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

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 Ito 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 Holder continuous path space .