Vol. 8, No. 3, 2015

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Low temperature asymptotics for quasistationary distributions in a bounded domain

Tony Lelièvre and Francis Nier

Vol. 8 (2015), No. 3, 561–628

We analyze the low temperature asymptotics of the quasistationary distribution associated with the overdamped Langevin dynamics (also known as 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 factors 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 widen the range of applicability, the usual assumption that the energy landscape is a Morse function has been relaxed as much as possible.

quasistationary distributions, Witten Laplacian, low temperature asymptotics and semiclassical asymptotics
Mathematical Subject Classification 2010
Primary: 58J10, 58J32, 60J65, 60J70, 81Q20
Received: 18 September 2013
Revised: 22 December 2014
Accepted: 18 February 2015
Published: 3 June 2015
Tony Lelièvre
Université Paris-Est
Joint Project-team INRIA Matherials
6 & 8 Avenue Blaise Pascal
F-77455 Marne-la-Vallée
Francis Nier
Université de Paris 13
Avenue Jean-Baptiste Clément
F-93430 Villetaneuse