Conditioned, quasi-stationary, restricted measures and escape from metastable states

TL;DR

This paper analyzes the asymptotic behavior of hitting times for Markov processes in complex traps, providing explicit descriptions and bounds without requiring reversibility or rare events, applicable to broad scenarios.

Contribution

It introduces a general framework for conditioned, quasi-stationary, restricted measures in Markov processes, enabling explicit analysis of escape times from traps without reversibility or rarity assumptions.

Findings

Hitting times are asymptotically exponential.

Explicit bounds on deviations from exponential distribution.

Applicable to broad classes of Markov processes.

Abstract

We study the asymptotic hitting time $\tau^{(n)}$ of a family of Markov processes $X^{(n)}$ to a target set $G^{(n)}$ when the process starts from a trap defined by very general properties. We give an explicit description of the law of $X^{(n)}$ conditioned to stay within the trap, and from this we deduce the exponential distribution of $\tau^{(n)}$. Our approach is very broad ---it does not require reversibility, the target $G$ does not need to be a rare event, and the traps and the limit on $n$ can be of very general nature--- and leads to explicit bounds on the deviations of $\tau^{(n)}$ from exponentially. We provide two non trivial examples to which our techniques directly apply.