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A convergence to Brownian motion on sub-Riemannian manifolds


Authors: Maria Gordina and Thomas Laetsch
Journal: Trans. Amer. Math. Soc. 369 (2017), 6263-6278
MSC (2010): Primary 60J65, 58G32; Secondary 58J65
DOI: https://doi.org/10.1090/tran/6831
Published electronically: March 1, 2017
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Abstract: This paper considers a classical question of approximation of
Brownian motion by a random walk in the setting of a sub-Riemannian manifold $ M$. To construct such a random walk we first address several issues related to the degeneracy of such a manifold. In particular, we define a family of sub-Laplacian operators naturally connected to the geometry of the underlying manifold. In the case when $ M$ is a Riemannian (non-degenerate) manifold, we recover the Laplace-Beltrami operator. We then construct the corresponding random walk, and under standard assumptions on the sub-Laplacian and $ M$ we show that this random walk converges (at the level of semigroups) to a process, horizontal Brownian motion, whose infinitesimal generator is the sub-Laplacian. An example of the Heisenberg group equipped with a standard sub-Riemannian metric is considered in detail, in which case the sub-Laplacian we introduced is shown to be the sum of squares (Hörmander's) operator.


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Additional Information

Maria Gordina
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269
Email: maria.gordina@uconn.edu

Thomas Laetsch
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269
Address at time of publication: Center for Data Science, 60 5th Avenue, New York, New York 10011
Email: thomas.laetsch@cims.nyu.edu

DOI: https://doi.org/10.1090/tran/6831
Keywords: Brownian motion, sub-Riemannian manifold, hypoelliptic operator, random walk
Received by editor(s): December 15, 2014
Received by editor(s) in revised form: September 19, 2015
Published electronically: March 1, 2017
Additional Notes: The first and second author’s research was supported in part by NSF Grant DMS-1007496.
Article copyright: © Copyright 2017 American Mathematical Society