Heat kernel fluctuations and quantitative homogenization for the one-dimensional Bouchaud trap model

TL;DR

This paper provides heat kernel estimates and quantitative homogenization results for the one-dimensional Bouchaud trap model, advancing understanding of random walks in random environments with precise probabilistic bounds.

Contribution

It introduces new heat kernel bounds and homogenization theorems, including Berry-Esseen and local limit results, for the Bouchaud trap model, a fundamental example in random environment studies.

Findings

Established on-diagonal heat kernel estimates

Proved quenched and annealed Berry-Esseen theorems

Derived a quantitative quenched local limit theorem

Abstract

We present on-diagonal heat kernel estimates and quantitative homogenization statements for the one-dimensional Bouchaud trap model. The heat kernel estimates are obtained using standard techniques, with key inputs coming from a careful analysis of the volume growth of the invariant measure of the process under study. As for the quantitative homogenization results, these include both quenched and annealed Berry-Esseen-type theorems, as well as a quantitative quenched local limit theorem. Whilst the model we study here is a particularly simple example of a random walk in a random environment, we believe the roadmap we provide for establishing the latter result in particular will be useful for deriving quantitative local limit theorems in other, more challenging, settings.