On soft capacities, quasi-stationary distributions and the pathwise approach to metastability

TL;DR

This paper extends the concept of soft capacities and soft measures to overlapping sets in metastable stochastic models, providing new variational characterizations and clarifying their relation to the pathwise approach.

Contribution

It generalizes the notion of capacities and soft measures to overlapping sets, offering simplified properties and linking them to variational principles and the pathwise approach.

Findings

Extended capacities and soft measures to overlapping sets.

Derived variational formulas for metastable transition times.

Clarified connection between capacities and the pathwise approach.

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 ($\kappa$, $\lambda$)-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.