Dynamic approach to a stochastic domination: The FKG and Brascamp-Lieb inequalities
HTML articles powered by AMS MathViewer
- by Tadahisa Funaki and Kou Toukairin PDF
- Proc. Amer. Math. Soc. 135 (2007), 1915-1922 Request permission
Abstract:
A coupling based on a pair of stochastic differential equations is introduced to show a stochastic domination for a system with continuous spins, from which the FKG and Brascamp-Lieb like inequalities follow.References
- L. Ambrosio, L. A. Caffarelli, Y. Brenier, G. Buttazzo, and C. Villani, Optimal transportation and applications, Lecture Notes in Mathematics, vol. 1813, Springer-Verlag, Berlin; Centro Internazionale Matematico Estivo (C.I.M.E.), Florence, 2003. Lectures from the C.I.M.E. Summer School held in Martina Franca, September 2–8, 2001; Edited by Caffarelli and S. Salsa. MR 2006302, DOI 10.1007/b12016
- Dominique Bakry and Dominique Michel, Sur les inégalités FKG, Séminaire de Probabilités, XXVI, Lecture Notes in Math., vol. 1526, Springer, Berlin, 1992, pp. 170–188 (French). MR 1231994, DOI 10.1007/BFb0084321
- Herm Jan Brascamp and Elliott H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Functional Analysis 22 (1976), no. 4, 366–389. MR 0450480, DOI 10.1016/0022-1236(76)90004-5
- Luis A. Caffarelli, Monotonicity properties of optimal transportation and the FKG and related inequalities, Comm. Math. Phys. 214 (2000), no. 3, 547–563. MR 1800860, DOI 10.1007/s002200000257
- G. Da Prato and J. Zabczyk, Ergodicity for infinite-dimensional systems, London Mathematical Society Lecture Note Series, vol. 229, Cambridge University Press, Cambridge, 1996. MR 1417491, DOI 10.1017/CBO9780511662829
- C. M. Fortuin, P. W. Kasteleyn, and J. Ginibre, Correlation inequalities on some partially ordered sets, Comm. Math. Phys. 22 (1971), 89–103. MR 309498
- Giambattista Giacomin, On stochastic domination in the Brascamp-Lieb framework, Math. Proc. Cambridge Philos. Soc. 134 (2003), no. 3, 507–514. MR 1981215, DOI 10.1017/S0305004102006552
- Y. Hariya, private communication, 2005, Oct.
- Richard Holley, Remarks on the $\textrm {FKG}$ inequalities, Comm. Math. Phys. 36 (1974), 227–231. MR 341552
- Kanji Ichihara and Hiroshi Kunita, A classification of the second order degenerate elliptic operators and its probabilistic characterization, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 30 (1974), 235–254. MR 381007, DOI 10.1007/BF00533476
- Nobuyuki Ikeda and Shinzo Watanabe, Stochastic differential equations and diffusion processes, 2nd ed., North-Holland Mathematical Library, vol. 24, North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989. MR 1011252
- Thomas M. Liggett, Interacting particle systems, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 276, Springer-Verlag, New York, 1985. MR 776231, DOI 10.1007/978-1-4613-8542-4
- C. J. Preston, A generalization of the $\textrm {FKG}$ inequalities, Comm. Math. Phys. 36 (1974), 233–241. MR 341553
- Daniel W. Stroock, Probability theory, an analytic view, Cambridge University Press, Cambridge, 1993. MR 1267569
Additional Information
- Tadahisa Funaki
- Affiliation: Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Tokyo 153-8914, Japan
- Email: funaki@ms.u-tokyo.ac.jp
- Kou Toukairin
- Affiliation: Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Tokyo 153-8914, Japan
- Address at time of publication: Lehman Brothers Japan Inc., Roppongi Hills, Tokyo
- Email: kou.toukairin@lehman.com
- Received by editor(s): April 10, 2006
- Published electronically: February 6, 2007
- Additional Notes: The first author was supported in part by JSPS Grants (B)14340029 and 17654020
- Communicated by: Edward C. Waymire
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 135 (2007), 1915-1922
- MSC (2000): Primary 82B31; Secondary 82B20, 60K35
- DOI: https://doi.org/10.1090/S0002-9939-07-08757-6
- MathSciNet review: 2286104