The Annals of Probability

Stochastic analysis on sub-Riemannian manifolds with transverse symmetries

Fabrice Baudoin

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We prove a geometrically meaningful stochastic representation of the derivative of the heat semigroup on sub-Riemannian manifolds with tranverse symmetries. This representation is obtained from the study of Bochner–Weitzenböck type formulas for sub-Laplacians on 1-forms. As a consequence, we prove new hypoelliptic heat semigroup gradient bounds under natural global geometric conditions. The results are new even in the case of the Heisenberg group which is the simplest example of a sub-Riemannian manifold with transverse symmetries.

Article information

Ann. Probab., Volume 45, Number 1 (2017), 56-81.

Received: February 2014
Revised: June 2014
First available in Project Euclid: 26 January 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58J65: Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60] 53C17: Sub-Riemannian geometry
Secondary: 60J60: Diffusion processes [See also 58J65]

Sample Brownian motion sub-Riemannian manifold Bochner–Weitzenböck formula


Baudoin, Fabrice. Stochastic analysis on sub-Riemannian manifolds with transverse symmetries. Ann. Probab. 45 (2017), no. 1, 56--81. doi:10.1214/14-AOP964.

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