Anchored Nash inequalities and heat kernel bounds for static and dynamic degenerate environments

TL;DR

This paper introduces anchored Nash inequalities to handle degenerate environments and derives heat kernel bounds for diffusions in such settings, including random walks influenced by exclusion processes.

Contribution

It develops anchored Nash inequalities that extend classical bounds to non-uniformly elliptic cases and applies them to dynamic random environments.

Findings

Established heat kernel upper bounds for degenerate static environments.

Extended analysis to dynamic environments with degenerate jump rates.

Applied results to random walks influenced by exclusion processes.

Abstract

We introduce anchored versions of the Nash inequality. They allow to control the $L^2$ norm of a function by Dirichlet forms that are not uniformly elliptic. We then use them to provide heat kernel upper bounds for diffusions in degenerate static and dynamic random environments. As an example, we apply our results to the case of a random walk with degenerate jump rates that depend on an underlying exclusion process at equilibrium.