The Hypoelliptic Laplacian

If X is a Riemannian manifold, the Laplacian is a second order elliptic operator on X. The hypoelliptic Laplacian bL is an operator acting on the total space  of the tangent bundle of X, that is supposed to interpolate between the elliptic Laplacian (when 0b→) and the geodesic flow (when b→+∞). Up to lower order terms, bL is a weighted sum of the harmonic oscillator along the fibre TX and of the generator of the geodesic flow. One expects that, in this deformation, there are conserved quantities.
In the talk, I will explain the underlying algebraic, I will explain the underlying algebraic, analytic and probabilistic aspects of its construction, and outline some of the applications obtained so far.