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.
Categories: math.PR