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Cartan Connections for Stochastic Developments on sub-Riemannian Manifolds

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Abstract

Analogous to the characterisation of Brownian motion on a Riemannian manifold as the development of Brownian motion on a Euclidean space, we construct sub-Riemannian diffusions on equinilpotentisable sub-Riemannian manifolds by developing a canonical stochastic process arising as the lift of Brownian motion to an associated model space. The notion of stochastic development we introduce for equinilpotentisable sub-Riemannian manifolds uses Cartan connections, which take the place of the Levi-Civita connection in Riemannian geometry. We first derive a general expression for the generator of the stochastic process which is the stochastic development with respect to a Cartan connection of the lift of Brownian motion to the model space. We further provide a necessary and sufficient condition for the existence of a Cartan connection which develops the canonical stochastic process to the sub-Riemannian diffusion associated with the sub-Laplacian defined with respect to the Popp’s volume. We illustrate the construction of a suitable Cartan connection for free sub-Riemannian structures with two generators and we discuss an example where the condition is not satisfied.

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Authors and Affiliations

  1. Centro de Investigação e Desenvolvimento em Matemática e Aplicações, Universidade de Aveiro, 3810-193, Aveiro, Portugal

    Ivan Beschastnyi

  2. Department of Statistics, University of Warwick, Coventry, CV4 7AL, United Kingdom

    Karen Habermann

  3. International School for Advanced Studies, via Bonomea 265, 34136, Trieste, Italy

    Alexandr Medvedev

Authors
  1. Ivan Beschastnyi
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  2. Karen Habermann
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  3. Alexandr Medvedev
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The first author was supported by the ANR project Quaco ANR-17-CE40-0007-01. The second author was supported by the Fondation Sciences Mathématiques de Paris.

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Beschastnyi, I., Habermann, K. & Medvedev, A. Cartan Connections for Stochastic Developments on sub-Riemannian Manifolds. J Geom Anal 32, 13 (2022). https://doi.org/10.1007/s12220-021-00743-9

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  • DOI: https://doi.org/10.1007/s12220-021-00743-9

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