Ergodicity of inhomogeneous Markov processes under general criteria
Zhenxin Liu, Di Lu
TL;DR
The paper develops a general theory for ergodicity of inhomogeneous Markov processes by introducing invariant measure families $\{\mu_s\}_{s\in\mathbb{R}}$ that satisfy $P^{*}_{s,t}\mu_s=\mu_t$, and proving existence via a generalized Krylov–Bogolyubov approach or a Lyapunov-function criterion. It establishes uniqueness and exponential ergodicity under a contraction/Doeblin framework on level sets of a Lyapunov function, with separate discrete-time and continuous-time formulations. The authors apply the results to Markov chains on countable spaces, diffusions on $\mathbb{R}^n$, and Brownian storage models with almost periodic release rules, showing that under suitable drift and ellipticity conditions, there exists a unique invariant measure family and explicit exponential convergence rates, with additional almost periodicity properties in time. These findings provide practical criteria for long-time behavior and open avenues for extensions to LLN/CLT in nonautonomous settings.
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.