Irreducibility and uniqueness of stationary distribution

TL;DR

This paper demonstrates that irreducibility in the fine topology guarantees the uniqueness of stationary distributions and shows it is a weaker condition than the combined strong Feller property and original topology irreducibility.

Contribution

It establishes a new connection between fine topology irreducibility and uniqueness of invariant measures, expanding the understanding of conditions for stationary distribution uniqueness.

Findings

Fine topology irreducibility implies unique invariant measures.

This irreducibility is weaker than the strong Feller plus original topology irreducibility.

The paper clarifies the hierarchy of irreducibility conditions in stochastic processes.

Abstract

In this paper, we shall prove that the irreducibility in the sense of fine topology implies the uniqueness of invariant probability measures. It is also proven that this irreducibility is strictly weaker than the strong Feller property plus irreducibility in the sense of original topology, which is the usual uniqueness condition.