Central limit theorem for an infinite lattice system of interacting diffusion processes (Q1107212)

The author considers the following stochastic differential equation: \[ dX\quad i_ t=b\quad i(X_ t)dt+dW\quad i_ t \] with \(0\leq t\leq 1\), \(i\in Z\) d, b i shiftinvariant, (W i) independent Wiener processes. Let F be a smooth functional on the path space. Then asymptotic normality of the standardized sum of shifts of F is shown provided the central limit theorem holds for the initial distribution. The main idea is to obtain an explicit expression for the integrand in the representation of F as a stochastic integral by generalizing the Haussmann formula to infinite dimensions. This makes it possible to check the condition of a standard central limit theorem for martingales. Moreover it is shown that an exponential decay of correlations for the initial distribution results in a similar decay for the distribution on the path space. Finally the stochastic differential equation of the fluctuation field is obtained.