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

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.

Theorems & Definitions (37)

• Theorem 1.1
• proof
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Theorem 1.5: Berry-Esseen theorem
• Remark 1.6
• Theorem 1.7: Quantitative local limit theorem
• Remark 1.8
• Theorem 2.1