Dirichlet eigenvalues and exit time moments for symmetric Markov processes

TL;DR

This paper explores the relationship between Dirichlet eigenvalues and exit time moments in symmetric Markov processes, providing theoretical insights and practical estimates for various examples like diffusions and stable processes.

Contribution

It establishes new relationships between eigenvalues and exit times for symmetric Markov processes, including applications to diffusions and stable processes.

Findings

Derived bounds for Dirichlet eigenvalues

Estimated exit time moments for specific processes

Illustrated results with concrete examples

Abstract

We give some relationships between the first Dirichlet eigenvalues and the exit time moments for the general symmetric Markov processes. As applications, we present some examples, including symmetric diffusions and $\alpha$-stable processes, and provide the estimates of their first Dirichlet eigenvalues and the exit time moments.