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 $\mathbb Z^d, d\ge 2$, 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 function.


Introduction
In this article we consider a random walk in a balanced uniformly-elliptic time-dependent 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 nearest-neighbor transition rates on Z d ) Equip P with the the product topology and the corresponding Borel σ-field. We denote by Ω ⊂ P R the set of all measurable functions ω : t → ω t from R to P and call every ω ∈ Ω a time-dependent environment. For ω ∈ 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) * Electronic address: deuschel@math.tu-berlin.de † Electronic address: xguo@math.wisc.edu 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 Ω ⊂ P R with the induced product topology and let P be a probability measure on the Borel σ-field B(Ω) of Ω. An environment ω ∈ Ω is said to be balanced if y ω t (x, y)(y − x) = 0, for all t ∈ R, x ∈ Z d 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(Ω) 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 Ω. 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.
We recall the quenched central limit theorem (QCLT) in [12]. Theorem 1. [12,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 .
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 [26]. It is then generalized to static random environments with weaker ellipticity assumptions in [18,7]. It is obtained in [12] that ρ > 0, P-almost surely. At the end of the proof of [12,Theorem 1.2], it is shown that E Q [g] ≤ C g L d+1 (P) for any bounded continuous function g, which implies Moreover, one of our main results (see Theorem 15(a) below) shows that there exists q = q(κ, d) such that the L −q (P) moment of ρ is also bounded.
Since Ω is equipped with a product σ-field, for any fixed ω ∈ Ω, the map R → Ω 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 can be found in [11,Appendix].
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. Let 0 := (0, 0) ∈ Z d × R. The proof of the LLT follows from Theorem 1 and a localization of the heat kernel q ω (0, ·), an argument already implemented in [6] 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 (cf. Corollary 24 below) of q ω (0, ·) but also very sharp heat kernel estimates. Note that for fixedx = (x, t), the function u(ŷ) = q ω (ŷ,x) satisfies L ω u = 0 in Z d × (−∞, t). However in our non-reversible model, we need to prove, instead of PHI for L ω , the PHI (Theorem 5) for the adjoint operator L * ω (defined below), since the heat kernel v ω (x) := q ω (0,x) solves Note that ω * is not necessarily a balanced environment anymore.
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 cB ≤ A ≤ CB, we write Our second main result is In PDE, the Harnack inequality for the adjoint of non-divergence form elliptic differential operators was first proved by Bauman [5], and was generalized to the parabolic setting by Escauriaza [14]. Our proof of Theorem 5 follows the main idea of [14].

Remark 6.
For time discrete random walks in a static environment, Theorem 5 was obtained by Mustapha [25]. His argument follows basically [14], and uses the PHI [23,Theorem 4.4] of Kuo and Trudinger in the time discrete situation. Moreover, in the static case, the volume-doubling property of the invariant distribution, which is the essential part of the proof of Theorem 5, is much simpler, see [16]. In our dynamical setting, a parabolic volume-doubling property (Theorem 9) is required. To this end, we adapt ideas of Safonov-Yuan [27] and results in the references therein [15,5,17] into our discrete space setting.
The main challenge in proving Theorem 5 is that L * ω is neither balanced nor uniformly elliptic, and so the PHI for L ω (Theorem 17) is not immediately applicable. This is the main difference with the random conductance model with symmetric jump rates where , and thus which PHI for L ω is the same as PHI for L * ω . See [1,9,10,2,19]. Let us explain the main idea for the proof of Theorem 5. An important observation is that solutions v of L * ω can be expressed in terms of hitting probabilities of the time-reversed process, cf. Lemma 21 below. Thus to compare values of the adjoint solution, 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" (Theorem 20) which compares L ω -harmonic functions near the boundary. We will also need a volume-doubling inequality for the invariant measure (Theorem 9) to control the change of probabilities due to time-reversal.
Recall the heat kernel q ω in (4). For any A ⊂ R d and s ∈ R, let Note that h(c 1 r, c 2 t) ≍ h(r, t) for constants c 1 , c 2 > 0. Our third main results are the following heat kernel estimates (HKE).

