%0 Journal Article
%T Metastable states, quasi-stationary distributions and soft measures
%+ Dipartimento di Matematica
%+ Institut de Mathématiques de Marseille (I2M)
%A Bianchi, Alessandra
%A Gaudilliere, Alexandre
%Z Supported by GDRE 224 GREFI-MEFI and the European Research Council through the Advanced Grant PTRELSS 228032.
%Z 32 pages
%< avec comité de lecture
%Z Supported by GDRE 224 GREFI-MEFI and the European Research Council through the Advanced Grant PTRELSS 228032.
%@ 0304-4149
%J Stochastic Processes and their Applications
%I Elsevier
%V 126
%N 6
%P 1622--1680
%8 2016-06
%D 2016
%Z 1103.1143
%R 10.1016/j.spa.2015.11.015
%K potential theory
%K Metastability
%K restricted ensemble
%K quasi-stationary measure
%K soft measures
%K exponential law
%K spectral gap
%K mixing time
%K potential theory.
%Z 2010 Mathematics Subject Classification. 82C26, 60J27, 60J75, 60J45.
%Z Mathematics [math]/Probability [math.PR]
%Z Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech]Journal articles
%X 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 measure, 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 Poincaré 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.
%G English
%Z Supported by GDRE 224 GREFI-MEFI and the European Research Council through the Advanced Grant PTRELSS 228032.
%2 https://hal.science/hal-00573852v2/document
%2 https://hal.science/hal-00573852v2/file/SPA-soft_revised5.pdf
%L hal-00573852
%U https://hal.science/hal-00573852
%~ CNRS
%~ UNIV-AMU
%~ EC-MARSEILLE
%~ I2M
%~ I2M-2014-