The parametrix method approach to diffusions in a turbulent Gaussian environment

In this paper we deal with the solutions of Itô stochastic differential equationfor a small parameter [epsilon]. We prove that for 0[less-than-or-equals, slant][alpha]<1 and V a divergence-free, Gaussian random field, sufficiently strongly mixing in t variable the family of processes {X[epsilon](t)}t[greater-or-equal, slanted]0, [epsilon]>0 converges weakly to a Brownian motion. The entries of the covariance matrix of the limiting Brownian motion are given by ai,j=2[delta]i,j+[integral operator]+[infinity]-[infinity]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).