Cartan connections for stochastic developments on sub-Riemannian manifolds

Ivan Beschastnyi (1, 2) , Karen Habermann (1, 2, 3) , Alexandr Medvedev (4)

1 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
2 CaGE - Control And GEometry
3 Department of Statistics [Warwick]
4 SISSA / ISAS - Scuola Internazionale Superiore di Studi Avanzati / International School for Advanced Studies

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 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

Domaines: Mathématiques [math]; Géométrie différentielle [math.DG]; Probabilités [math.PR]

Ivan Beschastnyi, Karen Habermann, Alexandr Medvedev. Cartan connections for stochastic developments on sub-Riemannian manifolds. 2020. ⟨hal-02885313⟩