Theorem 7 (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 Note that for a general ergodic environment, the density ρ ω does not have deterministic (positive) upper and lower bounds, thus one cannot expect deterministic quenched Gaussian bounds for p ω (0,x). However, our Theorem 7 shows that it has L (d+1)/d (P) and L −q (P) moment bounds. Furthermore, we can characterize asymptotics of the Green function of the RWRE. Recall the notations Σ (in Theorem 1 (b)), p Σ t , ⌊x⌋ (in Theorem 4), and h in (7). Corollary 8. The following statements are true for P-almost every ω.
As a consequence, the RWRE is recurrent when d = 2 and transient when d ≥ 3.
Similar results as Corollary 8(ii)(iii) are also obtained recently for the conductance model [4].
The organization of this paper is as follows. In Section 2, we prove a parabolic volume-doubling property and an A p inequality for ρ ω , and obtain a new proof of the PHI for L ω . In Section 3, we establish estimates of L ω -harmonic functions near the boundary, showing both the interior elliptictype and boundary PHI's. We prove the PHI for the adjoint operator (Theorem 5) in Section 4. Finally, with the adjoint PHI, we prove Theorems 4, 7, and Corollary 8 in Section 5. Section 6 contains probability estimates that are used in previous sections. Some estimates and standard arguments can be found in the Appendix of the longer arXiv version [11] of this paper.

A local volume-doubling property and its consequences
The purpose of this section is to obtain a parabolic volume-doubling property (Theorem 9) and a negative moment estimate (Theorem 15) for the density ρ ω . The proof relies crucially on a volume-doubling property for the hitting probabilities restricted in a finite ball (Theorem 10), which is an improved version of [27, 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 (Theorem 17). Our proof, which is of interest on its own, can be viewed as the parabolic version of Fabes and Stroock's [16] proof in the elliptic PDE setting.
In PDE setting, this type of estimate was first established by Fabes and Stroock [16] for adjoint solutions of non-divergence form elliptic operators, and then generalized by Escauriaza [14] to the parabolic case.
To obtain Theorem 9, a crucial estimate is a volume-doubling property (Theorem 10) for the hitting measure of the random walk, which we will prove by adapting some ideas of Safonov and Yuan [27, Theorem 1.1] in the PDE setting. Note that our proof of Theorem 10 relies on a probabilistic estimate (Lemma 27) rather than the Harnack inequality (Theorem 17).
Note that k 0 > 6 is a large constant to be determined. For any R ≤ ρ, by Figure 2: The shaded region is D k,ρ . Lemma 27 below, 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 (14) 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 By Lemma 27, g(R, R/2) := min x∈B R P x,s−R 2 ω (X ∆(B 2R ,s−(R/2) 2 ) ∈ B R/2 ) ≥ C. Further, by the Markov property and that R ρ < ρ, for R ∈ [2r, R ρ ), Since v ρ (X t ) is a martingale in B mkρ × (−∞, s) and that v ρ ≥ 0 in D k,Rρ , choosing n such that R/2 n ≤ r < R/2 n−1 , the above inequality yields where

Inequality (16) 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 (14) is proved, and therefore, min x∈B k √ s v √ s (x, 0) ≥ 0. The theorem follows.

Corollary 11.
Let ω ∈ Ω κ and k 0 as in Theorem 10. For any Proof. It suffices to consider r < √ s, because otherwise, by Lemma 27, the right side is bigger than a constant.
By Theorem 10, we can replace 4r in the above inequality by r.
Observing that (cf. Theorem 10) Proof of Theorem 9: Let k 0 ≥ 2 be as in Theorem 10. Recallω t , Q r , ∆ in (1), (6), (13). For fixed ξ ∈ Ω κ , define a probability measure where C R is a renormalization 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, Thus C R ≥ CR 2 and so C R ≍ R 2 . Next, since Ω is pre-compact, by Prohorov's theorem, there is a subsequence of Q R that converges weakly, as R → ∞, to a probability measureQ on Ω. We will show thatQ is an invariant measure of the process (ω t ). Indeed, let . Hence, using integration by parts, u| ∂ P D = 0. By the parabolic maximum principle [11, Theroem A.3.1], we get u(0, −R 2 ) ≤ CR (2d+1)/(d+1) . Hence, by (17), (18), and 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 12. By Theorem 9, for any r ≥ 1, Hence, by the multi-dimensional ergodic theorem, for P-almost every ω,

