Asymptotic Behaviour of the Fleming-Viot Process

Abstract

This thesis examines the Fleming-Viot process, a particle system which provides an approximation method for killed Markov processes conditioned on survival and their quasi-stationary distributions (QSDs). In the first part, we establish that the Fleming-Viot process also provides for an approximation method for killed McKean-Vlasov processes conditioned on survival and their QSDs. 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→ ∞, such a QSD may be obtained from the stationary distributions of the corresponding N-particle Fleming-Viot system as N→∞. The main techniques we employ are two coupling constructions, martingale methods, and an analysis of the dynamical historical processes, which together enable a compactness-uniqueness argument. In the second part, we fix our given killed Markov process as a normally reflected diffusion in a compact domain, killed according to a position-dependent Poisson clock. We obtain the Fleming-Viot super process as a scaling limit of the Fleming-Viot multi-colour process. This has three applications: it enables us to prove a conjecture due to Bieniek and Burdzy on the asymptotic distribution of the spine of the Fleming-Viot process, it provides for a particle representation for the principal right eigenfunction of the infinitesimal generator, and it provides an approximation method for the Q-process - the killed Markov process conditioned never to be killed. The main technique employed is to observe that tilting the empirical measure by the principal right eigenfunction of the infinitesimal generator allows for fast-variable elimination.