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Sobolev spaces and capacities theory on path spaces over a compact Riemannian manifold
Authors:Xiang Dong Li
Institution:(1) Mathematical Institute, University of Oxford, 24-29, St. Giles, Oxford, OX1 3LB, UK. e-mail: lix@maths.ox.ac.uk, GB
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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