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.
Manzo, F., Scoppola, E. (2019). Exact Results on the First Hitting via Conditional Strong Quasi-Stationary Times and Applications to Metastability. JOURNAL OF STATISTICAL PHYSICS, 174(6), 1239-1262 [10.1007/s10955-019-02233-3].