Calculus of Variations and Geometric Measure Theory
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A. Agrachev - U. Boscain - R. Neel - L. Rizzi

Intrinsic random walks in Riemannian and sub-Riemannian geometry via volume sampling

created by rizzi1 on 20 Jan 2016
modified on 17 Aug 2018


Published Paper

Inserted: 20 jan 2016
Last Updated: 17 aug 2018

Journal: ESAIM: COCV
Year: 2016
Doi: 10.1051/cocv/2017037

ArXiv: 1601.03304 PDF
Links: ArXiv preprint


We relate some basic 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 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.

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