Advertisement

Archive for Rational Mechanics and Analysis

, Volume 199, Issue 2, pp 563–596 | Cite as

Convex Sobolev Inequalities Derived from Entropy Dissipation

  • Daniel MatthesEmail author
  • Ansgar Jüngel
  • Giuseppe Toscani
Article

Abstract

We study families of convex Sobolev inequalities, which arise as entropy–dissipation relations for certain linear Fokker–Planck equations. Extending the ideas recently developed by the first two authors, a refinement of the Bakry–Émery method is established, which allows us to prove non-trivial inequalities even in situations where the classical Bakry–Émery criterion fails. The main application of our theory concerns the linearized fast diffusion equation in dimensions d ≧ 1, which admits a Poincaré, but no logarithmic Sobolev inequality. We calculate bounds on the constants in the interpolating convex Sobolev inequalities, and prove that these bounds are sharp on a specified range. In dimension d = 1, our estimates improve the corresponding results that can be obtained by the measure-theoretic techniques of Barthe and Roberto. As a by-product, we give a short and elementary alternative proof of the sharp spectral gap inequality first obtained by Denzler and McCann. In further applications of our method, we prove convex Sobolev inequalities for a mean field model for the redistribution of wealth in a simple market economy, and the Lasota model for blood cell production.

Keywords

Sobolev Inequality Planck Equation Logarithmic Sobolev Inequality Functional Inequality Optimal Constant 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arnold A., Dolbeault J.: Refined convex Sobolev inequalities. J. Funct. Anal. 225, 337–351 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Arnold A., Markowich P., Toscani G., Unterreiter A.: On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker–Planck type equations. Comm. Part. Differ. Equ. 26, 43–100 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bakry, D., Émery, M.: Diffusions hypercontractives. Séminaire de probabilités, XIX, 1983/84, pp. 177–206. Lecture Notes in Math., vol. 1123. Springer, Berlin, 1985Google Scholar
  4. 4.
    Barthe F.: Levels of concentration between exponential and Gaussian. Ann. Fac. Sci. Toulouse Math. 10(6), 393–404 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Barthe F., Cattiaux P., Roberto C.: Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry. Rev. Mat. Iberoam. 22, 993–1067 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Barthe F., Roberto C.: Sobolev inequalities for probability measures on the real line. Studia Math. 159, 481–497 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Bartier J.-P., Dolbeault J.: Convex Sobolev inequalities and spectral gap. C. R. Math. Acad. Sci. Paris 342, 307–312 (2006)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Beckner W.: A generalized Poincaré inequality for Gaussian measures. Proc. Am. Math. Soc. 105, 397–400 (1989)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Blanchet A., Bonforte M., Dolbeault J., Grillo G., Vázquez J.: Hardy- Poincaré inequalities and applications to nonlinear diffusions. C. R. Math. Acad. Sci. Paris 344, 431–436 (2007)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Blanchet A., Bonforte M., Dolbeault J., Grillo G., Vázquez J.: Asymptotics of the fast diffusion equation via entropy estimates. Arch. Rational Mech. Anal. 191, 347–385 (2009)zbMATHCrossRefADSGoogle Scholar
  11. 11.
    Bobkov S.G., Götze F.: Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163, 1–28 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Bobkov S.G., Tetali P.: Modified logarithmic Sobolev inequalities in discrete settings. J. Theoret. Probab. 19, 289–336 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Bolley F., Gentil I.: Phi-entropy inequalities for diffusion semigroups. Preprint (2010). arXiv:0812.0800Google Scholar
  14. 14.
    Bonforte M., Dolbeault J., Grillo G., Vázquez J.: Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities. Preprint (2009). arXiv:0907.2986Google Scholar
  15. 15.
    Bouchard J.P., Mézard M.: Wealth condensation in a simple model of economy. Physica A 282, 536–545 (2000)zbMATHCrossRefADSGoogle Scholar
  16. 16.
    Càceres M.J., Toscani G.: Kinetic approach to long time behavior of linearized fast diffusion equations. J. Stat. Phys. 128, 883–925 (2007)zbMATHCrossRefMathSciNetADSGoogle Scholar
  17. 17.
    Carrillo J.A., Lederman C., Markowich P.A., Toscani G.: Poincaré inequalities for linearizations of very fast diffusion equations. Nonlinearity 15, 565–580 (2002)zbMATHCrossRefMathSciNetADSGoogle Scholar
