Existence condition of strong stationary times for continuous time Markov chains on discrete graphs

TL;DR

This paper investigates the conditions under which strong stationary times exist for continuous-time Markov chains on discrete graphs, introducing a dual process and providing criteria for existence.

Contribution

It constructs a strong stationary dual process for Markov chains on complex graphs and derives equivalent conditions for the existence of strong stationary times.

Findings

Established the generator of the strong stationary dual process.

Derived an equivalent condition for the existence of strong stationary times.

Applied the theory to the case of the integer lattice Z.

Abstract

We consider a random walk on a discrete connected graph having some infinite branches plus finitely many vertices with finite degrees. We find the generator of a strong stationary dual in the sense of Fill, and use it to find some equivalent condition to the existence of a strong stationary time. This strong stationary dual process lies in the set of connected compact sets of the compactification of the graph. When this graph is Z, this is simply the set of (possibly infinite) segments of Z.