A p property and a new proof of the PHI for L ω
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 Lemma 13. Recall · D,p in (23) and the parabolic balls Q r in (6).
Proof. Denote the continuous balls of radius r by The lemma follows by taking supremum over all f with f Qr,d+1 = 1.
We say that a function w ∈ R Z d ×R satisfies the reverse Hölder inequality RH q (D), 1 < q < ∞, if for any parabolic subcube K of D, We say that w belongs to the if there exists A < ∞ such that, for any parabolic subcube K of D, Recall the stopping time ∆ in (13).
Proof. Note that ρ ω , v y are adjoint solutions with volume-doubling properties Theorem 9 and Corollary 11. The corollary follows from Lemma 13.

Remark 16.
The fact that (RH) implies (A p ) is a classical result in harmonic analysis. See e.g, [8, pg.246-249], [28, pg. 213-214]. In the elliptic non-divergence form PDE setting, the A p inequality for adjoint solutions was proved by Bauman [5], and estimate of form (27) was used by Fabes and Stroock [16] to obtain a short proof of the elliptic Harnack inequality.
In what follows, using (27), we will obtain a new proof of the parabolic Harnack inequality (Theorem 17). Our proof follows the ideas of [16]. Note that in our parabolic setting, the local volume-doubling property (Corollary 11) played a crucial role in the proof of (27).
We remark that in discrete time setting, (PHI) is obtained by Kuo and Trudinger for the so-called implicit form operators, see [23, (1.16)].
Proof of Theorem 17.

Estimates of solutions near the boundary
The purpose of this section is to establish estimates of L ω -harmonic functions near the parabolic boundary. For

An elliptic-type Harnack inequality
Theorem 18 (Interior elliptic-type Harnack inequality). Assume ω ∈ Ω κ . Let R ≥ 2 and u ≥ 0 satisfies . Figure 3: The values of u are comparable inside the region Q δ R .
To prove Theorem 18, 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 [17] (see also [15,Theorem 3.3]).

Proof of Theorem 18:
where we used Theorem 19 and Theorem 17 in the second inequality.

A boundary Harnack inequality
For positive harmonic functions with zero values on the spatial boundary, the following boundary Harnack inequality compares values near the spatial boundary and values inside, with time coordinates appropriately shifted.
Theorem 20 is a lattice version of [17, (3.9)]. In what follows we offer a probabilistic proof.
Proof of Theorem 20: 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 : where in the last inequality we used Lemma 29 and applied Theorem 17 (to a chain of parabolic balls). The lower bound is obtained.
To obtain the upper bound, note that forx ∈ (B 4R \B 3R ) × (−R 2 , R 2 ), where in the last inequality we applied Lemma 30 and used an iteration of the Harnack inequality (Theorem 17).

Proof of PHI for the adjoint operator (Theorem 5)
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

Proof. Write the two summations in the lemma as I and II. Clearly, II =
So it remains to show I = E y,s ω * [v(Ŷ τ R )1 τ R ≤s ]. We claim that for x ∈ ∂B R , (32) 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 (32) follows by Lebesgue's differentiation theorem. Applying (32) to we obtain I = E y,s ω * [v(Ŷ τ R )1 τ R ≤s ] with a change of variable.

Remark 23. It is clear that for static environments, the adjoint Harnack inequality (Theorem 5) follows immediately from Lemma 22. However, in
time-dependent case, we need the parabolic volume-doubling property of ρ ω .
Proof of Theorem 5. First, we will show that for all R > 0, Recall τ R at (31) and set g( Moreover, since |X t | 2 − d κ t and |X t | 2 − κt are super-and sub-martingales, Finally, Theorem 5 follows by Theorem 9 and the above inequality.

Proof of Theorem 7, Corollary 8 and Theorem 4 5.1 Proof of Theorem 7
Proof. First, using Theorem 5 and standard arguments, we will prove (8).

