Fluctuation-dissipation relations (FDR) were introduced in statistical physics to describe off-equilibrium dynamics; they express the linear response of a perturbed system as correlations for the un-perturbed system.

When applied to reversible diffusions in a random environment, they yield the so-called Einstein relation: the derivative of the effective drift of a diffusion in a random environment subject to a small external force equals the effective variance of the un-perturbed dynamics in the direction of the perturbation.

The aim of the course will be to explain the proof of FDR for reversible diffusions in a random environment with finite range of correlation. The proof also provides a full description of all the scaling limits of such processes.

Lectures 1 and 2 are introductory. Lectures 3 to 5 will concern the proof of FDR for diffusions.

**Lecture 1:**Central limit theorems

We shall first survey the martingale approach to establish the convergence towards Brownian motion of a reversible diffusion in a random environment.**Lecture 2:**Fluctuation-dissipation relations

The lecture will be devoted to a soft introduction to FDR for additive functionals of Markov processes.**Lecture 3:**A priori estimates on diffusions

We gather some PDE estimates for diffusions with a local drift.**Lecture 4:**Regeneration times and steady states

We construct a steady state for perturbed diffusions in a random environment with finite range of correlation and study its continuity.**Lecture 5:**FDR and scaling limits

End of the proof of FDR and the Einstein relation.

(References for the mini course)

I will discuss diffusions in a random environment, based on my papers with Piatnitski and Gantert.

Students will need some basic knowledge about diffusions i.e. existence of solutions of very nice sde’s, pde estimates like Aronson’s for operators in divergence form. Nothing advanced in that respect. I will treat in details the tricky pde estimates needed and assume only elementary results.

We’ll need some homogenization theory. I plan to give a sketch of the proof of the invariance principle for reversible diffusions. Basic results in homogenization can be found in the first chapters of the book of Oleinik-Jikov-Kozlov. We shall use the Kipnis-Varadhan approach and also the corrector approach.

The study of ballistic diffusions relies on regeneration techniques. Our construction of regeneration times is inspired by works of Sznitman and Zerner in the discrete case and Lian Shen for diffusions. I’ll recall the construction. (It differs a bit from Shen’s.) However one may want to have a look at the papers of Zerner-Sznitman and Lian Shen at least to get a feeling of how regeneration times are constructed and why they indeed give a regeneration property.

**[NOTE]** This series of lectures will be video-recorded and made available online. Please note that anyone in the front rows of the room can be captured by a video camera.