Recurrence and Ergodicity of Switching Diffusions with Past-Dependent Switching Having A Countable State Space

TL;DR

This paper investigates the recurrence and ergodic properties of switching diffusions with countably infinite states and past-dependent switching rates, providing conditions for recurrence and exploring PDE relationships.

Contribution

It introduces new sufficient conditions for recurrence and ergodicity in systems with past-dependent switching in countable state spaces.

Findings

Provided criteria for recurrence and ergodicity.

Established links between PDE systems and recurrence.

Analyzed systems with past-dependent switching rates.

Abstract

This work focuses on recurrence and ergodicity of switching diffusions consisting of continuous and discrete components, in which the discrete component takes values in a countably infinite set and the rates of switching at current time depend on the value of the continuous component over an interval including certain past history. Sufficient conditions for recurrence and ergodicity are given. Moreover, the relationship between systems of partial differential equations and recurrence when the switching is past-independent is established under suitable conditions.