  We are interested in the connection between a metastable continuous state
space Markov process (satisfying e.g. the Langevin or overdamped Langevin
equation) and a jump Markov process in a discrete state space. More precisely,
we use the notion of quasi-stationary distribution within a metastable state
for the continuous state space Markov process to parametrize the exit event
from the state. This approach is useful to analyze and justify methods which
use the jump Markov process underlying a metastable dynamics as a support to
efficiently sample the state-to-state dynamics (accelerated dynamics
techniques). Moreover, it is possible by this approach to quantify the error on
the exit event when the parametrization of the jump Markov model is based on
the Eyring-Kramers formula. This therefore provides a mathematical framework to
justify the use of transition state theory and the Eyring-Kramers formula to
build kinetic Monte Carlo or Markov state models.
