invariant measure of (ω t ).
Furthermore, we will show thatQ P. Notice that the function which, by the multi-dimensional ergodic theorem, yields Hence, for any measurable function f ≥ 0, |t| ≤ r 2 , and P-a.a. ξ , Since f is arbitrary, the theorem follows.

Remark 21
By Theorem 8, for any r ≥ 1, Hence, by the multi-dimensional ergodic theorem, for P-almost every ω, Display (23) will be used in the Proof of Theorem 4 in Sect. 6.

A p property and proof of the PHI for L !
The goal of this subsection is to obtain a negative moment bound for the density ρ ω and to prove the PHI for L ω . We will first obtain a reverse Hölder inequality for adjoint solutions, and then use it to imply the negative moment bounds and the PHI. We endow Z d with the discrete topology and counting measure, and equip Z d × R with the corresponding product topology and measure (where R has the usual topology and measure). For D ⊂ Z d × R, let |D| be its measure, and denote the integration over D by D f . For instance, and |D| = D 1. For p > 0, we define a norm We write f (D) := D f . . . . , x d , t), define parabolic cubes with side-length r > 0 as

Definition 22
We say that a function w ∈ R Z d ×R satisfies the reverse Hölder inequality R H q (D), 1 < q < ∞, if for any parabolic subcube K of D, We say that w belongs to the A p (D) class (with A p bound A), 1 < p < ∞, if there exists A < ∞ such that, for any parabolic subcube K of D, The following lemma is useful in the derivation of reverse Hölder inequalities for adjoint solutions.
Proof Note that ρ ω , v y are adjoint solutions with volume-doubling properties Theorem 8 and Corollary 20. The corollary follows from Lemma 23.

Lemma 25
Let K 0 ⊂ Z d × R be a parabolic cube with side-length r > 0. If a function w > 0 on K 0 satisfies R H q (K 0 ), q > 1, then With Lemma 25 and the reverse Hölder inequalities for ρ ω and g R (y, 0; ·), the following A p bounds and measure estimate follow immediately.

Theorem 26
Let ω, R, k 0 , v y be the same as in Corollary 24. There exist p = p(d, κ) > 1, A = A(d, κ) such that, for P-a.e. ω, (a) ρ ω ∈ A p (Z d × R) with A p bound A. As a consequence,

Remark 27
In the elliptic non-divergence form PDE setting, the A p inequality for adjoint solutions was proved by Bauman [6], and estimate of the form (30) was used by Fabes and Stroock [17] to obtain a short proof of the elliptic Harnack inequality.

Estimates of caloric functions near the boundary
The purpose of this section is to establish estimates (Propositions 28 and 30) of ωcaloric functions near the parabolic boundary. These estimates are important tools for our proof of the PHI for L * ω in Sect. 5.

An elliptic-type Harnack inequality
Proposition 28 (Interior elliptic-type Harnack inequality) Assume ω ∈ κ , R ≥ 2. Suppose u ≥ 0 is an ω-caloric function on Q R with u = 0 on ∂ B R × [0, R 2 ). Then Fig. 4 The values of u are comparable inside the region Q δ To prove Proposition 28, we need a so-called Carlson-type estimate. For parabolic differential operators in non-divergence form, this kind of estimate was first proved by Garofalo [18] (see also [16,Theorem 3.3]).

Lemma 29 (Carlson-type estimate) Assume
Then, with the convention sup ∅ = −∞, we have Indeed, for anyx = (x, t) ∈ D, we can find a sequence of n ≤ C log(r /d 1 (x)) balls with increasing radii r k := c2 k d 1 (x): such that x 1 = x, dist(x n , ∂ B R−r ) ≤ r /2, and t − r 2 n ≥ r 2 /2. By Proposition 5, where in the last inequality we applied Proposition 5 to a chain of parabolic balls with spatial centers at ∂ B R−r and radius cr . Display (32) is proved. Next, with γ as in (32), u. Display (33) is proved. Inequality (31) follows from (32) and (33).

