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Random walks and Laplacians in Riemannian and sub-Riemannian geometry

Course Type: 
PhD Course
Academic Year: 
Mar. - Apr.
20 h
In this lectures I will explain how to study an intrinsic diffusion process in a generalized Riemannian manifold.
1) Introduction. The problem of defining a macroscopic Laplacian starting from a notion of volume and a notion of gradient. The problem of defining a microscopic Laplacian as limit of geodesic random walks.
2) Generalities in sub-Riemannian geometry. Carnot-Caratheodoy distance. Rank-varying and non-equiregular structures. Basic examples.

3) Finiteness of the distance.

4) First order conditions. Normal and abnormal extremals.

5) Proof that normal extremals are always geodesics (i.e. such that small arcs are minimizers of the distance).

6) Crucial Examples: Heisenberg, Grushin, Martinet.

7) Basic properties of the macroscopic Laplacian. Self-adjointness, Stochastic completeness.

8) The microscopic Laplacian in the Riemannian context. The Ito and the Stratonovich random walks. Sampling the volume. 

9) The microscopic Laplacian in the sub-Riemannian context.  Surprising results in 5 dimensional contact manifolds.

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