Geometry of subelliptic diffusions
by Anton Thalmaier
Abstract
The lectures focus on some probabilistic aspects related to sub-Riemannian geometry. The main intention is to give an introduction to hypoelliptic and subelliptic diffusions. The notes are written from a geometric point of view trying to minimize the weight of "probabilistic baggage" necessary to follow the arguments. We discuss in particular the following topics: stochastic flows to second order differential operators; smoothness of transition probabilities under Hörmander's brackets condition; control theory and Stroock-Varadhan's support theorems; Malliavin calculus; Hörmander's theorem. The notes start from well-known facts in Geometric Stochastic Analysis and guide to recent on-going
research topics, like hypoelliptic heat kernel estimates; gradient
estimates and Harnack type inequalities for subelliptic diffusion
semigroups; notions of curvature related to sub-Riemannian
diffusions.
These are notes to a series of lectures given at the CIRM Summer School on "Sub-Riemannian manifolds: from geodesics to hypoelliptic diffusions" which took place at the Centre International de Rencontres Mathématiques in Marseille, as part of the IHP Trimester program "Geometry, Analysis and Dynamics on Sub-Riemannian manifolds" (Institut Henri Poincaré in Paris from Sept. 1 to Dec. 14, 2014).
The paper is available here:
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