The Fleming-Viot Process with McKean-Vlasov Dynamics

TL;DR

This paper extends the Fleming-Viot particle system to McKean-Vlasov dynamics, establishing a connection between particle systems and conditioned distributions for these complex stochastic processes.

Contribution

It introduces a novel extension of the Fleming-Viot system to McKean-Vlasov processes and proves the convergence of conditioned laws and QSDs via hydrodynamic limits.

Findings

Law conditioned on survival obtained from hydrodynamic limit.

QSD for McKean-Vlasov process derived from stationary distributions.

Extension of particle representation to McKean-Vlasov dynamics.

Abstract

The Fleming-Viot particle system consists of $N$ identical particles diffusing in a domain $U \subset \mathbb{R}^d$. Whenever a particle hits the boundary $\partial U$, 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 $t \rightarrow \infty$, such a QSD may be obtained from the stationary distributions of the corresponding $N$-particle Fleming-Viot system as $N\rightarrow\infty$.