Imprecise Markov semigroups and their ergodicity

¹ Department of Computer Science, The University of Manchester, Manchester M13 9PL, UK
² Department of Statistics, University of Warwick, Coventry CV4 7AL, UK

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 7 March 2026
Accepted: 19 May 2026

Abstract

We introduce the concept of an imprecise Markov semigroup Q, a tool that represents ambiguity around both the transition probabilities and the invariant measure of a continuous-time Markov process via a collection of Markov semigroups, each associated with a (possibly different) Markov process. We use techniques from topology, geometry, and probability to analyze ergodic limits under model uncertainty encoded by Q. We establish long-term bounds that are uniform in the initial state and identify regimes in which the imprecision in these bounds collapses asymptotically. Our results are proved in progressively more general settings. We first assume that Q is compact and that the state space is Euclidean or a Riemannian manifold, working with a fixed bounded observable. We then allow the state space to be standard Borel, while keeping Q compact and the observable fixed. Finally, we drop compactness and work on Polish metric spaces of finite diameter, where we treat arbitrary bounded Lipschitz observables. The importance of our findings for the fields of artificial intelligence and computer vision is also discussed at a high level; In particular, in the study of how the probability of an output evolves over time as we perturb the input of a convolutional autoencoder.