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

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