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

Annals of Probability

Volumn 27, Issue 2, 1999, Pages 664-683

  Fang, Shizan ^a  

^a UNIVERSITÉ DE BOURGOGNE   (France)

Author keywords

Integration by parts; Martingale representation; Stochastic parallel transport; Tangent processes

Indexed keywords

EID: 0039526911     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/aop/1022677382     Document Type: Article

References (30)

1. AIRAULT, H. and MALLIAVIN, P. (1992). Integration on loop groups II. Heat equation for the Wiener measure. J. Funct. Anal. 104 71-109.
2. BISMUT, J. M. (1984). Large Deviation and Malliavin Calculus. Birkhäuser, Boston.
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.
4. CRUZEIRO, A. B. and MALLIAVIN, P. (1996). Renormalized differential geometry on path space: structural equation, curvature. J. Funct. Anal. 139 119-181.
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.
7. DRIVER, B. K. (1997). Integration by parts and quasi-invariance for heat kernel measures on loop groups. J. Funct. Analysis 149 470-547
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.
9. DRIVER, B. K. and LOHRENTZ, T. (1996). Logarithmic Sobolev inequalities for pinned loop groups. J. Funct. Anal. 140 381-448.
10. ELWORTHY K. D. and LI, X. M. (1996). A class of integration by parts formulae in stochastic analysis I. In Itô 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.
12. FANG, S. (1994). Inégalité du type de Poincaré 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.
15. FREED, S. (1988). The geometry of loop groups. J. Differential Geom. 28 223-276.
16. GETZLER, E. (1989). Dirichlet forms on loop space. Bull. Sci. Math. 113 151-174.
17. GROSS, L. (1991). Logarithmic Sobolev inequality or loop groups. J. Funct. Anal. 102 268-313.
18. GROSS, L. (1993). Uniqueness of ground states for Schrödinger 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.
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.
23. MALLIAVIN, M. P. and MALLIAVIN, P. (1990). Integration on loop groups I. Quasi-invariant measures. J. Funct. Anal. 93 207-237.
24. MALLIAVIN, M. P. and MALLIAVIN, P. (1992). Integration on loop groups III. Asymptotic Peter-Weyl orthogonality. J. Funct. Anal. 108 13-46.
25. MALLIAVIN, P. (1978). Géométrie différentielle stochastique. Presses Univ. Montréal.
26. MALLIAVIN, P. (1991). Hypoellipticity in infinite dimension. In Diffusion Processes and Related Problems in Analysis (M. Pinsky, ed.) 1 17-33. Birkhaüser, Boston.
27. MALLIAVIN, P. (1997). Stochastic Analysis. Springer, New York.
28. REVUZ, D. and YOR, M. (1991). Continuous Martingales and Brownian Motion. Springer, New York.
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.
30. FEYEL, D. and DE LA PRADELLE, A. (1993). Sur les processus browniens de dimension infinie, C.R. Acad. Sci. Paris 316 149-152.