Conditioned, quasi-stationary, restricted measures and escape from metastable states

We study the asymptotic hitting time τ(n) of a family of Markov pro- cesses X(n) to a target set G(n) when the process starts from a “trap” defined by very general properties. We give an explicit description of the law of X(n) conditioned to stay within the trap, and from this we deduce the exponen- tial distribution of τ(n). Our approach is very broad—it does not require re- versibility, the target G does not need to be a rare event and the traps and the limit on n can be of very general nature—and leads to explicit bounds on the deviations of τ (n) from exponentially. We provide two nontrivial examples to which our techniques directly apply.