Publicação
Cartan connections for stochastic developments on sub-Riemannian manifolds
| 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 |
| _version_ | 1866173072811229184 |
|---|---|
| author | Beschastnyi, Ivan |
| author2 | Habermann, Karen Medvedev, Alexandr |
| author2_role | author author |
| author_facet | Beschastnyi, Ivan Habermann, Karen Medvedev, Alexandr |
| author_role | author |
| country_str | PT |
| creators_json_txt | [{\"Person.name\":\"Beschastnyi, Ivan\"},{\"Person.name\":\"Habermann, Karen\"},{\"Person.name\":\"Medvedev, Alexandr\"}] |
| datacite.creators.creator.creatorName.fl_str_mv | Beschastnyi, Ivan Habermann, Karen Medvedev, Alexandr |
| datacite.date.Accepted.fl_str_mv | 2022-01-01T00:00:00Z |
| datacite.date.available.fl_str_mv | 2023-02-10T17:55:41Z |
| datacite.date.embargoed.fl_str_mv | 2023-02-10T17:55:41Z |
| datacite.rights.fl_str_mv | http://purl.org/coar/access_right/c_abf2 |
| datacite.subjects.subject.fl_str_mv | Brownian motion Sub-Riemannian geometry Cartan geometry Stochastic development Popp’s volume Hypoelliptic operators |
| datacite.titles.title.fl_str_mv | Cartan connections for stochastic developments on sub-Riemannian manifolds |
| dc.creator.none.fl_str_mv | Beschastnyi, Ivan Habermann, Karen Medvedev, Alexandr |
| dc.date.Accepted.fl_str_mv | 2022-01-01T00:00:00Z |
| dc.date.available.fl_str_mv | 2023-02-10T17:55:41Z |
| dc.date.embargoed.fl_str_mv | 2023-02-10T17:55:41Z |
| dc.format.none.fl_str_mv | application/pdf |
| dc.identifier.none.fl_str_mv | http://hdl.handle.net/10773/36301 |
| dc.language.none.fl_str_mv | eng |
| dc.publisher.none.fl_str_mv | Springer |
| dc.rights.none.fl_str_mv | http://purl.org/coar/access_right/c_abf2 |
| dc.subject.none.fl_str_mv | Brownian motion Sub-Riemannian geometry Cartan geometry Stochastic development Popp’s volume Hypoelliptic operators |
| dc.title.fl_str_mv | Cartan connections for stochastic developments on sub-Riemannian manifolds |
| dc.type.none.fl_str_mv | http://purl.org/coar/resource_type/c_6501 |
| description | 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. |
| dirty | 0 |
| eu_rights_str_mv | openAccess |
| format | article |
| id | ria_6256ad3f5d70cc15073a02bbd61ba047 |
| identifier.url.fl_str_mv | http://hdl.handle.net/10773/36301 |
| instacron_str | ua |
| institution | Universidade de Aveiro |
| instname_str | Universidade de Aveiro |
| language | eng |
| network_acronym_str | ria |
| network_name_str | RIA - Repositório Institucional da Universidade de Aveiro |
| oai_identifier_str | oai:ria.ua.pt:10773/36301 |
| organization_str_mv | urn:organizationAcronym:ua |
| person_str_mv | Beschastnyi, Ivan Habermann, Karen Medvedev, Alexandr |
| publishDate | 2022 |
| publisher.none.fl_str_mv | Springer |
| reponame_str | RIA - Repositório Institucional da Universidade de Aveiro |
| repository_id_str | urn:repositoryAcronym:ria |
| service_str_mv | urn:repositoryAcronym:ria |
| spelling | pt_PTAnalogous 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.application/pdfengSpringerpt_PTCartan connections for stochastic developments on sub-Riemannian manifoldsBeschastnyi, IvanHabermann, KarenMedvedev, AlexandrHandlehttp://hdl.handle.net/10773/36301ISSNIsPartOf1050-6926DOIIsPartOf10.1007/s12220-021-00743-92023-02-10T17:55:41Z2022-01-01T00:00:00Z2022-01http://purl.org/coar/access_right/c_abf2open accesspt_PTBrownian motionpt_PTSub-Riemannian geometrypt_PTCartan geometrypt_PTStochastic developmentpt_PTPopp’s volumept_PTHypoelliptic operators329387 byteshttp://purl.org/coar/access_right/c_abf2application/pdffulltexthttps://ria.ua.pt/bitstream/10773/36301/1/2006.16135%5b1%5d.pdfliteraturehttp://purl.org/coar/resource_type/c_6501journal article |
| spellingShingle | Cartan connections for stochastic developments on sub-Riemannian manifolds Beschastnyi, Ivan Brownian motion Sub-Riemannian geometry Cartan geometry Stochastic development Popp’s volume Hypoelliptic operators |
| status | SINGLETON |
| subject.fl_str_mv | Brownian motion Sub-Riemannian geometry Cartan geometry Stochastic development Popp’s volume Hypoelliptic operators |
| title | Cartan connections for stochastic developments on sub-Riemannian manifolds |
| title_full | Cartan connections for stochastic developments on sub-Riemannian manifolds |
| title_fullStr | Cartan connections for stochastic developments on sub-Riemannian manifolds |
| title_full_unstemmed | Cartan connections for stochastic developments on sub-Riemannian manifolds |
| title_short | Cartan connections for stochastic developments on sub-Riemannian manifolds |
| title_sort | Cartan connections for stochastic developments on sub-Riemannian manifolds |
| topic | Brownian motion Sub-Riemannian geometry Cartan geometry Stochastic development Popp’s volume Hypoelliptic operators |
| topic_facet | Brownian motion Sub-Riemannian geometry Cartan geometry Stochastic development Popp’s volume Hypoelliptic operators |
| url | http://hdl.handle.net/10773/36301 |
| visible | 1 |