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

• 成果类型: Article
• 署名作者: Li, XD
• 署名单位: University of Oxford
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s004400200227
• 发表日期: 2003
• 页码: 96-134
• 关键词: loop-spaces; banach-space; functionals; DIFFUSIONS; calculus

摘要

We introduce Sobolev spaces and capacities on the path space P-m0 (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-m0 (M). Moreover, we prove the tightness of (r, p)-capacities on P-m0 (M), r is an element of N, p > 1, which generalises a result due to Airault-Malliavin and Sugitaon the Wiener space. Finally, we extend our results to the fractional Holder continuous path space p(m0)(2m,alpha) (M), m is an element of N, m greater than or equal to 2, alpha is an element of (1/2m, 1/2).