Ergodicity and regularity properties of ODEs with semi-Markov switching
Tobias Hurth, Edouard Strickler
TL;DR
The paper extends ergodicity analysis for switched ODEs to semi-Markov switching with general dwell-time laws. By augmenting the state to $(X, au,I)$ and establishing Feller properties only under continuous survival functions $G^i$, it develops bracket-condition-based criteria that yield Doeblin-type minorization and exponential ergodicity, even when switching times lack a density. The work proves uniqueness and absolute continuity of invariant measures under regularity or analyticity assumptions, and provides a thorough accessibility framework linking deterministic control paths to stochastic dynamics. Applications to ecological models illustrate how dwell-time and bracket conditions guarantee robust long-term behavior in realistic switching environments.
Abstract
This paper is devoted to the study of a stochastic process obtained by random switching between a finite collection of vector fields. Such processes have recently been the focus of much attention in the case where the switching times are exponentially distributed, i.e., Markovian switching. In this contribution, we admit any distribution on $\mathbb{R}_+$ as a law for the switching times. We show that whenever this law is not singular with respect to the Lebesgue measure, the stochastic process obtained from the random switching is Feller. More importantly, we give conditions on the switching and on the vector fields ensuring that the Lie bracket condition considered in the Markovian case in Bakhtin and Hurth (2012) and Benaïm, Le Borgne, Malrieu and Zitt (2015) still imply ergodicity of the process.