Homogenization of an advection equation with locally stationary random coefficients

TL;DR

This paper proves that solutions to a rapidly oscillating advection equation with random coefficients converge to a diffusion process, establishing a homogenization result under specific statistical assumptions on the coefficients.

Contribution

It demonstrates the homogenization of an advection equation with locally stationary, Gaussian, and strongly mixing random coefficients, deriving the limiting diffusion process.

Findings

Solutions converge in law to a diffusion process as ε→0

Averaged solutions satisfy the Kolmogorov backward equation

Homogenization holds under Gaussian, locally stationary, and mixing conditions

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 $u_\eps(T,x)=u_0(x)$, $x\in\bbR^d$. Here $\eps>0$ is some small parameter and the drift term $\left(V(t,x)\right)_{(t,x)\in \bbR^{1+d}}$ 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.