Tightness problem and stochastic evolution equation arising from fluctuation phenomena for interacting diffusions (Q1083123)

Let \(\xi\) and B(t) be a real valued random variable and an independent Brownian motion respectively, and \((\xi_ i,B_ i(t))\), \(i=1,...,n\) be independent copies of (\(\xi\),B(t)). Consider the n-particle diffusion process described by the stochastic equation \[ X^ n_ i(t)=\xi_ i+\int^{t}_{0}a[X^ n_ i(s),U^ n(s)]dB_ i(s)+\int^{t}_{0}b[X^ n_ i(s),U^ n(s)]ds,\quad i=1,...,n \] with \(U^ n(t)=1/n\sum^{n}_{i=1}\delta_{X^ n_ i(t)}\) converging in probability to a probability distribution \(u(t)=u(t,dx)\). The main result of the paper states that \[ S_ n(t)=\sqrt{n}(U^ n(t)- u(t)) \] converges weakly to a Gaussian process S(t) with values in the dual space \(\Phi\) ' of some nuclear space \(\Phi\) of infinitely differentiable functions. In addition a stochastic differential equation for the limit process S(t) is obtained.