The Fleming-Viot process with McKean-Vlasov dynamics

Abstract

The Fleming-Viot particle system consists of N identical particles diffusing in an open domain $D\subset {\mathbb{R}}^d$. Whenever a particle hits the boundary $\partial D$, that particle jumps onto another particle in the interior. It is known that this system provides a particle representation for both the Quasi-Stationary Distribution (QSD) and the distribution conditioned on survival for a given diffusion killed at the boundary of its domain. We extend these results to the case of McKean-Vlasov dynamics. We prove that the law conditioned on survival of a given McKean-Vlasov process killed on the boundary of its domain may be obtained from the hydrodynamic limit of the corresponding Fleming-Viot particle system. We then show that if the target killed McKean-Vlasov process converges to a QSD as $t\to \mathrm{\infty }$, such a QSD may be obtained from the stationary distributions of the corresponding N-particle Fleming-Viot system as $N\to \mathrm{\infty }$.