Abstract
We establish Talagrand’s T2-transportation inequalities for infinite dimensional dissipative diffusions with sharp constants, through Galerkin type’s approximations and the known results in the finite dimensional case. Furthermore in the additive noise case we prove also logarithmic Sobolev inequalities with sharp constants. Applications to Reaction-Diffusion equations are provided.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bakry, D. and Emery, M., Diffusions hypercontractives, Séminaire de Probabilités, Lect. Notes Math., 1123, Springer, 1985, 259-206.
Bobkov, S., Gentil, I. and Ledoux, M., Hypercontractivity of Hamilton-Jacobi equations, J. Math. Pure Appl., 80, 2001, 669-696.
Bobkov, S. and Götze, F., Exponential integrablity and transportation cost related to logarithmic Sobolev inequalities, J. Funct. Anal., 163, 1999, 1-28.
Djellout, H., Guillin, A. and Wu, L., Transportation cost-information inequalities and applications to random dynamical systems and diffusions, Ann. Probab., 32B(3), 2004, 2702-2732.
Da Prato, G., Debussche, A. and Goldys, B., Some properties of invariant measures of non-symmetric dissipative stochastic systems, Probab. Th. Rel. Fields, 123, 2002, 355-380.
Da Prato, G. and Zabczyk, J., Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and Its Applictions, Combridge University Press, 1992.
Da Prato, G. and Zabczyk, J., Ergodicity for Infinite Dimensional Systems, London Mathematical Society, Lecture Note Series 229, Combridge University Press, 1996.
Fang, S. and Shao, J., Transportation cost inequalities on path and loop groups, J. Funct. Anal., to appear.
Feyel, D. and Ustunel, A. S., Measure transport on Wiener space and Girsanov theorem, CRAS Serie I, 334, 2002, 1025-1028.
Feyel, D. and Ustunel, A. S., The Monge-Kantorovitch problem and Monge-Ampµere equation on Wiener space, Probab. Theor. Rel. Fields, 128(3), 2004, 347-385.
Gourcy, M. and Wu, L., Logarithmic Sobolev inequality for diffusions w.r.t. the L2-metric, preprint, 2004.
Gross, L., Logarithmic Sobolev inequalities, Amer. J. Math., 97, 1975, 1061-1083.
Ledoux, M., The Concentration of Measure Phenomenon, Mathematical Surveys and Monographs, Vol. 89, Amer. Math. Soc., 2001.
Marton, K., A measure concentration inequality for contracting Markov chains, Geom. Funct. Anal., 6, 1997, 556-571.
Marton, K., Bounding d̄-distance by information divergence: a method to prove measure concentration, Ann. Probab., 24, 1996, 857-866.
Otto, F. and Villani, C., Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal., 173, 2000, 361-400.
Pazy, A., Semigroups of Linear Operayors and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.
Talagrand, M., Transportation cost for Gaussian and other product measures, Geom. Funct. Anal., 6, 1996, 587-600.
Villani, C., Topics in optimal transportation, Grad. Stud. Math. Vol. 58, Amer. Math. Soc., 2003.
Wang, F. Y., Transportation cost inequalities on path spaces over Riemannian manifolds, Illinois J. Math., 46, 2002, 1197-1206.
Wang, F. Y., Probability distance inequalities on Riemaniann manifolds and path spaces, J. Funct. Anal., 206, 2004, 167-190.
Wu, L. and Zhang, Z. L., Talagrand’s T2-transportation inequality w.r.t. a uniform metric for diffusions, Acta Math. Appl. Sinica, English Series, 20(3), 2004, 357-364.
Author information
Authors and Affiliations
Corresponding author
Additional information
* Project supported by the Yangtze Scholarship Program.
Rights and permissions
About this article
Cite this article
Wu, L., Zhang, Z. Talagrand’s T2-Transportation Inequality and Log-Sobolev Inequality for Dissipative SPDEs and Applications to Reaction-Diffusion Equations*. Chin. Ann. Math. Ser. B 27, 243–262 (2006). https://doi.org/10.1007/s11401-005-0176-y
Received:
Issue Date:
DOI: https://doi.org/10.1007/s11401-005-0176-y
Keywords
- Stochastic partial differential equations (SPDEs)
- Logarithmic Sobolev inequality
- Talagrand’s transportation inequality
- Poincaré inequality