Weighted heat kernel estimates: rate of convergence in Kolmogorov distance

TL;DR

This paper investigates the convergence rate of random walks in random environments on integer lattices, providing heat kernel estimates and a Berry-Esseen bound demonstrating a specific polynomial rate of convergence.

Contribution

It introduces new heat kernel estimates for non-diagonal random matrices and establishes a Berry-Esseen bound with a convergence rate of t^{-1/10} in dimensions d≥3.

Findings

Heat kernel estimates for non-diagonal random matrices

Berry-Esseen upper bound with rate t^{-1/10}

Analysis based on ergodicity and logarithmic Sobolev inequalities

Abstract

This paper is concerned about random walks on random environments in the lattice $\mathbb{Z}^d$. This model is analyzed through ergodicity in the form of the logarithmic Sobolev inequality. We assume that the environments are random variables being independent and identically distributed. Here, we give heat kernel estimates for non-diagonal random matrices leading in dimension $d\geq 3$ a Berry-Esseen upper bound with a rate of convergence $t^{-\frac{1}{10}}$.