Diiusion-approximation for the Advection-diiusion of a Passive Scalar by a Space-time Gaussian Velocity Field 1

We study the asymptotic behavior, as goes to 0, of a passive scalar T (x;t) solution of the following advection-diiusion equation: @T @t = 2 4T + 1 V (x; t 2) rT ; t > 0 T (x;0) = T0(x); x 2 IR d where is a strictly positive diiusion constant and fV (x; t); x 2 IR d ; t 0g is a mean zero homogeneous Gaussian eld. We assume that the covariance is of the form: IEfV (x; t)V (y; s)g = ?(x ? y) exp(?ajt ? sj) and under some regularity assumption on ?, we prove that T (x; t) converges in distribution to the solution of a stochastic partial diierential equation. We derive the eeective diiusion coeecient from this result. This work is a generalization of previous works by Bouc-Pardoux 3] and Kushner-Huang 8] where the velocity eld is of the form 1 V (x; Z t== 2) for some nite dimensional ergodic noise process Z. Our situation is an example of innnite dimensional noise.