## Journal of the Mathematical Society of Japan

### Hypoelliptic Laplacian and probability

Jean-Michel BISMUT

#### Abstract

The purpose of this paper is to describe the probabilistic aspects underlying the theory of the hypoelliptic Laplacian, as a deformation of the standard elliptic Laplacian. The corresponding diffusion on the total space of the tangent bundle of a Riemannian manifold is a geometric Langevin process, that interpolates between the geometric Brownian motion and the geodesic flow. Connections with the central limit theorem for the occupation measure by the geometric Brownian motion are emphasized. Spectral aspects of the hypoelliptic deformation are also provided on tori. The relevant hypoelliptic deformation of the Laplacian in the case of Riemann surfaces of constant negative curvature is briefly described, in connection with Selberg's trace formula.

#### Article information

Source
J. Math. Soc. Japan, Volume 67, Number 4 (2015), 1317-1357.

Dates
First available in Project Euclid: 27 October 2015

https://projecteuclid.org/euclid.jmsj/1445951153

Digital Object Identifier
doi:10.2969/jmsj/06741317

Mathematical Reviews number (MathSciNet)
MR3417500

Zentralblatt MATH identifier
1334.35019

#### Citation

BISMUT, Jean-Michel. Hypoelliptic Laplacian and probability. J. Math. Soc. Japan 67 (2015), no. 4, 1317--1357. doi:10.2969/jmsj/06741317. https://projecteuclid.org/euclid.jmsj/1445951153

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