  We analyze the low temperature asymptotics of the quasi-stationary
distribution associated with the overdamped Langevin dynamics (a.k.a. the
Einstein-Smoluchowski diffusion equation) in a bounded domain. This analysis is
useful to rigorously prove the consistency of an algorithm used in molecular
dynamics (the hyperdynamics), in the small temperature regime. More precisely,
we show that the algorithm is exact in terms of state-to-state dynamics up to
exponentially small factor in the limit of small temperature. The proof is
based on the asymptotic spectral analysis of associated Dirichlet and Neumann
realizations of Witten Laplacians. In order to cover a reasonably large range
of applications, the usual assumptions that the energy landscape is a Morse
function has been relaxed as much as possible.
