Hypoelliptic Laplacian and Probability

Hypoelliptic Laplacian and Probability

-
Jean-Michel Bismut , Orsay
Fine Hall 214

The hypoelliptic Laplacian is a family of operators, indexed by b ∈ R_+, acting on the total space of the tangent bundle of a Riemannian manifold, that interpolates between the ordinary Laplacian as b → 0 and the generator of the geodesic flow as b → +∞. The probabilistic counterpart to the hypoelliptic Laplacian is a 1-parameter family of differential equations, known as geometric Langevin equations, that interpolates between Brownian motion and the geodesic flow. I will present some of the probabilistic ideas that explain some of its remarkable and often hidden properties. This will include the Ito formula for the corresponding hypoelliptic diffusion. I will also explain some of the applications of the hypoelliptic Laplacian that have been obtained so far.