Topological and metric recurrence for general Markov chains

TL;DR

This paper introduces topological and metric recurrence concepts for Markov chains, analyzing the size of recurrent point sets and establishing conditions under which non-recurrent points are negligible, with implications even in deterministic systems.

Contribution

It develops new topological and metric recurrence frameworks for Markov chains and provides criteria for the measure of non-recurrent points, extending to deterministic cases.

Findings

Non-recurrent points form a zero measure set under mild conditions.

Provides necessary and sufficient conditions for measures to assign zero to non-recurrent points.

Results are novel even in deterministic dynamical systems.

Abstract

Using ideas borrowed from topological dynamics and ergodic theory we introduce topological and metric versions of the recurrence property for general Markov chains. The main question of interest here is how large is the set of recurrent points. We show that under some mild technical assumptions the set of non recurrent points is of zero reference measure. Necessary and sufficient conditions for a reference measure $m$ (which needs not to be dynamically invariant) to satisfy this property are obtained. These results are new even in the purely deterministic setting.