Switching Diffusion Systems with Past-Dependent Switching and Countable State Space: Successful Couplings and Strong Ergodicity

TL;DR

This paper investigates switching diffusion systems with countable state spaces and history-dependent switching, establishing stability and ergodicity through successful coupling methods, and illustrating results with a mean-field model.

Contribution

It introduces a coupling approach for stability and ergodicity in past-dependent switching diffusions with countable states, extending existing theories.

Findings

Successful coupling establishes stability in total variation norm.

Proves strong ergodicity for the process.

Demonstrates results with a mean-field model.

Abstract

This work studies a class of switching diffusion systems where the switching component takes values in a countable state space and its transition rates depend on the history of the continuous component. Under suitable conditions, we construct a successful coupling that establishes stability of the underlying process in the total variation norm. The coupling approach also enables us to derive strong ergodicity for the underlying process. Finally, we illustrate the main results with an $N$-body mean-field model featuring past-dependent switching and a countable state space.