Proof of Theorem 4
As a standard consequence of the PHI for L * ω , we first state the following Hölder estimate. (See a proof in [11,Section A.2].) Corollary 24. There exists γ = γ(d, κ) ∈ (0, 1] such that for P-almost all ω, where ⌊x⌋ is as in Theorem 4. Note that L * ω v = 0 in Z d × (0, ∞). By Corollary 24 and Theorem 7, for anyŷ = (y, s) when t > t 0 (ω) is big enough. Here in the last inequality we used Corollary 8(i) which is an immediate consequence of Theorem 7 and (21).
To prove Theorem 4, it suffices to show that for any K > T , Indeed, for any ǫ > 0, there exists K = K(T, ǫ, d, κ) > 0 such that, writing Hence Theorem 4 follows provided that (42) is proved.
Proof of Theorem 4. As we discussed in the above, it suffices to prove (42). It suffices to consider the case T < K < 2T . For any ǫ > 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 , where in the second inequality we used Corollary 8(i) and (41). Further, by an ergodic theorem of Krengel and Pyke [21, Theorem 1] and (2), Display (44) follows. Next, forx = (x, t), by writing B n (x, ǫ) as We claim that B n (x, ǫ) is approximately equicontinuous (with order ǫ γ ). That is, there exist N, δ depending on (ǫ, ω, d, κ, T, K) such that, whenever n ≥ N andx 1 = (x 1 , t 1 ), It suffices to show that B 1 n (x, ǫ) is approximately equicontinuous. Indeed, by (44) and (41), when n ≥ N is large andx 1 , The approximate equicontinuity of B 1 n (x, ǫ) follows. To prove (46), we choose a finite sequence Since lim n→∞ max 1≤i≤M B n (x i ) = 0 by the QCLT (Theorem 1), display (46) follows by the approximate equicontinuity.

Auxiliary probability estimates
This section contains probability estimates that are useful in the rest of the paper. It does not rely on results in the previous sections, and can be read independently. Recall the definition of the function h(r, t) in (7).
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 ½ r<9c 2 0 t + r log( r t )½ r≥9c 2 0 t , we obtain (r, t)).
Proof. By uniform ellipticity, it suffices to prove the lemma for all r ≥ 10.
Recall the definition of the stopping time ∆ in (13).
Next, set (Recall the stopping time ∆ in (13).) Then by the optional stopping theorem, we have for x ∈ B R \B βR , Therefore, for any x ∈ B R \B βR , Our proof of Lemma 30 is complete.

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 Ω, 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 [12, Theorem 1.2].
(iii) By (49) 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. For any (22).
Lemma A.5.1. Let K 0 ⊂ Z d × R be a parabolic cube with side-length r > 0. If a function w > 0 on K 0 satisfies RH q (K 0 ), q > 1, then (i) w ∈ A p (K 0 ) for some 1 < p < ∞; Proof. First, we claim that there exist constants γ, δ ∈ (0, 1) such that w(E) > γw(K) implies |E| > δ|K| for all E ⊂ K where K = ∅ is a subcube of K 0 . Indeed, this is a simple consequence of Hölder's inequality: where q ′ = q/(q − 1) denotes the conjugate of q. Assume K 0 = K r . Let M k (K r ), k > 1 be the family of nonempty subcubes of K r of the form where m i , n i 's are integers. Elements in M k (K r ) are called k-level dyadic subcubes of K r . Note that every k-level cube K is contained in a unique (k − 1)-level "parent" denoted by K −1 . Since the class A p is invariant under constant multiplication, we may assume that w(K 0 )/|K 0 | = 1.
Let f := w −1 ½ K 0 and define a maximal function where the supremum is taken over all dyadic subcubes K of K 0 . Consider the level sets where N is a big constant to be determined. Notice that by assumption, E 0 is comprised of dyadic subcubes strictly smaller than K 0 . Since w is volume-doubling, there exists a constant c 0 > 0 such that for any maximal dyadic subcube K of E k−1 , Moreover, for the same K, we have 2 N k w(E k ∩ K) ≤ K f w and so, by the inequality above, w(E k ∩ K) ≤ 2 c 0 −N w(K). We now take N to be large enough that w(E k ∩ K) ≤ (1 − γ)w(K) which implies |E k ∩ K| ≤ (1 − δ)|K|. Summing over all such K's, we have |E k | ≤ (1 − δ)|E k−1 |, k ≥ 1. Thus |E k | ≤ δ k |E 0 | ≤ δ k |K 0 |, k = 0, . . . and so, for p > 1 chosen so that p ′ = p/(p − 1) is sufficiently close to 1, (i) is proved. (ii) then follows from Hölder's inequality and the A p inequality.