First hitting and strong metastability times

TL;DR

This paper derives exact results for the distribution of first hitting times in non-reversible Markov chains, introducing a new concept of strong metastability time via a generalized strong stationary time, applicable to general initial distributions.

Contribution

It introduces the concept of strong metastability time through a generalized quasi-stationary time, providing a new framework for understanding metastability in non-reversible Markov chains.

Findings

Exact distribution results for first hitting times.

Introduction of the conditionally strong quasi-stationary time (CSQST).

Clarification of the role of initial distribution on exponential hitting time laws.

Abstract

In the setting of non-reversible Markov chains on finite or countable state space, exact results on the distribution of the first hitting time to a given set $G$ are obtained. A new notion of "strong metastability time" is introduced to describe the local relaxation time. This time is defined via a generalization of the strong stationary time to a "conditionally strong quasi-stationary time"(CSQST). Rarity of the target set $G$ is not required and the initial distribution can be completely general. The results clarify the the role played by the initial distribution on the exponential law; they are used to give a general notion of metastability and to discuss the relation between the exponential distribution of the first hitting time and metastability.