A Dirichlet principle for non reversible Markov chains and some recurrence theorems

TL;DR

This paper extends the Dirichlet principle to non-reversible Markov chains, providing variational formulas for the Poisson equation and proving recurrence theorems, including for two-dimensional cycle random walks.

Contribution

It introduces variational formulas for the Poisson equation in non-reversible Markov processes and applies them to establish new recurrence results.

Findings

Derived variational formulas for capacity and Poisson equation solutions.

Proved recurrence of two-dimensional cycle random walks under specific conditions.

Extended Dirichlet principle to non-reversible Markov chains.

Abstract

We extend the Dirichlet principle to non-reversible Markov processes on countable state spaces. We present two variational formulas for the solution of the Poisson equation or, equivalently, for the capacity between two disjoint sets. As an application we prove a some recurrence theorems. In particular, we show the recurrence of two-dimensional cycle random walks under a second moment condition on the winding numbers.