Approximation and Homogenization of a Transport Process

We consider a transport process such that the mean free path between two successive jumps is of order $\varepsilon $ and such that the structure of the medium is periodic with period $\delta = \varepsilon ^k ( k < 1 )$ in each direction. We show that this process converges weakly (when $\varepsilon \to 0$) to a diffusion which is identical with the one obtained when we let $\delta $ go to 0 (homogenization) after making the classical approximation of a transport process by a diffusion.