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Small-time fluctuations for sub-Riemannian diffusion loops
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Habermann, Karen (2018) Small-time fluctuations for sub-Riemannian diffusion loops. Probability Theory and Related Fields, 171 (3-4). pp. 617-652. doi:10.1007/s00440-017-0788-7
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Official URL: http://dx.doi.org/10.1007/s00440-017-0788-7
Abstract
We study the small-time fluctuations for diffusion processes which are conditioned by their initial and final positions, under the assumptions that the diffusivity has a sub-Riemannian structure and that the drift vector field lies in the span of the sub-Riemannian structure. In the case where the endpoints agree and the generator of the diffusion process is non-elliptic at that point, the deterministic Malliavin covariance matrix is always degenerate. We identify, after a suitable rescaling, another limiting Malliavin covariance matrix which is non-degenerate, and we show that, with the same scaling, the diffusion Malliavin covariance matrices are uniformly non-degenerate. We further show that the suitably rescaled fluctuations of the diffusion loop converge to a limiting diffusion loop, which is equal in law to the loop we obtain by taking the limiting process of the unconditioned rescaled diffusion processes and condition it to return to its starting point. The generator of the unconditioned limiting rescaled diffusion process can be described in terms of the original generator.
| Item Type: | Journal Article | ||||||||
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| Subjects: | Q Science > QA Mathematics | ||||||||
| Divisions: | Faculty of Science > Statistics | ||||||||
| Library of Congress Subject Headings (LCSH): | Diffusion processes, Limit theorems (Probability theory), Malliavin calculus, Hypoelliptic operators | ||||||||
| Journal or Publication Title: | Probability Theory and Related Fields | ||||||||
| Publisher: | Springer | ||||||||
| ISSN: | 0178-8051 | ||||||||
| Official Date: | 2018 | ||||||||
| Dates: |
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| Volume: | 171 | ||||||||
| Number: | 3-4 | ||||||||
| Page Range: | pp. 617-652 | ||||||||
| DOI: | 10.1007/s00440-017-0788-7 | ||||||||
| Status: | Peer Reviewed | ||||||||
| Publication Status: | Published | ||||||||
| Access rights to Published version: | Open Access | ||||||||
| RIOXX Funder/Project Grant: |
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