Diffusion approximation for the convection-diffusion equation with random drift

We consider the asymptotic behavior of the solutions ofscaled convection-diffusion equations ∂tuɛ(t, x) = κΔx(t, x) + 1/ɛV(t/ɛ2,xɛ) ·∇xuɛ(t, x) with the initial condition uɛ(0,x) = u0(x) as the parameter ɛ↓ 0. Under the assumptions that κ > 0 and V(t, x), (t, x) ∈Rd is a d-dimensional,stationary, zero mean, incompressible, Gaussian random field, Markovian and mixing in t we show that the laws of uɛ(t,·), t≥ 0 in an appropriate functional space converge weakly, as ɛ↓ 0, to a δ-type measureconcentrated on a solution of a certain constant coefficient heat equation.