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.

Accepted:

DOI: 10.1051/cocv/2017037

Keywords: Sub-Riemannian geometry, diffusion processes, Brownian motion, random walk

^{1}; Boscain, Ugo

^{1}; Neel, Robert

^{1}; Rizzi, Luca

^{1}

@article{COCV_2018__24_3_1075_0, author = {Agrachev, Andrei and Boscain, Ugo and Neel, Robert and Rizzi, Luca}, title = {Intrinsic random walks in {Riemannian} and {sub-Riemannian} geometry via volume sampling}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {1075--1105}, publisher = {EDP-Sciences}, volume = {24}, number = {3}, year = {2018}, doi = {10.1051/cocv/2017037}, zbl = {1481.53041}, mrnumber = {3877194}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2017037/} }

TY - JOUR AU - Agrachev, Andrei AU - Boscain, Ugo AU - Neel, Robert AU - Rizzi, Luca TI - Intrinsic random walks in Riemannian and sub-Riemannian geometry via volume sampling JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2018 SP - 1075 EP - 1105 VL - 24 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2017037/ DO - 10.1051/cocv/2017037 LA - en ID - COCV_2018__24_3_1075_0 ER -

%0 Journal Article %A Agrachev, Andrei %A Boscain, Ugo %A Neel, Robert %A Rizzi, Luca %T Intrinsic random walks in Riemannian and sub-Riemannian geometry via volume sampling %J ESAIM: Control, Optimisation and Calculus of Variations %D 2018 %P 1075-1105 %V 24 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2017037/ %R 10.1051/cocv/2017037 %G en %F COCV_2018__24_3_1075_0

Agrachev, Andrei; Boscain, Ugo; Neel, Robert; Rizzi, Luca. Intrinsic random walks in Riemannian and sub-Riemannian geometry via volume sampling. ESAIM: Control, Optimisation and Calculus of Variations, Volume 24 (2018) no. 3, pp. 1075-1105. doi : 10.1051/cocv/2017037. http://archive.numdam.org/articles/10.1051/cocv/2017037/

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