The parametrix method approach to diffusions in a turbulent Gaussian environment

In this paper we deal with the solutions of Itô stochastic differential equationdXε(t)=1εVtε2,Xε(t)εαdt+2 dB(t),for a small parameter ε. We prove that for 0⩽α<1 and V a divergence-free, Gaussian random field, sufficiently strongly mixing in t variable the family of processes {Xε(t)}t⩾0, ε>0 converges weakly to a Brownian motion. The entries of the covariance matrix of the limiting Brownian motion are given by ai,j=2δi,j+∫+∞−∞Ri,j(t,0) dt, i,j=1,…,d, where [Ri,j(t,x)] is the covariance matrix of the field V. To obtain this result we apply a version of the parametrix method for a linear parabolic PDE (see Friedman, 1963).