Criteria for asymptotic stability of eventually continuous Markov-Feller semigroups
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 -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.
Keywords
Cite
@article{arxiv.2502.13448,
title = {Criteria for asymptotic stability of eventually continuous Markov-Feller semigroups},
author = {Ting Li and Xianming Liu},
journal= {arXiv preprint arXiv:2502.13448},
year = {2025}
}
Comments
The authors are sincerely grateful to Dr. Ziyu Liu for pointing out the reference [8] to us. Noticing an overlap between our first version and Reference [8], we have revised in the current version