Central limit theorem for a Gaussian incompressible flow with additional Brownian noise

Abstract

We generalize the result of T. Komorowski and G. Papanicolaou. We consider the solution of stochastic differential equation $dX(t)=V(t,X(t))dt+\sqrt{2\kappa}dB(t)$ where $B(t)$ is a standard $d$-dimensional Brownian motion and $V(t,x)$, $(t,x)\in R \times R^{d}$ is a $d$-dimensional, incompressible, stationary, random Gaussian field decorrelating in finite time. We prove that the weak limit as \ep\downarrow 0 of the family of rescaled processes $X_{\epsilon}(t)=\epsilon X(\frac{t}{\epsilon^{2}})$ exists and may be identified as a certain Brownian motion