  18. 18.
    Cattiaux P., Gentil I., Guillin A.: Weak logarithmic Sobolev inequalities and entropic convergence. Probab. Theory Relat. Fields 139, 563–603 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Chafaï D.: Entropies, convexity, and functional inequalities: on Φ-entropies and Φ-Sobolev inequalities. J. Math. Kyoto Univ. 44, 325–363 (2004)MathSciNetGoogle Scholar
  20. 20.
    Cox J.C., Ingersoll J.E., Ross S.A.: A theory of the term structure of interest rates. Econometrica 53, 385–407 (1985)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Denzler J., McCann R.: Fast diffusion to self-similarity: complete spectrum, long-time asymptotics, and numerology. Arch. Rational Mech. Anal. 175, 301–342 (2005)zbMATHCrossRefMathSciNetADSGoogle Scholar
  22. 22.
    Dolbeault J., Gentil I., Guillin A., Wang F.-Y.: L q-functional inequalities and weighted porous media equations. Potential Anal. 28, 35–59 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Dolbeault J., Nazaret B., Savaré G.: On the Bakry–Emery criterion for linear diffusions and weighted porous media equations. Commun. Math. Sci. 6, 477–494 (2008)zbMATHMathSciNetGoogle Scholar
  24. 24.
    Düring B., Matthes D., Toscani G.: Kinetic equations modelling wealth redistribution: a comparison of approaches. Phys. Rev. E 78, 050801 (2008)CrossRefGoogle Scholar
  25. 25.
    Feller W.: Diffusion processes in genetics. In: Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pp. 227–246. University of California Press, Berkeley and Los Angeles, 1951Google Scholar
  26. 26.
    Feller W.: Two singular diffusion problems. Ann. Math. 54, 173–182 (1951)CrossRefMathSciNetzbMATHGoogle Scholar
  27. 27.
    Gearhart W.B., Martelli M.: A blood cell population model, dynamical diseases, and chaos. UMAP J. 11, 309–338 (1990)Google Scholar
  28. 28.
    Henry D.: Geometric theory of semilinear parabolic equations. In: Lecture Notes in Mathematics, vol. 840. Springer, Berlin, 1981Google Scholar
  29. 29.
    Jüngel A., Matthes D.: An algorithmic construction of entropies in higher-order nonlinear PDEs. Nonlinearity 19, 633–659 (2006)zbMATHCrossRefMathSciNetADSGoogle Scholar
  30. 30.
    Jüngel A., Matthes D.: The Derrida-Lebowitz-Speer-Spohn equation: existence, nonuniqueness, and decay rates of the solutions. SIAM J. Math. Anal. 39, 1996–2015 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Lasota A.: Ergodic problems in biology. Astérisque (Soc. Math. France) 50, 239–250 (1977)MathSciNetGoogle Scholar
  32. 32.
    Latala R., Oleszkiewicz K.: Between Sobolev and Poincaré. In: Geometric Aspects of Functional Analysis, pp. 147–168. Lecture Notes in Math., vol. 1745. Springer, Berlin, 2000Google Scholar
  33. 33.
    Ledoux M.: On an integral criterion for hypercontractivity of diffusion semigroups and extremal functions. J. Funct. Anal. 105, 444–465 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Löfberg J.: YALMIP : A toolbox for Modeling and Optimization in MATLAB. In: Proceedings of the CACSD Conference, Taipei, Taiwan, 2004Google Scholar
  35. 35.
    Muckenhoupt B.: Hardy’s inequality with weights. Studia Math. 44, 31–38 (1972)zbMATHMathSciNetGoogle Scholar
  36. 36.
    Pareschi L., Toscani G.: Self-similarity and power-like tails in nonconservative kinetic models. J. Stat. Phys. 124, 747–779 (2006)zbMATHCrossRefMathSciNetADSGoogle Scholar
  37. 37.
    Rothaus O.S.: Hypercontractivity and the Bakry-Émery criterion for compact Lie groups. J. Funct. Anal. 65, 358–367 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Solomon S.: Stochastic Lotka-Volterra systems of competing auto-catalytic agents lead generically to truncated Pareto power wealth distribution, truncated Levy distribution of market returns, clustered volatility, booms and crashes. In: Computational Finance, vol. 97 (Eds. A.-P.N. Refenes, A.N. Burgess, and J.E. Moody). Kluwer Academic Publishers, Amsterdam, 1998Google Scholar
  39. 39.
    Tarski A.: A Decision Method for Elementary Algebra and Geometry, 2nd edn. University of California Press, Berkeley and Los Angeles, 1951Google Scholar
  40. 40.
    Vázquez J.-L.: The Porous Medium Equation. Mathematical Theory. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Daniel Matthes
    • 1
    Email author
  • Ansgar Jüngel
    • 1
  • Giuseppe Toscani
    • 2
  1. 1.Institute for Analysis and Scientific ComputingVienna University of TechnologyViennaAustria
  2. 2.Department of MathematicsUniversity of PaviaPaviaItaly

Personalised recommendations