The parametrix method approach to diffusions in a turbulent Gaussian environment

In this paper we deal with the solutions of Ito stochastic differential equation dX(epsilon)(t) = 1/epsilon V (t/epsilon(2), X-epsilon(t)/epsilon(alpha)) dt + root 2 dB(t), for a small parameter epsilon. We prove that for 0 less than or equal to 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 than or equal to 0), epsilon > 0 converges weakly to a Brownian motion. The entries of the covariance matrix of the limiting Brownian motion are given by a(i,j) = 2 delta(i,j) + integral(-infinity)(+infinity) R-i,R-j(t, 0) dt, i, i = 1,..., d, where [R-i,R-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). (C) 1998 Elsevier Science B.V. All rights reserved.