The e-property of asymptotically stable Markov semigroups
Abstract
The relations between asymptotic stability, the eventual e-property and the e-property of Markov semigroups, acting on measures defined on general (Polish) metric spaces, are studied. While usually much attention is paid to asymptotic stability (and the e-property has been for years verified only to establish it), it should be noted that the e-property itself is also important as it, e.g., ensures that numerical errors in simulations are negligible. Here, it is shown that any asymptotically stable Markov-Feller semigroup with an invariant measure such that the interior of its support is non-empty satisfies the eventual e-property. Moreover, we prove that any Markov-Feller semigroup, which is strongly stochastically continuous, and which possesses the eventual e-property, also has the e-property. We also present an example highlighting that strong stochastic continuity cannot be replaced by its weak counterpart, unless a state space of a process corresponding to a Markov semigroup is a compact metric space.
Keywords
Cite
@article{arxiv.2211.16424,
title = {The e-property of asymptotically stable Markov semigroups},
author = {Ryszard Kukulski and Hanna Wojewódka-Ściążko},
journal= {arXiv preprint arXiv:2211.16424},
year = {2024}
}
Comments
19 pages, This preprint has not undergone peer review or any post-submission improvements or corrections. The Version of Record of this article is published in Results in Mathematics, and is available online at https://doi.org/10.1007/s00025-024-02134-2