Selection principle for the Fleming–Viot particle system on the positive half-line with constant negative drift

Abstract

We consider the Fleming–Viot particle system consisting of $N$ identical particles evolving in ${\mathbb{R}}_{>0}$ as Brownian motions with constant negative drift. Whenever a particle hits $0$, it jumps onto another particle in the interior. We prove the following microscopic selection principle: the empirical measure of the $N$-particle stationary distribution converges, as $N\to \infty$, to the minimal quasi-stationary distribution of Brownian motion with the specified constant negative drift, killed instantaneously at $0$. Such a selection principle has long been known in bounded domains dating back to Burdzy, Hołyst and March’s original paper in 2000, and in unbounded domains with a sufficiently strong drift towards $0$ using Lyapunov function techniques. The simple case of constant negative drift had been a longstanding open problem, however, with previously developed techniques not applying. For instance, Lyapunov function techniques imply a property which does not hold here. We instead introduce a completely different strategy using a variety of new techniques to prove the selection principle. As we shall explain, this strategy and these techniques are applicable to other particle motions, opening the door to proving the selection principle in generality. Moreover, the selection principle for this particular Fleming–Viot particle system is closely connected to the longstanding selection problem in the travelling wave setting. Since the publication of the original arXiv version of this paper, J. Berestycki and the present author have established such a selection principle for the $N$-branching Brownian motion with selection. That proof uses the same overall proof structure, combining techniques and ideas developed in this paper with key additional ideas.