Irreducible recurrence, ergodicity, and extremality of invariant measures for resolvents

TL;DR

This paper provides a unified potential-theoretic framework to analyze recurrence, irreducibility, and extremality of invariant measures for sub-Markovian resolvents, extending classical results to non-symmetric cases.

Contribution

It introduces a new characterization of invariant functions, links irreducibility with extremality of invariant measures, and extends Fukushima's ergodic theorem to sub-Markovian resolvents.

Findings

Characterization of harmonic functions via weak dual resolvent

Equivalence between irreducibility and extremality of invariant measures

Extension of Fukushima's ergodic theorem to non-symmetric resolvents

Abstract

We analyze the transience, recurrence, and irreducibility properties of general sub- Markovian resolvents of kernels and their duals, with respect to a fixed sub-invariant measure $m$. We give a unifying characterization of the invariant functions, revealing the fact that an $L^p$-integrable function is harmonic if and only if it is harmonic with respect to the weak dual resolvent. Our approach is based on potential theoretical techniques for resolvents in weak duality. We prove the equivalence between the $m$-irreducible recurrence of the resolvent and the extremality of $m$ in the set of all invariant measures, and we apply this result to the extremality of Gibbs states. We also show that our results can be applied to non-symmetric Dirichlet forms, in general and in concrete situations. A second application is the extension of the so called Fukushima ergodic theorem for symmetric Dirichlet forms to the case of sub-Markovian resolvents of kernels.