Proof of Proposition 28:
where we used Lemma 29 and Proposition 5 in the second inequality.

A boundary Harnack inequality
For positive caloric functions with zero values on the spatial boundary, the following boundary PHI compares values near the spatial boundary and values inside, with time coordinates appropriately shifted.

Proposition 30 (Boundary PHI) Let R > 0. Suppose u is a nonnegative ω-caloric
Proposition 30 is a lattice version of [18, (3.9)]. In what follows we offer a probabilistic proof.

Proof of Proposition 30:
Our proof uses the fact that u(X t ) is a martingale before exiting the region D : For the lower bound, let τ 3,4 := inf{s > 0 : X s / ∈ B 4R \B 3R }. By the optional stopping lemma, u(x) = Ex ω [u(X τ 3,4 ∧0.5R 2 )], and so where in the last inequality we used Lemma 17 and applied Proposition 5 (to a chain of parabolic balls). The lower bound is obtained.
To obtain the upper bound, note that forx where in the last inequality we applied Lemma 18 and used an iteration of the PHI for ω-caloric functions (Proposition 5).

Proof of the PHI for the adjoint operator (Theorem 6)
In this section we will prove the PHI for L * ω . Our proof relies on a representation formula for ω * -caloric functions (Proposition 31), the parabolic volume-doubling property of ρ ω (Theorem 8), the PHI (Proposition 5) and boundary PHI (Proposition 30) for ω-caloric functions.
We defineŶ t = (Y t , S t ) to be the continuous-time Markov chain on Z d × R with generator L * ω . The processŶ t can be interpreted as the time-reversal ofX t . Denote by P y,s ω * the quenched law ofŶ · starting fromŶ 0 = (y, s) and by E y,s ω * the corresponding expectation. Note that S t = S 0 − t.
and τ R (Ŷ ) is defined similarly. Note that Proposition 31 For anyŷ = (y, s) ∈ B R × (0, ∞) and any non-negative ω * -caloric Proof Write the two summations in the proposition as I and II. Clearly, So it remains to show I = E y,s Indeed, for h > 0 small enough, x ∈ ∂ B R and almost every t ∈ (0, s), Dividing both sides by 2h and taking h → 0, display (35) follows by Lebesgue's differentiation theorem. Applying (35) to with a change of variable. and u(x, t) = 0 when x ∈ ∂ B 2R or t > s. By Proposition 30 and Proposition 28, for any ( and, for any ( .

Remark 33
It is clear that for static environments, the adjoint Harnack inequality (Theorem 6) follows immediately from Lemma 32. However, in time-dependent case, we need the parabolic volume-doubling property of ρ ω .

Proof of Theorem 6
First, we will show that for all R > 0, Recall τ R at (34) and set g( s)ρ(x, s).
Finally, Theorem 6 follows by Theorem 8 and the above inequality.

Proof of Theorems 4, 11, and Corollary 12
The goal of this section is to prove the LLT (Theorem 4), the HKE (Theorem 11), and Corollary 12. With the QCLT and the Hölder regularity for ω * -caloric functions, the LLT, quenched heat kernel bound (6), and the Green's function asymptotics Corollary 12(ii)(iii) all follow from rather standard arguments, which have been successfully implemented for random conductance models, e.g., [2,3,5,7]. Our main novelty in this section is the bounds (7)(8) of positive and negative moments for the heat kernel.

