Local Central Limit Theorem for diffusions in a degenerate and unbounded Random Medium

TL;DR

This paper proves a local central limit theorem for symmetric diffusions in complex environments with unbounded and degenerate coefficients, using a novel parabolic Harnack inequality approach.

Contribution

It establishes a quenched local CLT for diffusions in unbounded, degenerate media, extending previous results to more general environments.

Findings

Proves a quenched local CLT for diffusions in complex media.

Develops a local parabolic Harnack inequality via Moser iteration.

Handles unbounded and degenerate coefficients under certain moment conditions.

Abstract

We study a symmetric diffusion $X$ on $\mathbb{R}^d$ in divergence form in a stationary and ergodic environment, with measurable unbounded and degenerate coefficients. We prove a quenched local central limit theorem for $X$, under some moment conditions on the environment; the key tool is a local parabolic Harnack inequality obtained with Moser iteration technique.