Local Central Limit Theorem for Reflecting Diffusions in a Continuum Percolation Cluster

TL;DR

This paper proves a local central limit theorem for reflecting diffusions on continuum percolation clusters under certain geometric conditions, advancing understanding of stochastic processes in complex random environments.

Contribution

It establishes a quenched local CLT for reflecting diffusions on continuum percolation clusters with specific geometric assumptions, a novel result in this setting.

Findings

Quenched local CLT proven for reflecting diffusions

Results depend on volume regularity and isoperimetric conditions

Advances understanding of diffusion behavior in random media

Abstract

Reflecting diffusions on continuum percolation clusters are considered. Assuming that the occupied region has a unique unbounded cluster and the cluster satisfies geometrical conditions such as volume regularity, isoperimetric conditions, and a hole size condition, we prove a quenched local central limit theorem for reflecting diffusions on the cluster.