Dynamic approach to a stochastic domination: The FKG and Brascamp-Lieb inequalities

Funaki, Tadahisa; Toukairin, Kou

doi:10.1090/S0002-9939-07-08757-6

Dynamic approach to a stochastic domination: The FKG and Brascamp-Lieb inequalities
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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