Homogenization of an advection equation with locally stationary random coefficients

Abstract

In the paper we consider the solution of an advection equation with rapidly changing coefficients $\partial_t u_\eps+(1/\eps)V(t\eps^{-2},x/{\eps})\cdot\nabla_x u_\eps=0$ for $t<T$ and $u_\eps(T,x)=u_0(x)$, $x\in {\mathbb{R}}^d$. Here $\eps>0$ is some small parameter and the drift term ${\left(V(t,x)\right)}_{(t,x)\in {\mathbb{R}}^{1+d}}$ is assumed to be a $d$-dimensional, vector valued random field with incompressible spatial realizations. We prove that when the field is Gaussian, locally stationary, quasi-periodic in the $x$ variable and strongly mixing in time the solutions $u_\eps(t,x)$ converge in law, as $\eps\to0$, to $u_0(x(T;t,x))$, where ${\left(x(s;t,x)\right)}_{s\ge t}$ is a diffusion satisfying $x(t;t,x)=x$. The averages of $u_\eps(T,x)$ converge then to the solution of the corresponding Kolmogorov backward equation.