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

We generalize the result of T. Komorowski and G. Papanicolaou published in [7]. We consider the solution of stochastic differential equation dX(t) = V(t,X(t))dt+ 2κdB(t) where B(t) is a standard d-dimensional Brownian motion and V(t,x), (t,x) ∈ R × Rd is a d-dimensional, incompressible, stationary, random Gaussian field decorrelating in finite time. We prove that the weak limit as ε ↓ 0 of the family of rescaled processes Xε(t) = εX( t ε2) exists and may be identified as a certain Brownian motion.