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

We study the asymptotic hitting time .(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 .(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 .(n) from exponentially. We provide two nontrivial examples to which our techniques directly apply.