Small-time fluctuations for the bridge of a sub-Riemannian diffusion
Abstract
We consider small-time asymptotics for diffusion processes conditioned by their initial and final positions, under the assumption that the diffusivity has a sub-Riemannian structure, not necessarily of constant rank. We show that, if the endpoints are joined by a unique path of minimal energy, and lie outside the sub-Riemannian cut locus, then the fluctuations of the conditioned diffusion from the minimal energy path, suitably rescaled, converge to a Gaussian limit. The Gaussian limit is characterized in terms of the bicharacteristic flow, and also in terms of a second variation of the energy functional at the minimal path, the formulation of which is new in this context.
- Publication:
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arXiv e-prints
- Pub Date:
- May 2015
- DOI:
- 10.48550/arXiv.1505.03464
- arXiv:
- arXiv:1505.03464
- Bibcode:
- 2015arXiv150503464B
- Keywords:
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- Mathematics - Probability;
- Mathematics - Analysis of PDEs;
- Mathematics - Differential Geometry;
- 58J65
- E-Print:
- Reorganized. Some material removed and developed further in a companion paper