2023/06/06 by Tough, Oliver
#35C07 #35K57 #35Q92 #60J80 #60J85 #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2306.03585
We consider the Fleming-Viot particle system consisting of N identical particles evolving in ℝ>0 as Brownian motions with constant drift -1. Whenever a particle hits 0, it jumps onto another particle in the interior. It is known that this particle system has a hydrodynamic limit as N→∞ given by Brownian motion with drift -1 conditioned not to hit 0. This killed Brownian motion has an infinite family of quasi-stationary distributions (QSDs), with a Yaglom limit given by the unique QSD minimising the survival probability. On the other hand, for fixed N<∞, this particle system converges to a unique stationary distribution as time t→∞. We prove the following selection principle: the empirical measure of the N-particle stationary distribution converges to the aforedescribed Yaglom limit as N→∞. The selection problem for this particular Fleming-Viot process is closely connected to the microscopic selection problem in front propagation, in particular for the N-branching Brownian motion. The proof requires neither fine estimates on the particle system nor the use of Lyapunov functions.