Volume 27, 2023
|Page(s)||867 - 912|
|Published online||31 October 2023|
Exponential quasi-ergodicity for processes with discontinuous trajectories
Université Paris-Saclay, INRAE, MaIAGE, Domaine de Vilvert, 78350 Jouy-en-Josas, France
* Corresponding author: email@example.com
Accepted: 11 September 2023
This paper tackles the issue of establishing an upper-bound on the asymptotic ratio of survival probabilities between two different initial conditions, asymptotically in time for a given Markov process with extinction. Such a comparison is a crucial step in recent techniques for proving exponential convergence to a quasi-stationary distribution. We introduce a weak form of the Harnack’s inequality as the essential ingredient for such a comparison. This property is actually a consequence of the convergence property that we intend to prove. Its complexity appears as the price to pay for the level of flexibility required by our applications, notably for processes with jumps on a multidimensional state-space. We show in our illustrations how simply and efficiently it can be used nonetheless. As illustrations, we consider two continuous-time processes on ℝd that do not satisfy the classical Harnack’s inequality, even in a local version. The first one is a piecewise deterministic process while the second is a pure jump process with restrictions on the directions of its jumps.
Mathematics Subject Classification: 37A30 / 28D10 / 60J25 / 92D15
Key words: Continuous-time and continuous-space Markov process / jumps / quasi-stationary distribution / survival capacity / Q-process / Harris recurrence
© The authors. Published by EDP Sciences, SMAI 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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