×

Integration by parts formula and logarithmic Sobolev inequality on the path space over loop groups. (English) Zbl 0946.60053

Let \(G\) be a connected compact Lie group, and let \({\mathcal L}_e(G)=\{l\colon[0,1]\to G\) continuous; \(l(0)=l(1)=e\}\) be the based loop group, where \(e\) is the unit element of \(G\). Let \(\mu\) denote the Wiener measure on the path space over \({\mathcal L}_e(G)\) defined by a Brownian motion on \({\mathcal L}_e(G)\). The author proves an integration by parts formula with respect to \(\mu\) for adapted vector fields (for constant vector fields, it was established by B. K. Driver). This is used to obtain the Clark-Ocone representation formula. Then, following the martingale method developed by M. Capitaine, E. P. Hsu, and M. Ledoux, the logarithmic Sobolev inequality on the path space over \({\mathcal L}_e(G)\) is proved. As a particular case, by taking one-point cylindric functions, the Driver-Lohrenz heat kernel logarithmic Sobolev inequalities over \({\mathcal L}_e(G)\) are obtained.

MSC:

60H07 Stochastic calculus of variations and the Malliavin calculus
60H30 Applications of stochastic analysis (to PDEs, etc.)
58J65 Diffusion processes and stochastic analysis on manifolds
PDFBibTeX XMLCite
Full Text: DOI

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. · Zbl 0787.22021
[2] Bismut, J. M. (1984). Large Deviation and Malliavin Calculus. Birkhäuser, Boston. · Zbl 0537.35003
[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. · Zbl 0890.60045
[4] Cruzeiro, A. B. and Malliavin, P. (1996). Renormalized differential geometry on path space: structural equation, curvature. J. Funct. Anal. 139 119-181. · Zbl 0869.60060
[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. · Zbl 0765.60064
[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. · Zbl 0821.60021
[7] Driver, B. K. (1997). Integration by parts and quasi-invariance for heat kernel measures on loop groups. J. Funct. Analysis 149 470-547. · Zbl 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.
[9] Driver, B. K. and Lohrentz, T. (1996). Logarithmic Sobolev inequalities for pinned loop groups. J. Funct. Anal. 140 381-448. · Zbl 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. · Zbl 0881.60052
[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. · Zbl 0847.58080
[12] Fang, S. (1994). Inégalité du type de Poincaré sur l’espace des chemins Riemanniens. C.R. Acad. Sci. Paris 318 257-260. · Zbl 0805.60056
[13] Fang, S. and Franchi, J. (1998). De Rham-Hodge-Kodaira operator on loop groups. J. Funct. Anal. · Zbl 0901.58067
[14] Fang, S. and Malliavin, P. (1993). Stochastic analysis on the path space of a Riemannian manifold. J. Funct. Anal. 118 249-274. · Zbl 0798.58080
[15] Freed, S. (1988). The geometry of loop groups. J. Differential Geom. 28 223-276. · Zbl 0619.58003
[16] Getzler, E. (1989). Dirichlet forms on loop space. Bull. Sci. Math. 113 151-174. · Zbl 0683.31002
[17] Gross, L. (1991). Logarithmic Sobolev inequality on loop groups. J. Funct. Anal. 102 268- 313. · Zbl 0742.22003
[18] Gross, L. (1993). Uniqueness of ground states for Schrödinger operators over loop groups. J. Funct. Anal. 112 373-441. · Zbl 0774.60059
[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. · Zbl 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. · Zbl 0495.60005
[22] Kuo, H. H. (1975). Gaussian measures in Banach spaces. Lecture Notes in Math. 463. Springer, Berlin. · Zbl 0306.28010
[23] Malliavin, M. P. and Malliavin, P. (1990). Integration on loop groups I. Quasi-invariant measures. J. Funct. Anal. 93 207-237. · Zbl 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. · Zbl 0762.22019
[25] Malliavin, P. (1978). Géométrie différentielle stochastique. Presses Univ. Montréal. · Zbl 0393.60062
[26] Malliavin, P. (1991). Hypoellipticity in infinite dimension. In Diffusion Processes and Related Problems in Analysis (M. Pinsky, ed.) 1 17-33. Birkha üser, Boston.
[27] Malliavin, P. (1997). Stochastic Analysis. Springer, New York. · Zbl 0878.60001
[28] Revuz, D. and Yor, M. (1991). Continuous Martingales and Brownian Motion. Springer, New York. · Zbl 0731.60002
[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. · Zbl 0771.60024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.