Abstract
In this paper, the dimensional-free Harnack inequalities are established on infinite-dimensional spaces. More precisely, we establish Harnack inequalities for heat semigroup on based loop group and for Ornstein-Uhlenbeck semigroup on the abstract Wiener space. As an application, we establish the HWI inequality on the abstract Wiener space, which contains three important quantities in one inequality, the relative entropy “H”, Wasserstein distance “W”, and Fisher information “I”.
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Li, P., Yau, S. T.: On the parabolic heat kernel of the Schrödinger operator. Acta Math., 156, 153–201 (1986)
Wang, F. Y.: Logarithmic Sobolev inequalities on noncompact Riemannian manifolds. Probab. Theory Relat. Fields, 109, 417–424 (1997)
Wang, F. Y.: Equivalence of dimension-free Harnack inequality and curvature condition. Integral Equat. Operator Theory, 48, 547–552 (2004)
Röckner, M., Wang, F. Y.: Harnack and functional inequalities for generalized Mehler semigroups. J. Funct. Anal., 203, 237–261 (2003)
Aida, S., Kawabi, H.: Short time asymptotics of a certain infinite dimensional diffusion process. Stochastic Analysis and Related Topics, VII (1998), Progress in Probability 48, Birkhäuser, Boston, 2001, 77–124
Warner, F. W.: Foundation of Differentiable Manifolds and Lie Groups, China Academic Publishers and Springer-Verlag, Berlin, 1983
Otto, F., Villani, C.: Generalization of an inequality by Talagrand, and links with the lograrithmic Sobolev inequality. J. Funct. Anal., 173, 361–400 (2000)
Bobkov, S., Gentil, I., Ledoux, M.: Hypercontractivity of Hamilton-Jacobi equations. J. Math. Pure Appl., 80, 669–696 (2001)
Driver, B. K., Lohrenz, T.: Logarithmic Sobolev inequalities for pinned loop groups. J. Funct. Anal., 140, 381–448 (1996)
Fang, S.: Integration by parts for heat measures over loop groups. J. Math. Pures Appl., 78, 877–894 (1999)
Fang, S., Zhang, T.: Large deviations for the Brownian motion on loop groups. J. Theoret. Probab., 14, 463–483 (2001)
Fang, S., Shao, J.: Distance riemannienne, Théorème de Rademacher et Inégalité de transport sur le groupe des lacets. C. R. Acad. Sci. Paris. Ser. I Math., 341, 445–450 (2005)
Enchev, O., Stroock, D. W.: Rademacher’s theorem for Wiener functionals. Annal. Prob., 21, 25–33 (1993)
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Supported by National Natural Science Foundation of China (Grant No. 10721091) and the 973-Project (Grant No. 2006CB805901)
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Shao, J.H. Harnack and HWI inequalities on infinite-dimensional spaces. Acta. Math. Sin.-English Ser. 27, 1195–1204 (2011). https://doi.org/10.1007/s10114-011-8021-6
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DOI: https://doi.org/10.1007/s10114-011-8021-6