Markov properties of the fluctuation limit of a particle system with death

Abstract

Consider a sequence of independent Brownian motions in R d whose initial positions are distributed according to a probability measure μ on R d × R + and disappear after an exponentially distributed lifetime with parameter λ. It is shown that the fluctuation process of the empirical distribution of the first n motions converges weakly in the Skorokhod spaces D([0, T 0), L′( R d )) and D([ T 0, ∞), L′( R d )) as n → ∞, where T 0 = inf{ t: μ( R d × [ t, ∞)) = 0}, to a continuous, centered Gaussian process X. Conditions for X to be Markovian are determined in terms of μ. The cases λ > 0 and λ = 0 (i.e., the motions go on forever) exhibit different Markovian properties.