Variational formulas for the exit time of Hunt processes generated by semi-Dirichlet forms

TL;DR

This paper derives variational formulas for the Laplace transform of exit times of Hunt processes from open sets, extending known results to semi-Dirichlet forms and providing applications like comparison theorems and relations to Poincaré inequalities.

Contribution

It introduces new variational formulas for exit times of Hunt processes generated by semi-Dirichlet forms, broadening the scope beyond symmetric cases.

Findings

Variational formulas for Laplace transforms of exit times are established.

Comparison theorems for exit times are derived.

Relations between exponential moments and Poincaré inequalities are provided.

Abstract

Variational formulas for the Laplace transform of the exit time from an open set of a Hunt process generated by a regular lower bounded semi-Dirichlet form are established. While for symmetric Markov processes, variational formulas are derived for the exponential moments of the exit time. As applications, we provide some comparison theorems and quantitative relations of the exponential moments and Poincar\'e inequalities.