Metastable states, quasi-stationary and soft measures, mixing time asymptotics via variational principles

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 “(κ,λ)-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.