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

TL;DR

This paper establishes a quenched local central limit theorem for continuous-time random walks in a time-dependent, balanced, ergodic environment on integer lattices, providing Gaussian bounds and Green function asymptotics.

Contribution

It proves a quenched local CLT in a time-dependent environment and derives Gaussian bounds and Green function asymptotics, extending previous static results.

Findings

Proved quenched local CLT for time-dependent environments

Established Gaussian upper and lower bounds for transition probabilities

Derived asymptotics for the discrete Green function

Abstract

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.