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A generalization of Caffarelli’s contraction theorem via (reverse) heat flow

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Abstract

A theorem of L. Caffarelli implies the existence of a map, pushing forward a source Gaussian measure to a target measure which is more log-concave than the source one, which contracts Euclidean distance (in fact, Caffarelli showed that the optimal-transport Brenier map T opt is a contraction in this case). We generalize this result to more general source and target measures, using a condition on the third derivative of the potential, by providing two different proofs. The first uses a map T, whose inverse is constructed as a flow along an advection field associated to an appropriate heat-diffusion process. The contraction property is then reduced to showing that log-concavity is preserved along the corresponding diffusion semi-group, by using a maximum principle for parabolic PDE. In particular, Caffarelli’s original result immediately follows by using the Ornstein–Uhlenbeck process and the Prékopa–Leindler Theorem. The second uses the map T opt by generalizing Caffarelli’s argument, employing in addition further results of Caffarelli. As applications, we obtain new correlation and isoperimetric inequalities.

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References

  1. Alvarez O., Lasry J.-M., Lions P.-L.: Convex viscosity solutions and state constraints. J. Math. Pures Appl. (9) 76(3), 265–288 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bakry, D.: Transformations de Riesz pour les semi-groupes symétriques. II. Étude sous la condition Γ2 ≥  0. In: Séminaire de probabilités, XIX, 1983/84. Lecture Notes in Math., vol. 1123, pp. 145–174. Springer, Berlin (1985)

  3. Barthe F.: Log-concave and spherical models in isoperimetry. Geom. Funct. Anal. 12(1), 32–55 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Barthe F., Cattiaux P., Roberto C.: Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry. Rev. Mat. Iberoamericana 22(3), 993–1067 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Barthe, F., Cattiaux, P., Roberto, C.: Isoperimetry between exponential and Gaussian. Electron. J. Probab. 12(44), 1212–1237 (2007, electronic)

    Google Scholar 

  6. Bian B., Guan P.: A microscopic convexity principle for nonlinear partial differential equations. Invent. Math. 177(2), 307–335 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Borell C.: Brownian motion in a convex ring and quasiconcavity. Comm. Math. Phys. 86(1), 143–147 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  8. Borell Ch.: The Brunn–Minkowski inequality in Gauss spaces. Invent. Mathematicae 30, 207–216 (1975)

    Article  MathSciNet  Google Scholar 

  9. Brascamp H.J., Lieb E.H.: 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. Funct. Anal. 22(4), 366–389 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  10. Brenier Y.: Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math. 44(4), 375–417 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  11. Caffarelli L.A.: Interior W 2,p estimates for solutions of the Monge–Ampère equation. Ann. Math. (2) 131(1), 135–150 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  12. Caffarelli L.A.: Some regularity properties of solutions of Monge–Ampère equation. Comm. Pure Appl. Math. 44(8–9), 965–969 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  13. Caffarelli L.A.: The regularity of mappings with a convex potential. J. Am. Math. Soc. 5(1), 99–104 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  14. Caffarelli L.A.: Monotonicity properties of optimal transportation and the FKG and related inequalities. Comm. Math. Phys. 214(3), 547–563 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Caffarelli L.A., Spruck J.: Convexity properties of solutions to some classical variational problems. Comm. Partial Differ. Equ. 7(11), 1337–1379 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  16. Colesanti A., Salani P.: Quasi-concave envelope of a function and convexity of level sets of solutions to elliptic equations. Math. Nachr. 258, 3–15 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Cordero-Erausquin D.: Some applications of mass transport to Gaussian-type inequalities. Arch. Ration. Mech. Anal. 161(3), 257–269 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Cordero-Erausquin D., Fradelizi M., Maurey B.: The (B) conjecture for the Gaussian measure of dilates of symmetric convex sets and related problems. J. Funct. Anal. 214(2), 410–427 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Demange, J.: Des équations à diffusion rapide aux inégalités de Sobolev sur les modèles de géométrie. PhD thesis, Université Paul Sabatier, Toulouse (2006)

  20. Diaz J.I., Kawohl B.: On convexity and starshapedness of level sets for some nonlinear elliptic and parabolic problems on convex rings. J. Math. Anal. Appl. 177(1), 263–286 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  21. Friedman A.: Partial differential equations of parabolic type. Prentice-Hall Inc., Englewood Cliffs (1964)

    MATH  Google Scholar 

  22. Gallot S., Hulin D., Lafontaine J.: Riemannian geometry second edition Universitext. Springer, Berlin (1990)

    Google Scholar 

  23. Giga Y., Goto S., Ishii H., Sato M.-H.: Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains. Indiana Univ. Math. J. 40(2), 443–470 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  24. Greco A., Kawohl B.: Log-concavity in some parabolic problems. Electron. J. Differ. Equ. 1999(19), 1–12 (1999)

    MathSciNet  Google Scholar 

  25. Hargé G.: A particular case of correlation inequality for the Gaussian measure. Ann. Probab. 27(4), 1939–1951 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  26. Hargé G.: A convex/log-concave correlation inequality for Gaussian measure and an application to abstract Wiener spaces. Probab. Theory Related Fields 130(3), 415–440 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  27. Huet N.: Isoperimetry for spherically symmetric log-concave probability measures. Rev. Mat. Iberoamericana 27(1), 93–122 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ishige K., Salani P.: Is quasi-concavity preserved by heat flow?. Arch. Math. (Basel) 90(5), 450–460 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. Jordan R., Kinderlehrer D., Otto F.: The variational formulation of the Fokker–Planck equation. SIAM J. Math. Anal. 29(1), 1–17 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  30. Kannan R., Lovász L., Simonovits M.: Isoperimetric problems for convex bodies and a localization lemma. Discrete Comput. Geom. 13(3–4), 541–559 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  31. Kawohl, B.: Rearrangements and convexity of level sets in PDE. In: Lecture Notes in Mathematics, vol. 1150. Springer, Berlin (1985)

