Pierre Mathieu: Scaling limits of diffusions in a random environment

The scaling limit of a reversible diffusion in a random environment is a Brownian motion. That is a classical homogenization result. In the talk, I’ll discuss scaling limits of reversible diffusions in a random environment with an extra local drift in some fixed direction. When the local drift is small enough, the scaling limit is still diffusive (Lebowitz-Rost ’94). In joint work with A. Piatnitski and N. Gantert (CPAM 2012), we investigated the ballistic regime and completed the picture if the environment has finite range of correlations. Eventually one obtains a linear relation between the effective diffusivity and the mobility called the ‘Einstein relation’.