Ergodicity of inhomogeneous Markov processes under general criteria

TL;DR

This paper investigates the ergodic behavior of inhomogeneous Markov processes, establishing conditions for invariant measure families, their uniqueness, and exponential ergodicity, supported by practical examples.

Contribution

It introduces new criteria for invariant measures of inhomogeneous Markov processes, extending classical methods and demonstrating exponential ergodicity under contraction conditions.

Findings

Invariant measure families exist under generalized conditions.

Uniqueness and exponential ergodicity are proven with contraction assumptions.

Practical examples include Markov chains, diffusion, and storage processes.

Abstract

This paper is concerned with ergodic properties of inhomogeneous Markov processes. Since the transition probabilities depend on initial times, the existing methods to obtain invariant measures for homogeneous Markov processes are not applicable straightforwardly. We impose some appropriate conditions under which invariant measure families for inhomogeneous Markov processes can be studied. Specifically, the existence of invariant measure families is established by either a generalization of the classical Krylov-Bogolyubov method or a Lyapunov criterion. Moreover, the uniqueness and exponential ergodicity are demonstrated under a contraction assumption of the transition probabilities on a large set. Finally, three examples, including Markov chains, diffusion processes and storage processes, are analyzed to illustrate the practicality of our method.