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Lévy processes on smooth manifolds with a connection
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Mijatović, Aleksandar and Mramor, Veno (2021) Lévy processes on smooth manifolds with a connection. Electronic Journal of Probability, 26 . doi:10.1214/21-ejp702 ISSN 1083-6489.
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Official URL: https://doi.org/10.1214/21-ejp702
Abstract
We define a Lévy process on a smooth manifold M with a connection as a projection of a solution of a Marcus stochastic differential equation on a holonomy bundle of M, driven by a holonomy-invariant Lévy process on a Euclidean space. On a Riemannian manifold, our definition (with Levi-Civita connection) generalizes the Eells-Elworthy-Malliavin construction of the Brownian motion and extends the class of isotropic Lévy process introduced in Applebaum and Estrade [3]. On a Lie group with a surjective exponential map, our definition (with left-invariant connection) coincides with the classical definition of a (left) Lévy process given in terms of its increments.
Our main theorem characterizes the class of Lévy processes via their generators on M, generalizing the fact that the Laplace-Beltrami operator generates Brownian motion on a Riemannian manifold. Its proof requires a path-wise construction of the stochastic horizontal lift and anti-development of a discontinuous semimartingale, leading to a generalization of Pontier and Estrade [32] to smooth manifolds with non-unique geodesics between distinct points.
| Item Type: | Journal Article | |||||||||||||||
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| Subjects: | Q Science > QA Mathematics | |||||||||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Statistics | |||||||||||||||
| SWORD Depositor: | Library Publications Router | |||||||||||||||
| Library of Congress Subject Headings (LCSH): | Lévy processes, Manifolds (Mathematics), Diffusion processes, Stochastic analysis, Markov processes, Holonomy groups | |||||||||||||||
| Journal or Publication Title: | Electronic Journal of Probability | |||||||||||||||
| Publisher: | Institute of Mathematical Statistics | |||||||||||||||
| ISSN: | 1083-6489 | |||||||||||||||
| Official Date: | 1 January 2021 | |||||||||||||||
| Dates: |
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| Volume: | 26 | |||||||||||||||
| DOI: | 10.1214/21-ejp702 | |||||||||||||||
| Status: | Peer Reviewed | |||||||||||||||
| Publication Status: | Published | |||||||||||||||
| Access rights to Published version: | Open Access (Creative Commons) | |||||||||||||||
| Date of first compliant deposit: | 11 January 2022 | |||||||||||||||
| Date of first compliant Open Access: | 12 January 2022 | |||||||||||||||
| RIOXX Funder/Project Grant: |
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| Is Part Of: | 1 |
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