  We consider the exit event from a metastable state for the overdamped
Langevin dynamics $dX_t = -\nabla f(X_t) dt + \sqrt{h} dB_t$. Using tools from
semiclassical analysis, we prove that, starting from the quasi stationary
distribution within the state, the exit event can be modeled using a jump
Markov process parametrized with the Eyring-Kramers formula, in the small
temperature regime $h \to 0$. We provide in particular sharp asymptotic
estimates on the exit distribution which demonstrate the importance of the
prefactors in the Eyring-Kramers formula. Numerical experiments indicate that
the geometric assumptions we need to perform our analysis are likely to be
necessary. These results also hold starting from deterministic initial
conditions within the well which are sufficiently low in energy. From a
modelling viewpoint, this gives a rigorous justification of the transition
state theory and the Eyring-Kramers formula, which are used to relate the
overdamped Langevin dynamics (a continuous state space Markov dynamics) to
kinetic Monte Carlo or Markov state models (discrete state space Markov
dynamics). From a theoretical viewpoint, our analysis paves a new route to
study the exit event from a metastable state for a stochastic process.
