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Sobolev spaces and capacities theory on path spaces over a compact Riemannian manifold
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  • Published: January 2003

Sobolev spaces and capacities theory on path spaces over a compact Riemannian manifold

  • Xiang Dong Li1 

Probability Theory and Related Fields volume 125, pages 96–134 (2003)Cite this article

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  • 6 Citations

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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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Authors and Affiliations

  1. Mathematical Institute, University of Oxford, 24-29, St. Giles, Oxford, OX1 3LB, UK. e-mail: lix@maths.ox.ac.uk, , , , , , GB

    Xiang Dong Li

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  1. Xiang Dong Li
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Additional information

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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  • Issue Date: January 2003

  • DOI: https://doi.org/10.1007/s004400200227

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Keywords

  • Manifold
  • Riemannian Manifold
  • Sobolev Space
  • Compact Riemannian Manifold
  • Path Space
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