Non-extinction of a Fleming-Viot particle model

Abstract

We consider a branching particle model in which particles move inside a Euclidean domain according to the following rules. The particles move as independent Brownian motions until one of them hits the boundary. This particle is killed but another randomly chosen particle branches into two particles, to keep the population size constant. We prove that the particle population does not approach the boundary simultaneously in a finite time in some Lipschitz domains. This is used to prove a limit theorem for the empirical distribution of the particle family.

Quasistationary Distributions and Fleming-Viot Processes in Finite Spaces

“…These two questions have been answered affirmatively for Brownian motion (the empirical profile of the invariant measure converges in this case to the first eigenfunction of the Laplacian with homogeneous Dirichlet boundary conditions) and more general diffusions (see [1], [2], and [5]). For countable spaces, under the condition that …”

“…This model and generalizations of it were studied in several papers; see, e.g. [1], [2], [5], [6], and [7], which dealt with diffusions in bounded or unbounded domains. These works had to address the serious problem of nonexplosion of the number of hits of the boundary, and this required sophisticated analysis.…”

The Parallel Replica Method for Simulating Long Trajectories of Markov Chains

“…In this step, the reference trajectory is allowed to evolve until it spends a sufficiently long time in a single state. At the termination of the decorrelation step, the distribution of the reference trajectory should be, according to (2), close to that of the QSD (see Theorem 4.2 in Section 4.1).…”

“…While we have described a simple rejection sampling algorithm, there is another technique [3] based on a branching and interacting particle process sometimes called the Fleming-Viot particle process [11]. See [2,9,12,19,21] for studies of this process, and [3] for a discussion of how the Fleming-Viot particle process may be used in ParRep.…”

Convergence of the Fleming-Viot process toward the minimal quasi-stationary distribution

“…Hence, contrarily to the hard killing case [4], there is no risk for the particle system (X 1 t , X 2 t , . .…”