Invariance Principle for symmetric Diffusions in a degenerate and unbounded stationary and ergodic Random Medium

TL;DR

This paper proves a quenched invariance principle for symmetric diffusions in a stationary, ergodic, and possibly degenerate environment, using Dirichlet forms and Moser's iteration to handle unbounded coefficients.

Contribution

It establishes a quenched invariance principle for diffusions with unbounded, degenerate coefficients in a stationary ergodic setting, extending previous results to more general environments.

Findings

Proves a quenched invariance principle for the diffusion.

Uses sublinearity of the corrector via Moser's iteration.

Handles unbounded and degenerate coefficients in the environment.

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 $a^\omega$. The diffusion is formally associated with $L^\omega u = \nabla\cdot(a^\omega\nabla u)$, and we make sense of it through Dirichlet forms theory. We prove for $X$ a quenched invariance principle, under some moment conditions on the environment; the key tool is the sublinearity of the corrector obtained by Moser's iteration scheme.