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

We study the asymptotic hitting time $\tau^{(n)}$ of a family of Markov processes $X^{(n)}$ 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 exponential distribution of $\tau^{(n)}$. Our approach is very broad ---it does not require reversibility, 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 $\tau^{(n)}$ from exponentially. We provide two non trivial examples to which our techniques directly apply.