Homogenization of transport equations: A simple PDE approach to the Kubo formula

We are interested in the behavior with respect to the small parameter ϵ>0 of solutions ρϵ of the conservative transport(-diffusion) equation ∂tρϵ+∇x(ρϵuϵ)=ηΔxρϵ, with η⩾0, driven by a large random velocity field: |uϵ|=O(1/ϵ). Assuming that the velocity does not have long-time memory we justify the convergence of the expectation Eρϵ to the solution of a diffusion equation. This question has been widely investigated; here we present a simple proof which only relies on PDE tools.