Asymptotic stability for non-equicontinuous Markov semigroups
Fuzhou Gong, Yong Liu, Yuan Liu, Ziyu Liu
TL;DR
The paper addresses the problem of determining asymptotic stability for non-equicontinuous Markov-Feller semigroups. It introduces eventual continuity as a practical regularity and proves their equivalence with asymptotic stability under a lower-bound condition, namely there exists $z$ in the support of the unique invariant measure $\mu$ such that for any $\varepsilon>0$, $\inf_{x\in\mathcal X} \liminf_{t\to\infty} P_t(x,B(z,\varepsilon))>0$. The authors extend lower-bound criteria from $e$-processes to eventual continuity and provide a non-trivial randomized example that is asymptotically stable and eventually continuous but does not satisfy the $e$-property. An iterated function system–style construction with place-dependent randomness demonstrates the phenomenon, highlighting that eventual continuity can guarantee stability even when the classical $e$-property fails. These results broaden ergodicity tools for non-equicontinuous dynamics and offer constructive models illustrating the separation between eventual continuity and equicontinuity.
Abstract
We formulate a new criterion of the asymptotic stability for some non-equicontinuous Markov semigroups, the so-called eventually continuous semigroups. In particular, we provide a non-equicontinuous Markov semigroup example with essential randomness, which is asymptotically stable.