Variational principles for asymptotic variance of general Markov processes

TL;DR

This paper derives a variational formula for the asymptotic variance of general Markov processes and applies it to compare reversible and non-reversible diffusions, providing bounds on mean exit times.

Contribution

It introduces a new variational formula for asymptotic variance applicable to all Markov processes, enabling novel bounds and comparisons.

Findings

Derived a variational formula for asymptotic variance

Established an upper bound for mean exit time of reversible processes

Provided comparison theorems between reversible and non-reversible diffusions

Abstract

A variational formula for the asymptotic variance of general Markov processes is obtained. As application, we get a upper bound of the mean exit time of reversible Markov processes, and some comparison theorems between the reversible and non-reversible diffusion processes.