Integration by parts formula and logarithmic Sobolev inequality on the path space over loop groups

• 成果类型: Article
• 署名作者: Fang, SZ
• 署名单位: Universite Bourgogne Europe
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1022677382
• 发表日期: 1999
• 页码: 664-683
• 关键词: compact riemannian manifold; quasi-invariance; wiener measure; equation; geometry

摘要

The geometric stochastic analysis on the Riomannian path space developed recently gives rise to the concept of tangent processes. Roughly speaking, it is the infinitesimal version of the Girsanov theorem. Using this concept, we shall establish a formula of integration by parts on the path space over a loop group. Following the martingale method developed in Capitaine, Hsu and Ledoux, we shall prove that the logarithmic Sobolev inequality holds on the full paths. As a particular case of our result, vue obtain the Driver-Lohrenz's heat kernel logarithmic Sobolev inequalities over loop groups. The stochastic parallel transport introduced by Driver will play a crucial role.