Open Access
October, 2015 Hypoelliptic Laplacian and probability
Jean-Michel BISMUT
J. Math. Soc. Japan 67(4): 1317-1357 (October, 2015). DOI: 10.2969/jmsj/06741317


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.


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Jean-Michel BISMUT. "Hypoelliptic Laplacian and probability." J. Math. Soc. Japan 67 (4) 1317 - 1357, October, 2015.


Published: October, 2015
First available in Project Euclid: 27 October 2015

zbMATH: 1334.35019
MathSciNet: MR3417500
Digital Object Identifier: 10.2969/jmsj/06741317

Primary: 35H10 , 58J65

Keywords: diffusion processes and stochastic analysis on manifolds , hypoelliptic equations

Rights: Copyright © 2015 Mathematical Society of Japan

Vol.67 • No. 4 • October, 2015
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