Interacting particle processes and approximation of Markov processes conditioned to not be killed

Denis Villemonais (1)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

We prove an approximation method for general strong Markov processes conditioned to not be killed. The method is based on a Fleming-Viot type interacting particle system, whose particles evolve as independent copies of the original strong Markov process and jump onto each others instead of being killed. We only assume that the number of jumps of the Fleming-Viot type system doesn't explode in finite time almost surely, and that the survival probability at fixed time of the original process is positive. We also give a speed of convergence for the approximation method. A criterion for the non-explosion of the number of jumps is then given for general systems of time and environment dependent diffusion particles, which includes the case of the Fleming-Viot type system of the approximation method. The proof of the criterion uses an original non-attainability of (0,0) result for a pair of non-negative semi-martingales with positive jumps.

Mots clés: diffusion process; interacting particle system; empirical process; quasi-stationary distribution; Yaglom limit

Domaines: Probabilités [math.PR]

Denis Villemonais. Interacting particle processes and approximation of Markov processes conditioned to not be killed. 2011. ⟨hal-00598085v1⟩