Genealogical Particle Analysis of Rare Events

In this paper an original interacting particle system approach is developed for studying Markov chains in rare event regimes. The proposed particle system is theoretically studied through a genealogical tree interpretation of Keynman-Kac path measures. The algorithmic implementation of the particle system is presented. An efficient estimator for the probability of ocurrence of a rare event is proposed and its variance is computed. Applications and numerical implementations are discussed. First, we apply the particle system technique to a toy model (a Gaussian random walk), which permits to illustrate the theoretical predictions. Second, we address a physically relevant problem consisting in the estimation of the outage probability due to polarization-mode dispersion in optical fibers.