Metastable states, quasi-stationary distributions and soft measures

We establish metastability in the sense of Lebowitz and Penrose under practical and simple hypotheses for Markov chains on a finite configuration space in some asymptotic regime. By comparing restricted ensembles and quasi-stationary measures, and introducing soft measures as an interpolation between the two, we prove an asymptotic exponential exit law and, on a generally different time scale, an asymptotic exponential transition law. By using potential-theoretic tools, and introducing "(k,lambda)-capacities", we give sharp estimates on relaxation time, as well as mean exit time and transition time. We also establish local thermalization on shorter time scales. (C) 2015 Elsevier B.V. All rights reserved.