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

TL;DR

This paper investigates how random walk approximations on Riemannian and sub-Riemannian manifolds relate to stochastic analysis, focusing on the influence of volume sampling and the passage to diffusions in the geometric setting.

Contribution

It introduces volume-influenced random walks on manifolds to connect geometric structures with stochastic processes, especially in sub-Riemannian geometry where canonical operators are lacking.

Findings

Random walks approximate diffusions in the parabolic limit.

Volume sampling influences the stochastic behavior and generators.

Insights into stochastic differential equations in geometric contexts.

Abstract

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/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.