On metastability
type: Item
instance of: scholarly article
title: On metastability
zbMATH Open document ID: 1500.60046
author: Laurent Miclo
publication date: 2022-10-24T00:00:00Z
review text: The goal of this work is to introduce a quantity which measures the metastability of an absorbed Markov chain on a finite set, without referring to the asymptotic behaviour of some parameter such as a temperature. This quantity is presented as a spectral criterion, in the sense that it is defined from the spectrum of sub-Markovian chains which are obtained from the initial one by adding a regeneration mechanism according to the quasi-stationary state and then by removing one point in the boundary of the chain (i.e., the points at which the absorbing rate is positive). The metastability is quantified in terms of the difference between the law of the absorbing time and an exponential law. The first main result is that the distance between the repartition functions of these two laws is bounded uniformly in time and in the initial condition by the spectral quantity times the absorbing rate. A similar control is proven for the joint law of the exit time and and the exit position (where metastability means that they are approximately independent, with laws independent from the initial condition). As a consequence, a metastable behaviour is observed as soon as the spectral quantity is small with respect to the inverse of the absorbing rate. The results are tested for toy models (two or three states) in a low temperature regime.
reviewed by: Pierre Monmarché
Mathematics Subject Classification ID: 60J27
Mathematics Subject Classification ID: 60E15
Mathematics Subject Classification ID: 15A42
Mathematics Subject Classification ID: 60J35
Mathematics Subject Classification ID: 60J45
zbMATH DE Number: 7606004
zbMATH Keywords: metastability
zbMATH Keywords: finite absorbing Markov process
zbMATH Keywords: exit time
zbMATH Keywords: exit position
zbMATH Keywords: spectral quantitative criterion
zbMATH Keywords: first Dirichlet eigenvalue
zbMATH Keywords: quasi-stationary distribution
zbMATH Keywords: Poisson equation
zbMATH Keywords: eigentime identity
zbMATH Keywords: small temperature
MaRDI profile type: Publication
full work available at URL: s00440-022-01147-8
OpenAlex ID: W4281627056
cites work: Inequalities for rare events in time-reversible Markov chains. I.
cites work: A martingale approach to metastability
cites work: Metastable states, quasi-stationary distributions and soft measures
cites work: Metastability
cites work: An exercise(?) in Fourier analysis on the Heisenberg group
cites work: Quasi-Stationary Distributions
cites work: Sharp asymptotics of the first exit point density
cites work: The cutoff phenomenon in finite Markov chains.
cites work: Strong stationary times via a new form of duality
cites work: On times to quasi-stationarity for birth and death processes
cites work: The passage time distribution for a birth-and-death chain: Strong stationary duality gives a first stochastic proof
cites work: Q3672830
cites work: Simulated annealing via Sobolev inequalities
cites work: Asymptotics of exit times for Markov jump processes. II: Applications to Jackson networks
cites work: Perturbation theory for linear operators.
cites work: Dirichlet's and Thomson's principles for non-selfadjoint elliptic operators with application to non-reversible metastable diffusion processes
cites work: Exact results on the first hitting via conditional strong quasi-stationary times and applications to metastability
cites work: Q4028964
cites work: On discrete inhomogeneous exit problems
cites work: On absorption times and Dirichlet eigenvalues
cites work: Q3452082
cites work: On the Markovian Similarity
DOI: 10.1007/S00440-022-01147-8
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 7606004