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

24 pages, 1 figure.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 essentials 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 time, in terms of objects satisfying two-sided variational principles. We also make the connection with the classical "pathwise approach"