The Annals of Probability

Integration by Parts Formula and Logarithmic Sobolev Inequality on the Path Space Over Loop Groups

Shizan Fang

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Abstract

The geometric stochastic analysis on the Riemannian 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, we 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.

Article information

Source
Ann. Probab., Volume 27, Number 2 (1999), 664-683.

Dates
First available in Project Euclid: 29 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022677382

Digital Object Identifier
doi:10.1214/aop/1022677382

Mathematical Reviews number (MathSciNet)
MR1698951

Zentralblatt MATH identifier
0946.60053

Subjects
Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus
Secondary: 58G32 60H30: Applications of stochastic analysis (to PDE, etc.)

Keywords
Tangent processes stochastic parallel transport integration by parts martingale representation

Citation

Fang, Shizan. Integration by Parts Formula and Logarithmic Sobolev Inequality on the Path Space Over Loop Groups. Ann. Probab. 27 (1999), no. 2, 664--683. doi:10.1214/aop/1022677382. https://projecteuclid.org/euclid.aop/1022677382


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