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

Xiaoqin Guo,Jean-Dominique Deuschel

doi:10.1007/s00440-021-01097-7

Abstract

We prove a quenched local central limit theorem for continuous-time random walks in {mathbb {Z}}^d, dge 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’s function.

Highlights

In this article we consider a random walk in a balanced uniformly-elliptic timedependent random environment on Zd, d ≥ 2
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 volume-doubling property (VDP) 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 local limit theorem (LLT), quenched heat kernel estimates (HKE), positive and negative L p bounds for the heat kernel, and Green’s function asymptotics for the RWRE
The purpose of this section is to obtain the parabolic VDP (Theorem 8) and a negative moment estimate (Theorem 26) for the density ρω. 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

Summary

Introduction

In this article we consider a random walk in a balanced uniformly-elliptic timedependent random environment on Zd , d ≥ 2. (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. Some classical estimates and standard arguments can be found in the Appendix

Auxiliary probability estimates

A local volume-doubling property and its consequences

Volume-doubling properties

Ap property and proof of the PHI for L!

Estimates of caloric functions near the boundary

A boundary Harnack inequality

Proof of Theorem 11

Proof of Theorem 4 and Corollary 12

A Appendix

Proof of Corollary 7

Parabolic maximum principle

Mean value inequality