Journal of Statistical Physics

, Volume 176, Issue 5, pp 1172–1184 | Cite as

Exponential Decay of Rényi Divergence Under Fokker–Planck Equations

  • Yu CaoEmail author
  • Jianfeng Lu
  • Yulong Lu


We prove the exponential convergence to the equilibrium, quantified by Rényi divergence, of the solution of the Fokker–Planck equation with drift given by the gradient of a strictly convex potential. This extends the classical exponential decay result on the relative entropy for the same equation.


Rényi divergence Fokker–Planck equation Exponential convergence Gradient flow 



The work of YC and JL is supported in part by the National Science Foundation under grant DMS-1454939.


  1. 1.
    Ambrosio, L., Gigli, N., Savare, G.: Gradient Flows: In Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics, 2nd edn. ETH Zürich, Birkhäuser, Basel (2008)zbMATHGoogle 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. Commun. Partial Differ. Equ. 26(1–2), 43–100 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Atar, R., Chowdhary, K., Dupuis, P.: Robust bounds on risk-sensitive functionals via Rényi divergence. SIAM/ASA J. Uncertain. Quantif. 3(1), 18–33 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Audenaert, K.M.R., Datta, N.: \(\alpha \)-\(z\)-Rényi relative entropies. J. Math. Phys. 56(2), 022202 (2015)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bakry, D., Émery, M.: Diffusions hypercontractives. Séminaire de Probabilités de Strasbourg. Lecture Notes in Mathematics, vol. 19, pp. 177–206. Springer, New York (1985)Google Scholar
  6. 6.
    Bakry, D., Gentil, I., Ledoux, M.: Analysis and Geometry of Markov Diffusion Operators. Springer, New York (2014)zbMATHCrossRefGoogle Scholar
  7. 7.
    Bégin, L., Germain, P., Laviolette, F., Roy, J.F.: PAC-Bayesian bounds based on the Rényi divergence. In: AISTATS (2016)Google Scholar
  8. 8.
    Benamou, J.D., Brenier, Y.: A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math. 84(3), 375–393 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Brandão, F., Horodecki, M., Ng, N., Oppenheim, J., Wehner, S.: The second laws of quantum thermodynamics. Proc. Natl. Acad. Sci. USA 112(11), 3275–3279 (2015)ADSCrossRefGoogle Scholar
  10. 10.
    Cao, Y., Lu, J., Lu, Y.: Gradient flow structure and exponential decay of the sandwiched Rényi divergence for primitive Lindblad equations with GNS-detailed balance. J. Math. Phys. 60, 052202 (2019). ADSMathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Carlen, E.A., Frank, R.L., Lieb, E.H.: Inequalities for quantum divergences and the Audenaert-Datta conjecture. J. Phys. A 51(48), 483001 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Carlen, E.A., Maas, J.: An analog of the 2-Wasserstein metric in non-commutative probability under which the Fermionic Fokker–Planck equation is gradient flow for the entropy. Commun. Math. Phys. 331(3), 887–926 (2014)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Carlen, E.A., Maas, J.: Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance. J. Funct. Anal. 273(5), 1810–1869 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Carrillo, J., Toscani, G.: Rényi entropy and improved equilibration rates to self-similarity for nonlinear diffusion equations. Nonlinearity 27, 3159 (2014)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Dolbeault, J., Nazaret, B., Savaré, G.: A new class of transport distances between measures. Calc. Var. Partial Differ. Equ. 34(2), 193–231 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Dupuis, P., Katsoulakis, M.A., Pantazis, Y., Rey-Bellet, L.: Sensitivity Analysis for Rare Events Based on Rényi Divergence. arXiv:1805.06917 [math] (2018)
  17. 17.