  32. Kennington A.U.: Convexity of level curves for an initial value problem. J. Math. Anal. Appl. 133(2), 324–330 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  33. Klartag, B.: Marginals of geometric inequalities. In: Geometric aspects of functional analysis. Lecture Notes in Math., vol. 1910, pp. 133–166. Springer, Berlin (2007)

  34. Kolesnikov A.V.: On diffusion semigroups preserving the log-concavity. J. Funct. Anal. 186(1), 196–205 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  35. Kolesnikov, A.V.: On global Hölder estimates for optimal transportation. http://arxiv.org/abs/0810.5043 (2008)

  36. Korevaar N.J.: Convex solutions to nonlinear elliptic and parabolic boundary value problems. Indiana Univ. Math. J. 32(4), 603–614 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  37. Ladyženskaja, O.A., Solonnikov, V.A., Ural’ ceva, N.N.: Linear and quasilinear equations of parabolic type. In: Translated from the Russian by S. Smith. Translations of Mathematical Monographs, vol. 23. American Mathematical Society, Providence (1967)

  38. Latała R., Wojtaszczyk J.O.: On the infimum convolution inequality. Studia Math. 189(2), 147–187 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  39. Ledoux, M.: Concentration of measure and logarithmic Sobolev inequalities. In: Séminaire de Probabilités, XXXIII, Lecture Notes in Math., vol. 1709, pp. 120–216. Springer, Berlin, (1999)

  40. Ledoux, M.: Spectral gap, logarithmic Sobolev constant, and geometric bounds. In: Surveys in differential geometry. vol. IX, pp. 219–240. Int. Press, Somerville (2004)

  41. Lee K.-A., Vázquez J.L.: Parabolic approach to nonlinear elliptic eigenvalue problems. Adv. Math. 219(6), 2006–2028 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  42. Lewis J.L.: Capacitary functions in convex rings. Arch. Rational Mech. Anal. 66(3), 201–224 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  43. Lieberman G.M.: Second order parabolic differential equations. World Scientific, Singapore (1996)

    Book  MATH  Google Scholar 

  44. Lions P.-L., Musiela M.: Convexity of solutions of parabolic equations. C. R. Math. Acad. Sci. Paris 342(12), 915–921 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  45. Lunardi A.: Analytic semigroups and optimal regularity in parabolic problems. In: Progress in Nonlinear Differential Equations and their Applications, 16. Birkhäuser Verlag, Basel (1995)

    Book  Google Scholar 

  46. McCann R.J.: A convexity principle for interacting gases. Adv. Math. 128(1), 153–179 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  47. Milman E.: On the role of convexity in functional and isoperimetric inequalities. Proc. Lond. Math. Soc. 99(3), 32–66 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  48. Milman E.: On the role of convexity in isoperimetry, spectral-gap and concentration. Invent. Math. 177(1), 1–43 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  49. Otto F., Villani C.: Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173(2), 361–400 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  50. Schechtman G., Schlumprecht Th., Zinn J.: On the Gaussian measure of the intersection. Ann. Probab. 26(1), 346–357 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  51. Sudakov, V.N., Cirelc′son [Tsirelson], B.S.: Extremal properties of half-spaces for spherically invariant measures. Problems in the theory of probability distributions, II. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 41, 14–24, 165 (1974, in Russian); J. Sov. Math. 9(1), 9–18 (1978, in English)

  52. Valdimarsson S.I.: On the Hessian of the optimal transport potential. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6(3), 441–456 (2007)

    MathSciNet  MATH  Google Scholar 

  53. Villani C.: Topics in optimal transportation. In: Graduate Studies in Mathematics, vol. 58. American Mathematical Society, Providence (2003)

    Google Scholar 

  54. Villani C.: Optimal transport—old and new. In: Grundlehren der Mathematischen Wissenschaften, vol. 338 [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (2009)

    Google Scholar 

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Author notes

  1. Emanuel Milman

    Present address: Department of Mathematics, Technion-Israel Institute of Technology, 32000, Haifa, Israel

Authors and Affiliations

  1. Department of Mathematics, University of British Columbia, Vancouver, BC, Canada

    Young-Heon Kim

  2. Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, ON, M5S 2E4, Canada

    Emanuel Milman

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  1. Young-Heon Kim
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Correspondence to Emanuel Milman.

Additional information

Y.-H. Kim and E. Milman were partially supported by the Institute for Advanced Study through NSF Grant DMS-0635607. Y.-H. Kim is also supported by Canadian NSERC discovery Grant 371642-09. E. Milman is also supported by ISF, GIF and the Taub Foundation (Landau Fellow).

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Kim, YH., Milman, E. A generalization of Caffarelli’s contraction theorem via (reverse) heat flow. Math. Ann. 354, 827–862 (2012). https://doi.org/10.1007/s00208-011-0749-x

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  • DOI: https://doi.org/10.1007/s00208-011-0749-x

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