Top Global Course Special Lectures by Prof. Pierre Mathieu
Top Global Course Special Lectures by Prof. Pierre Mathieu (Kyoto University / Université d'AixMarseille) will take place as follows:
 Course Title
 Top Global Course Special Lectures 6
 Date & Time
 November 10 to December 1, 2017 (5 lectures)

 Friday, November 10, 13:0015:00
 Friday, November 17, 14:3016:30
 Friday, November 24, 14:3016:30
 Wednesday, November 29, 13:0015:00
 Friday, December 1, 13:0015:00
 Venue
 127 Conference room, Faculty of Science Bldg. #3, Kyoto University
 Title
 Fluctuationdissipation relations for reversible diffusions in a random environment
 Abstract
 Fluctuationdissipation relations (FDR) were introduced in statistical physics to describe offequilibrium dynamics; they express the linear response of a perturbed system as correlations for the unperturbed system.
 When applied to reversible diffusions in a random environment, they yield the socalled 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 unperturbed 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: Fluctuationdissipation 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.
 Lecture 1: Central limit theorems
 (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 OleinikJikovKozlov. We shall use the KipnisVaradhan 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 ZernerSznitman and Lian Shen at least to get a feeling of how regeneration times are constructed and why they indeed give a regeneration property.
 Language
 English
 Note
 This series of lectures will be videorecorded and made available online.
Please note that anyone in the front rows of the room can be captured by a video camera.