  Let $(X_t)_{t\ge 0}$ be the stochastic process solution to the overdamped
Langevin dynamics
  $$dX_t=-\nabla f(X_t) \, dt +\sqrt h \, dB_t$$ and let $\Omega \subset
\mathbb R^d $ be the basin of attraction of a local minimum of $f: \mathbb R^d
\to \mathbb R$. Up to a small perturbation of $\Omega$ to make it smooth, we
prove that the exit rates of $(X_t)_{t\ge 0}$ from $\Omega$ through each of the
saddle points of $f$ on $\partial \Omega$ can be parametrized by the celebrated
Eyring-Kramers laws, in the limit $h \to 0$. This result provides firm
mathematical grounds to jump Markov models which are used to model the
evolution of molecular systems, as well as to some numerical methods which use
these underlying jump Markov models to efficiently sample metastable
trajectories of the overdamped Langevin dynamics.
