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Metastable states, quasi-stationary distributions and soft measures

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 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.
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Contributor : Alexandre Gaudillière Connect in order to contact the contributor
Submitted on : Friday, January 29, 2016 - 6:18:29 PM
Last modification on : Tuesday, October 19, 2021 - 10:49:28 PM
Long-term archiving on: : Friday, November 11, 2016 - 9:21:06 PM


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Alessandra Bianchi, Alexandre Gaudilliere. Metastable states, quasi-stationary distributions and soft measures. Stochastic Processes and their Applications, Elsevier, 2016, 126 (6), pp.1622--1680. ⟨10.1016/⟩. ⟨hal-00573852v2⟩



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