Markov properties of the fluctuation limit of a particle system

AbstractLet Bk, k = 1, 2, …, be a sequence of independent Brownian particles in Rd, whose initial point is distributed according to a probability measure μ on Rd × 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, T0), S′(Rd)) and (D[T0, ∞), S′(Rd)) as n → ∞, where μRd 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, T0] if and only if μRd x(a,b])=0. These results extend an example discussed by K. Itô where T0 = 0