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Cartan connections for stochastic developments on sub-Riemannian manifolds

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Resumo: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.
Autores principais:Beschastnyi, Ivan
Outros Autores:Habermann, Karen; Medvedev, Alexandr
Assunto:Brownian motion Sub-Riemannian geometry Cartan geometry Stochastic development Popp’s volume Hypoelliptic operators
Ano:2022
País:Portugal
Tipo de documento:artigo
Tipo de acesso:acesso aberto
Instituição associada:Universidade de Aveiro
Idioma:inglês
Origem:RIA - Repositório Institucional da Universidade de Aveiro

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