Abstract
We introduce Sobolev spaces and capacities on the path space P m 0 (M) over a compact Riemannian manifold M. We prove the smoothness of the Itô map and the stochastic anti-development map in the sense of stochastic calculus of variation. We establish a Sobolev norm comparison theorem and a capacity comparison theorem between the Wiener space and the path space P m 0 (M). Moreover, we prove the tightness of (r, p)-capacities on P m 0 (M), , which generalises a result due to Airault-Malliavin and Sugita on the Wiener space. Finally, we extend our results to the fractional Hölder continuous path space .
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Received: 14 December 2000 / Revised version: 9 June 2002 / Published online: 24 October 2002
Research partially supported by the National Nature Science Foundation of China, the Foundation of Science and Technology of Portugal, and a Postdoctoral Fellowship of Oxford University.
Mathematics Subject Classification (2000): 58J65, 60H07, 60H30, 31C45
Key words or phrases: Sobolev norm – Capacity – Itô map – Tightness
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Li, X. Sobolev spaces and capacities theory on path spaces over a compact Riemannian manifold. Probab Theory Relat Fields 125, 96–134 (2003). https://doi.org/10.1007/s004400200227
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DOI: https://doi.org/10.1007/s004400200227
Keywords
- Manifold
- Riemannian Manifold
- Sobolev Space
- Compact Riemannian Manifold
- Path Space