Exact Results on the First Hitting via Conditional Strong Quasi-Stationary Times and Applications to Metastability

In the setting of non-reversible Markov chains on finite or countable state space, exact results on the distribution of the first hitting time to a given set G are obtained. A new notion of “conditional strong quasi stationary time” is introduced to describe the local relaxation time. This time is defined via a generalization of the strong stationary time. Rarity of the target set G is not required and the initial distribution can be completely general. The results clarify the the role played by the initial distribution on the exponential law; they are used to give a general notion of metastability and to discuss the relation between the exponential distribution of the first hitting time and metastability.