    Erbar, M., Maas, J.: Gradient flow structures for discrete porous medium equations. Discrete Contin. Dyn. Syst. Ser. A 34(4), 1355–1374 (2014)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Frank, R.L., Lieb, E.H.: Monotonicity of a relative Rényi entropy. J. Math. Phys. 54(12), 122201 (2013)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Gilardoni, G.L.: On Pinsker’s and Vajda’s type inequalities for Csiszár’s \(f\)-divergences. IEEE Trans. Inf. Theory 56(11), 5377–5386 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Gross, L.: Hypercontractivity and logarithmic Sobolev inequalities for the Clifford-Dirichlet form. Duke Math. J. 42(3), 383–396 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Gross, L.: Logarithmic Sobolev inequalities. Am. J. Math. 97(4), 1061 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Harremoës, P.: Interpretations of Rényi entropies and divergences. Phys. A 365(1), 57–62 (2006)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Jordan, R., Kinderlehrer, D., Otto, F.: The variational formulation of the Fokker–Planck equation. SIAM J. Math. Anal. 29(1), 1–17 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Maas, J.: Gradient flows of the entropy for finite Markov chains. J. Funct. Anal. 261(8), 2250–2292 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Markowich, P.A., Villani, C.: On the trend to equilibrium for the Fokker-Planck equation: An interplay between physics and functional analysis. In: Physics and Functional Analysis, Matematica Contemporanea (SBM) 19, pp. 1–29 (1999)Google Scholar
  26. 26.
    Masi, M.: A step beyond Tsallis and Rényi entropies. Phys. Lett. A 338(3), 217–224 (2005)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Müller-Hermes, A., Franca, D.S.: Sandwiched Rényi convergence for quantum evolutions. Quantum 2, 55 (2018)CrossRefGoogle Scholar
  28. 28.
    Müller-Lennert, M., Dupuis, F., Szehr, O., Fehr, S., Tomamichel, M.: On quantum Rényi entropies: a new generalization and some properties. J. Math. Phys. 54(12), 122203 (2013)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Nielsen, F., Nock, R.: A closed-form expression for the Sharma-Mittal entropy of exponential families. J. Phys. A 45(3), 032003 (2012). arXiv:1105.3259 ADSMathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Commun. Partial Differ. Equ. 26(1–2), 101–174 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Petz, D.: Quasi-entropies for finite quantum systems. Rep. Math. Phys. 23(1), 57–65 (1986)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Raginsky, M., Sason, I.: Concentration of measure inequalities in information theory, communications, and coding. Found. Trends Commun. Inf. Theory 10(1–2), 1–246 (2013). arXiv:1212.4663 zbMATHCrossRefGoogle Scholar
  34. 34.
    Rényi, A.: On measures of entropy and information. In: Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics, pp. 547–561. University of California Press (1961)Google Scholar
  35. 35.
    Shayevitz, O.: On Rényi measures and hypothesis testing. In: Proceedings of 2011 IEEE ISIT, pp. 894–898. IEEE (2011)Google Scholar
  36. 36.
    Toscani, G.: Entropy production and the rate of convergence to equilibrium for the Fokker–Planck equation. Q. Appl. Math. 57(3), 521–541 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Tsallis, C.: Possible generalization of Boltzmann-Gibbs statistics. J Stat. Phys. 52(1), 479–487 (1988)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    van Erven, T., Harremos, P.: Rényi divergence and Kullback-Leibler divergence. IEEE Trans. Inf. Theory 60(7), 3797–3820 (2014)zbMATHCrossRefGoogle Scholar
  39. 39.
    Villani, C.: Topics in Optimal Transportation, vol. 58. American Mathematical Soc., Providence (2003)zbMATHGoogle Scholar
  40. 40.
    Villani, C.: Entropy production and convergence to equilibrium. In: Golse, F., Olla, S. (eds.) Entropy Methods for the Boltzmann Equation, pp. 1–70. Springer, New York (2008)zbMATHGoogle Scholar
  41. 41.
    Villani, C.: Optimal Transport: Old and New. Springer, New York (2009)zbMATHCrossRefGoogle Scholar
  42. 42.
    Wilde, M.M., Winter, A., Yang, D.: Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched Rényi relative entropy. Commun. Math. Phys. 331(2), 593–622 (2014)ADSzbMATHCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsDuke UniversityDurhamUSA
  2. 2.Department of Mathematics, Department of Physics, and Department of ChemistryDuke UniversityDurhamUSA

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