Metastable states, quasi-stationary distributions and soft measures

TL;DR

This paper investigates metastability in finite Markov chains, establishing conditions for quasi-stationary distributions, exponential exit laws, and introducing soft measures to analyze transition times with explicit bounds.

Contribution

It introduces soft measures for interpolating between ensemble and quasi-stationary measures, providing new bounds and inequalities for metastability analysis.

Findings

Proves asymptotic exponential exit law.

Establishes a new Poincaré inequality with sharp estimates.

Provides explicit bounds on relaxation and transition times.

Abstract

We establish metastability in the sense of Lebowitz and Penrose under practical and simple hypothesis for (families of) Markov chains on finite configuration space in some asymptotic regime, including the case of configuration space size going to infinity. By comparing restricted ensemble and quasi-stationary measures, we study point-wise convergence velocity of Yaglom limits and prove asymptotic exponential exit law. We introduce soft measures as interpolation between restricted ensemble and quasi-stationary measure to prove an asymptotic exponential transition law on a generally different time scale. By using potential theoretic tools, we prove a new general Poincar\'e inequality and give sharp estimates via two-sided variational principles on relaxation time as well as mean exit time and transition time. We also establish local thermalization on a shorter time scale and give mixing time asymptotics up to a constant factor through a two-sided variational principal. All our asymptotics are given with explicit quantitative bounds on the corrective terms.