Volume 24, Number 3, July–September 2018
|Page(s)||1075 - 1105|
|Published online||11 July 2018|
Intrinsic random walks in Riemannian and sub-Riemannian geometry via volume sampling
SISSA, Trieste & Steklov Math. Inst., Moscow.
2 Inria, team GECO & CMAP, École Polytechnique, CNRS, Université Paris-Saclay, Palaiseau, France.
3 Department of Mathematics, Lehigh University, Bethlehem, PA, USA.
4 Univ. Grenoble Alpes, CNRS, Institut Fourier, 38000 Grenoble, France.
a Corresponding author: email@example.com
Accepted: 25 April 2017
We relate some constructions of stochastic analysis to differential geometry, via random walk approximations. We consider walks on both Riemannian and sub-Riemannian manifolds in which the steps consist of travel along either geodesics or integral curves associated to orthonormal frames, and we give particular attention to walks where the choice of step is influenced by a volume on the manifold. A primary motivation is to explore how one can pass, in the parabolic scaling limit, from geodesics, orthonormal frames, and/or volumes to diffusions, and hence their infinitesimal generators, on sub-Riemannian manifolds, which is interesting in light of the fact that there is no completely canonical notion of sub-Laplacian on a general sub-Riemannian manifold. However, even in the Riemannian case, this random walk approach illuminates the geometric significance of Ito and Stratonovich stochastic differential equations as well as the role played by the volume.
Mathematics Subject Classification: 53C17 / 60J65 / 58J65
Key words: Sub-Riemannian geometry / diffusion processes / Brownian motion, random walk
© EDP Sciences, SMAI 2018
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