Markov properties of the fluctuation limit of a particle system

Abstract

Let B k , k = 1, 2, …, be a sequence of independent Brownian particles in R d , whose initial point is distributed according to a probability measure μ on R d × R +. It is shown that the fluctuation process of the empirical distribution of the first n particles converges weakly in the Skorokhod spaces ( D[0, T 0), S ′( R d )) and ( D[ T 0, ∞), S ′( R d )) as n → ∞, where μ R d x[t,00))=0 , to continuous, centered Gaussian, Markov processes X and Y, respectively, and the corresponding Langevin equations are derived. A “strict” Markov property is defined, and it is shown that this property is satisfied by the process X in an interval [ a, b] ⊂ [0, T 0] if and only if μ R d x(a,b])=0 . These results extend an example discussed by K. Itô where T 0 = 0.