Rate functions for symmetric markov processes via heat kernel

TL;DR

This paper uses heat kernel estimates to establish integral tests for zero-one laws related to the upper and lower bounds of symmetric Markov process sample paths, applicable to various Dirichlet forms.

Contribution

It introduces new integral tests for zero-one laws of rate bounds in symmetric Markov processes, covering both local and non-local Dirichlet forms and critical/subcritical settings.

Findings

Established integral tests for upper rate bounds.

Analyzed lower rate bounds in subcritical and critical settings.

Results applicable to a broad class of symmetric Markov processes.

Abstract

By making full use of heat kernel estimates, we establish the integral tests on the zero-one laws of upper and lower bounds for the sample path ranges of symmetric Markov processes. In particular, these results concerning on upper rate bounds are applicable for local and non-local Dirichlet forms, while lower rate bounds are investigated in both subcritical setting and critical setting.