On Soft Capacities, Quasi-stationary Distributions and the Pathwise Approach to Metastability
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On Soft Capacities, Quasi-stationary Distributions and the Pathwise Approach to Metastability A. Bianchi1 · A. Gaudillière2
· P. Milanesi3
Received: 19 June 2019 / Accepted: 23 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Motivated by the study of the metastable stochastic Ising model at subcritical temperature and in the limit of a vanishing magnetic field, we extend the notion of (κ, λ)-capacities between sets, as well as the associated notion of soft-measures, to the case of overlapping sets. We recover their essential properties, sometimes in a stronger form or in a simpler way, relying on weaker hypotheses. These properties allow to write the main quantities associated with reversible metastable dynamics, e.g. asymptotic transition and relaxation times, in terms of objects that are associated with two-sided variational principles. We also clarify the connection with the classical “pathwise approach” by referring to temporal means on the appropriate time scale. Keywords Soft capacity · Soft measure · Quasi-stationary measure · Restricted ensemble · Metastability · Potential theory · Relaxation time Mathematics Subject Classification Primary 60J27 · 60J45 · 60J75; Secondary 82C20
Communicated by Yvan Velenik.
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A. Gaudillière [email protected] A. Bianchi [email protected] P. Milanesi [email protected]
1
Dipartimento di Matematica, Università degli Studi di Padova, Via Trieste, 63, 35121 Padua, Italy
2
Aix Marseille University, CNRS, Centrale Marseille, I2M, 39 rue Joliot-Curie, 13013 Marseille, France
3
Aix Marseille University, CNRS, UTLN, LIS, 52 Av. Escadrille Normandie Niemen, 13397 Marseille Cedex 20, France
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A. Bianchi et al.
1 Model and Main Results In the present paper we consider a general Markovian model for a dynamics exhibiting a metastable behavior, and we show how “soft measures and capacities” associated with a covering of the configuration space allow for a description of the metastable state. By these techniques, and under mild hypotheses, we provide sharp estimates of the relaxation time, fast relaxation to local equilibria, and we establish the asymptotic exponential law of the transition time to equilibrium. The ultimate motivation of this paper is to provide a solid theoretical background to analyze the metastable kinetic Ising model at any subcritical temperature and in the limit of a vanishing magnetic field. Indeed, by working out the model-dependent part of the proof, in the companion paper [18] we study the transition time to equilibrium for this model, and establish its convergence in law to an exponential distribution. In addition, we present an explicit connection with the classical “pathwise approach to metastability”, and we provide a comparison with the techniques appeared in the recent literature of abstract metastable dynamics. Before stating the main results, and to better illustrate the ideas leading to the present research, we start by discussing the qualitative behavior of a concrete
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