We study the asymptotic hitting time τ (n) τ(n) of a family of Markov processes X (n) X(n) to a target set G (n) 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) X(n) conditioned to stay within the trap, and from this we deduce the exponential distribution of τ (n) τ(n). Our approach is very broad—it does not require reversibility, the target G G does not need to be a rare event and the traps and the limit on n n can be of very general nature—and leads to explicit bounds on the deviations of τ (n) τ(n) from exponentially. We provide two nontrivial examples to which our techniques directly apply.
Fernández, R., Manzo, F., Nardi, F. R., Scoppola, E., & Sohier, J. (2016). Conditioned, quasi-stationary, restricted measures and escape from metastable states. The Annals of Applied Probability, 26(2), 760-793. https://doi.org/10.1214/15-AAP1102