Criteria for asymptotic stability of eventually continuous Markov-Feller semigroups

TL;DR

This paper introduces practical criteria for the asymptotic stability of Markov-Feller semigroups, focusing on local conditions like eventual continuity at a single point, simplifying stability analysis.

Contribution

It provides new, localized criteria for asymptotic stability of Markov-Feller semigroups based on eventual continuity at a single point, improving practical applicability.

Findings

Criteria for convergence in total variation to a unique invariant measure.

Two new criteria for asymptotic stability based on local eventual continuity.

An explicit example demonstrating the ease of verifying local conditions.

Abstract

In this paper, we establish three criteria for the asymptotic behavior of Markov-Feller semigroups. First, we present a criterion for convergence in total variation to a unique invariant measure, requiring only $TV$-eventual continuity of the semigroup at a single point. Second, we propose two new criteria for asymptotic stability that require eventual continuity at a single point. This localized condition is more practical and easier to check. To illustrate the advantages of our framework, we provide an explicit example where verifying eventual continuity at a single point is straightforward, whereas establishing the corresponding global property is challenging.