On the length of one-dimensional reactive paths

Frédéric Cérou (1) , Arnaud Guyader (1, 2) , Tony Lelièvre (3, 4) , Florent Malrieu (1, 2)

1 ASPI - Applications of interacting particle systems to statistics
2 IRMAR - Institut de Recherche Mathématique de Rennes
3 MICMAC - Methods and engineering of multiscale computing from atom to continuum
4 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique

Motivated by some numerical observations on molecular dynamics simulations, we analyze metastable trajectories in a very simplecsetting, namely paths generated by a one-dimensional overdamped Langevin equation for a double well potential. More precisely, we are interested in so-called reactive paths, namely trajectories which leave definitely one well and reach the other one. The aim of this paper is to precisely analyze the distribution of the lengths of reactive paths in the limit of small temperature, and to compare the theoretical results to numerical results obtained by a Monte Carlo method, namely the multi-level splitting approach.

Mots clés: Reactive path; Gumbel distribution; h-transform

Domaines: Probabilités [math.PR]

Frédéric Cérou, Arnaud Guyader, Tony Lelièvre, Florent Malrieu. On the length of one-dimensional reactive paths. ALEA : Latin American Journal of Probability and Mathematical Statistics, 2013, 10 (1), pp.359-389. ⟨hal-00704704⟩