Proof of Theorem 11
Proof First, using Theorem 6 and standard arguments, we will prove (6).
Next, we will prove the moment bounds (7) and (8), which, by (6) and (22), are equivalent to showing that, for r := √ t > 0, where Q r is as defined in (18). Indeed, using the translation-invariance of P and the volume-doubling property of ρ, for q := (d + 1)/d, where we used the Reverse Hölder inequality (Corollary 24(i)) in the last inequality.
where x is as in Theorem 4. Note that L * ω v = 0 in Z d × (0, ∞). By Corollary 7 and Theorem 11, for anyŷ = (y, when t > t 0 (ω) is big enough. Here in the last inequality we used Corollary 12(i) which is an immediate consequence of Theorem 11 and (23).
To prove Theorem 4, it suffices to show that for any K > T , Indeed, for any > 0, there exists K = K (T , , d, κ) > 0 such that, writing D : Hence Theorem 4 follows provided that (45) is proved.

Proof of Theorem 4
As we discussed in the above, it suffices to prove (45). For any > 0, where First, we will show that To this end, note that there exists N = N (T , ω, d, κ) such that for n ≥ N , Display (47) follows. Next, forx = (x, t), by writing B n (x, ) as We claim that B n (x, ) is approximately equicontinuous (with order γ ). That is, It suffices to show that B 1 n (x, ) is approximately equicontinuous. Indeed, by (47) and (44), when n ≥ N is large andx 1 The approximate equicontinuity of B 1 n (x, ) follows. To prove (49), we choose a finite sequence Since lim n→∞ max 1≤i≤M B n (x i ) = 0 by the QCLT (Theorem 1), display (49) follows by the approximate equicontinuity.
Clearly, lim →0 supx ∈O K ×[T ,K ] A(x, ) = 0. This, together with (47) and (49), yields the uniform convergence of (46) by sending first n → ∞ and then → 0. Our proof of Theorem 4 is complete. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

A Appendix
In Sect. A.1 we will show properties (i)-(iii) of ρ ω in Remark 3. In Sect. A.2, we prove the Hölder estimate (Corollary 7) for ω * -caloric functions. Sections A.3 and A.4 are devoted to the maximum principle (which is used in various places in the paper) and the mean value inequality (used in the proof of the PHI Proposition 5) for L ω , respectively. Section A.5 contains the proof of Lemma 25. Corollary 12(ii)(iii) are proved in Sect. A.6.

A.1 Properties (i)-(iii) in Remark 3
Proof (i)Since Q is an invariant measure for (ω t ), we have for any bounded measurable function f on , y ∈ Z d ,x = (x, t), and s < t, whereŷ = (y, s) and we used the translation-invariance of P in the last equality. Moreover, by Fubini's theorem, for any bounded compactly-supported continuous function φ : R → R, Thus we have that P-almost surely, for any such test function φ on R, which (together with the translation-invariance of P) implies that P-almost surely, ρ ω (x, t)δ x dt is an invariant measure for the process (X t ) t≥0 .
(ii) We have ρ ω > 0 since the measures Q and P are equivalent. The uniqueness follows from the uniqueness of Q in [14, Theorem 2.1(iii)].
(iii) By (50) and Fubini's theorem, we also have that P-almost surely, for any test function φ(t) as in (i) and any h > 0, Dividing both sides by h and letting h → 0, we obtain (3) with ∂ t ρ ω replaced by the weak derivative. Note that the weak differentiability of ρ ω in t implies that it has an absolutely continuous (in t) version. Since ρ ω is only used as a density, we may always assume that P-almost surely, ρ ω (x, ·) is continuous and almost-everywhere differentiable in t.

A.3 Parabolic maximum principle
In what follows we will prove a maximum principle for parabolic difference operators under the discrete space and continuous time setting. Its statement is a tiny modification from [14,Theorem 5.1] where its proof, which follows verbatim the lines of [14, Theorem 2.2], was omitted. For the purpose of completeness we will include a full proof in the below. First, we claim that This will be proved by showing that for any (ξ, h) ∈ , we have (ξ, h) ∈ χ(x 1 , t 1 ) for some (x 1 , t 1 ) ∈ + . Indeed, fix (ξ, h) ∈ and define φ(x, t) := u(x, t) − ξ · x − h.