Heat kernel estimates for Markov processes of direction-dependent type

TL;DR

This paper establishes sharp heat kernel estimates for a class of symmetric Markov processes that combine local and nonlocal behaviors across different coordinates, extending classical results to more complex, direction-dependent processes.

Contribution

It introduces a robustness theorem for heat kernel estimates in mixed local-nonlocal Markov processes, generalizing Aronson's classical estimate to new, direction-dependent settings.

Findings

Proves sharp pointwise heat kernel bounds for mixed local-nonlocal processes.

Shows invariance of pointwise estimates between translation-invariant and general Dirichlet forms.

Extends classical heat kernel estimates to processes with anisotropic, direction-dependent features.

Abstract

We prove sharp pointwise heat kernel estimates for symmetric Markov processes associated with symmetric Dirichlet forms that are local with respect to some coordinates and nonlocal with respect to the remaining coordinates. The main theorem is a robustness result like the famous estimate for the fundamental solution of second order differential operators, obtained by Donald G. Aronson. Analogous to his result, we show that the corresponding translation-invariant process and the one given by the general Dirichlet form share the same pointwise points.