from geodesics to hypoelliptic diffusion

Four courses of 6 hours each will be organized.

Download here a**preliminary program** of the CIRM School.

## Courses

Download here a

- Geometric control and sub-Riemannian geodesics (L. Rifford, University of Nice).

This will be an introduction to sub-Riemannian geometry from the point of view of control theory. We will define sub-Riemannian structures and prove the Chow Theorem. We will describe normal and abnormal geodesics and discuss the completeness of the Carnot-Carathéodory distance (Hopf-Rinow Theorem). Several examples will be given (Heisenberg group, Martinet distribution, Grusin plane).

Filmed lectures - The geometry of subelliptic diffusions (A. Thalmaier, University of Luxembourg).

We discuss hypoelliptic and subelliptic diffusions; the lectures include the following topics: Malliavin calculus; Hormander's theorem; smoothness of transition probabilities under Hormander's brackets condition; control theory and Stroock-Varadhan's support theorems; hypoelliptic heat kernel estimates; gradient estimates and Harnack type inequalities for subelliptic diffusion semi-groups; notions of curvature related to sub-Riemannian diffusions.

Filmed lectures - Differential forms and the Hölder equivalence problem (P. Pansu, University of Paris Sud).

In 1993, Gromov asked for which exponents $\alpha$ there exist $C^\alpha$-Hölder continuous (local) homeomorphisms of Euclidean spaces to sub-Riemannian Carnot groups. I will explain some of Gromov's partial results on this question. They rely on Rumin's theory of differential forms adapted to sub-Riemannian spaces. If time permits, other approaches to the Hölder equivalence problem will be discussed. Recommended reading : Gromov's book "Carnot-Caratheodory spaces seen from within".

Filmed lectures - Hypoelliptic operators and analysis on Carnot-Carathéodory spaces (N. Garofalo, University of Padova).

In this course, we will define the sub-Laplacian associated with a sub-Riemannian structure, and we will describe its hypoellipticity under the Hormander condition. We will introduce the main tools for the study of sub-elliptic PDEs.

Filmed lectures

A seminar session will be organized on Friday afternoon, with the following talks:

- Hausdorff volumes in sub-Riemannian geometry (Frédéric JEAN, ENSTA Paris Tech).

- Hypoelliptic random walks (Laurent MICHEL, LJAD - Nice).

- Wasserstein space of a negatively curved space (Jérôme BERTRAND, Institut de Mathématiques de Toulouse)