A stationary Fleming–Viot type Brownian particle system

Abstract

We consider a system ${\{X_1,\ldots,X_N\}}$ of N particles in a bounded d-dimensional domain D. During periods in which none of the particles ${X_1,\ldots,X_N}$ hit the boundary ${\partial D}$ , the system behaves like N independent d-dimensional Brownian motions. When one of the particles hits the boundary ${\partial D}$ , then it instantaneously jumps to the site of one of the remaining N − 1 particles with probability (N − 1)⁻¹. For the system ${\{X_1,\ldots,X_N\}}$ , the existence of an invariant measure ${\nu\mskip-12mu \nu}$ has been demonstrated in Burdzy et al. [Comm Math Phys 214(3):679–703, 2000]. We provide a structural formula for this invariant measure ${\nu\mskip-12mu \nu}$ in terms of the invariant measure m of the Markov chain ${\xi}$ which returns the sites the process ${X:=(X_1,\ldots,X_N)}$ jumps to after hitting the boundary ${\partial D^N}$ . In addition, we characterize the asymptotic behavior of the invariant measure m of ${\xi}$ when N → ∞. Using the methods of the paper, we provide a rigorous proof of the fact that the stationary empirical measure processes ${\frac1N\sum_{i=1}^N\delta_{X_i}}$ converge weakly as N → ∞ to a deterministic constant motion. This motion is concentrated on the probability measure whose density with respect to the Lebesgue measure is the first eigenfunction of the Dirichlet Laplacian on D. This result can be regarded as a complement to a previous one in Grigorescu and Kang [Stoch Process Appl 110(1):111–143, 2004].