Integration by Parts Formula and Logarithmic Sobolev Inequality on the Path Space Over Loop Groups
Shizan Fang
Source: Ann. Probab. Volume 27, Number 2
(1999), 664-683.
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.
First Page:
Show
Hide
Keywords: Tangent processes; stochastic parallel transport; integration by parts; martingale representation
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aop/1022677382
Mathematical Reviews number (MathSciNet): MR1698951
Digital Object Identifier: doi:10.1214/aop/1022677382
Zentralblatt MATH identifier: 0946.60053
References
[1] Airault, H. and Malliavin, P. (1992). Integration on loop groups II. Heat equation for the Wiener measure. J. Funct. Anal. 104 71-109.
Zentralblatt MATH: 0787.22021
[2] Bismut, J. M. (1984). Large Deviation and Malliavin Calculus. Birkh¨auser, Boston.
Mathematical Reviews (MathSciNet): MR86f:58150
[3] Capitaine, M., Hsu, E. P. and Ledoux, M. (1997). Martingale representation and a simple proof of logarithmic Sobolev inequalities on path spaces. Electron. Comm. Probab. 2 71-81.
Mathematical Reviews (MathSciNet): MR99b:60136
[4] Cruzeiro, A. B. and Malliavin, P. (1996). Renormalized differential geometry on path space: structural equation, curvature. J. Funct. Anal. 139 119-181.
Mathematical Reviews (MathSciNet): MR97h:58175
[5] Driver, B. K. (1992). A Cameron-Martin type quasi-invariant theorem for Brownian motion on a compact Riemannian manifold. J. Funct. Anal. 110 272-376.
[6] Driver, B. K. (1995). Towards calculus and geometry on path spaces. In Proceedings of the Symposium on Pure Math. Stochastic Analysis 405-422. Amer. Math. Soc., Providence, RI.
Mathematical Reviews (MathSciNet): MR96e:60097
[7] Driver, B. K. (1997). Integration by parts and quasi-invariance for heat kernel measures on loop groups. J. Funct. Analysis 149 470-547.
Mathematical Reviews (MathSciNet): MR99a:60054a
Zentralblatt MATH: 0887.58062
[8] Driver, B. K. (1998). Correction to Integration by parts and quasi-invariance for heat kernel measures on loop groups. J. Funct. Anal. 155 297-301.
Mathematical Reviews (MathSciNet): MR99a:60054b
[9] Driver, B. K. and Lohrentz, T. (1996). Logarithmic Sobolev inequalities for pinned loop groups. J. Funct. Anal. 140 381-448.
Mathematical Reviews (MathSciNet): MR97h:58176
Zentralblatt MATH: 0859.22012
[10] Elworthy K. D. and Li, X. M. (1996). A class of integration by parts formulae in stochastic analysis I. In It o Stochastic Calculus and Probability Theory (N. Ikeda, ed.) 15-30. Springer, New York.
[11] Enchev, O. and Stroock, D. (1995). Towards a Riemannian geometry on the path space over a Riemannian manifold. J. Funct. Anal. 134 392-416.
Mathematical Reviews (MathSciNet): MR96m:58270
Zentralblatt MATH: 0847.58080
[12] Fang, S. (1994). In´egalit´e du type de Poincar´e sur l'espace des chemins Riemanniens. C.R. Acad. Sci. Paris 318 257-260.
[13] Fang, S. and Franchi, J. (1998). De Rham-Hodge-Kodaira operator on loop groups. J. Funct. Anal. To appear.
[14] Fang, S. and Malliavin, P. (1993). Stochastic analysis on the path space of a Riemannian manifold. J. Funct. Anal. 118 249-274.
Zentralblatt MATH: 0798.58080
[15] Freed, S. (1988). The geometry of loop groups. J. Differential Geom. 28 223-276.
Mathematical Reviews (MathSciNet): MR89k:22036
[16] Getzler, E. (1989). Dirichlet forms on loop space. Bull. Sci. Math. 113 151-174.
Mathematical Reviews (MathSciNet): MR90g:58146
Zentralblatt MATH: 0683.31002
[17] Gross, L. (1991). Logarithmic Sobolev inequality on loop groups. J. Funct. Anal. 102 268- 313.
[18] Gross, L. (1993). Uniqueness of ground states for Schr¨odinger operators over loop groups. J. Funct. Anal. 112 373-441.
[19] Hsu, E. P. (1995). Quasi-invariance of the Wiener measure on the path space over a compact Riemannian manifold. J. Funct. Anal. 134 417-450.
Zentralblatt MATH: 0847.58082
[20] Hsu, E. P. (1997). Analysis on path and loop spaces. In IAS/Park City Mathematics Series (E. P. Hsu and S. R. S. Varadhan, eds.) 5 Amer. Math. Soc. and Institute for Advanced Study, Princeton.
[21] Ikeda, N. and Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam.
[22] Kuo, H. H. (1975). Gaussian measures in Banach spaces. Lecture Notes in Math. 463. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR57:1628
[23] Malliavin, M. P. and Malliavin, P. (1990). Integration on loop groups I. Quasi-invariant measures. J. Funct. Anal. 93 207-237.
Zentralblatt MATH: 0715.22024
[24] Malliavin, M. P. and Malliavin, P. (1992). Integration on loop groups III. Asymptotic Peter-Weyl orthogonality. J. Funct. Anal. 108 13-46.
Zentralblatt MATH: 0762.22019
[25] Malliavin, P. (1978). G´eom´etrie diff´erentielle stochastique. Presses Univ. Montr´eal.
[26] Malliavin, P. (1991). Hypoellipticity in infinite dimension. In Diffusion Processes and Related Problems in Analysis (M. Pinsky, ed.) 1 17-33. Birkha ¨user, Boston.
[27] Malliavin, P. (1997). Stochastic Analysis. Springer, New York.
Mathematical Reviews (MathSciNet): MR99b:60073
[28] Revuz, D. and Yor, M. (1991). Continuous Martingales and Brownian Motion. Springer, New York.
Mathematical Reviews (MathSciNet): MR92d:60053
[29] Shigekawa, I. (1997). Differential calculus on a based loop group. In New Trends in Stochastic Analysis (K. D. Elworthy, S. Kusuoka and I. Shigekawa, eds.). World Scientific, Singapore.
Mathematical Reviews (MathSciNet): MR99k:60146
[30] Feyel, D. and de La Pradelle, A. (1993). Sur les processus browniens de dimension infinie, C.R. Acad. Sci. Paris 316 149-152.
Mathematical Reviews (MathSciNet): MR93j:60115
