The Annals of Applied Probability

Discrete Beckner inequalities via the Bochner–Bakry–Emery approach for Markov chains

Ansgar Jüngel and Wen Yue

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Abstract

Discrete convex Sobolev inequalities and Beckner inequalities are derived for time-continuous Markov chains on finite state spaces. Beckner inequalities interpolate between the modified logarithmic Sobolev inequality and the Poincaré inequality. Their proof is based on the Bakry–Emery approach and on discrete Bochner-type inequalities established by Caputo, Dai Pra and Posta and recently extended by Fathi and Maas for logarithmic entropies. The abstract result for convex entropies is applied to several Markov chains, including birth-death processes, zero-range processes, Bernoulli–Laplace models, and random transposition models, and to a finite-volume discretization of a one-dimensional Fokker–Planck equation, applying results by Mielke.

Article information

Source
Ann. Appl. Probab., Volume 27, Number 4 (2017), 2238-2269.

Dates
Received: November 2015
Revised: August 2016
First available in Project Euclid: 30 August 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1504080032

Digital Object Identifier
doi:10.1214/16-AAP1258

Mathematical Reviews number (MathSciNet)
MR3693525

Zentralblatt MATH identifier
1374.60144

Subjects
Primary: 60J27: Continuous-time Markov processes on discrete state spaces 39B62: Functional inequalities, including subadditivity, convexity, etc. [See also 26A51, 26B25, 26Dxx] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Time-continuous Markov chain functional inequality entropy decay discrete Beckner inequality stochastic particle systems

Citation

Jüngel, Ansgar; Yue, Wen. Discrete Beckner inequalities via the Bochner–Bakry–Emery approach for Markov chains. Ann. Appl. Probab. 27 (2017), no. 4, 2238--2269. doi:10.1214/16-AAP1258. https://projecteuclid.org/euclid.aoap/1504080032


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References

  • [1] Ané, C., Blachère, S., Chafaï, D., Fougères, P., Gentil, I., Malrieu, F., Roberto, C. and Scheffer, G. (2000). Sur les inégalités de Sobolev logarithmiques. Panoramas et Synthèses [Panoramas and Syntheses] 10. Société Mathématique de France, Paris.
  • [2] Arnold, A., Markowich, P., Toscani, G. and Unterreiter, A. (2001). On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker–Planck type equations. Comm. Partial Differential Equations 26 43–100.
  • [3] Bakry, D. and Émery, M. (1985). Diffusions hypercontractives. In Séminaire de Probabilités, XIX, 1983/84. Lecture Notes in Math. 1123 177–206. Springer, Berlin.
  • [4] Bakry, D., Gentil, I. and Ledoux, M. (2014). Analysis and Geometry of Markov Diffusion Operators. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 348. Springer, Cham.
  • [5] Beckner, W. (1989). A generalized Poincaré inequality for Gaussian measures. Proc. Amer. Math. Soc. 105 397–400.
  • [6] Bobkov, S. and Tetali, P. (2006). Modified logarithmic Sobolev inequalities in discrete settings. J. Theoret. Probab. 19 289–336.
  • [7] Bobkov, S. G. and Ledoux, M. (1998). On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures. J. Funct. Anal. 156 347–365.
  • [8] Bochner, S. (1946). Vector fields and Ricci curvature. Bull. Amer. Math. Soc. 52 776–797.
  • [9] Boudou, A.-S., Caputo, P., Dai Pra, P. and Posta, G. (2006). Spectral gap estimates for interacting particle systems via a Bochner-type identity. J. Funct. Anal. 232 222–258.
  • [10] Burdzy, K. and Kendall, W. (2000). Efficient Markovian couplings: Examples and counterexamples. Ann. Appl. Probab. 10 362–409.
  • [11] Caputo, P., Dai Pra, P. and Posta, G. (2009). Convex entropy decay via the Bochner–Bakry–Emery approach. Ann. Inst. Henri Poincaré Probab. Stat. 45 734–753.
  • [12] Chainais-Hillairet, C., Jüngel, A. and Schuchnigg, S. (2016). Entropy-dissipative discretization of nonlinear diffusion equations and discrete Beckner inequalities. Math. Model. Numer. Anal. 50 135–162.
  • [13] Chen, G.-Y. and Saloff-Coste, L. (2014). Spectral computations for birth and death chains. Stochastic Process. Appl. 124 848–882.
  • [14] Chen, M. (1996). Estimation of spectral gap for Markov chains. Acta Math. Sinica (N.S.) 12 337–360.
  • [15] Chen, M. F. (2003). Variational formulas of Poincaré-type inequalities for birth-death processes. Acta Math. Sin. (Engl. Ser.) 19 625–644.
  • [16] Daneri, S. and Savaré, G. (2008). Eulerian calculus for the displacement convexity in the Wasserstein distance. SIAM J. Math. Anal. 40 1104–1122.
  • [17] del Molino, L., Chleboun, P. and Grosskinsky, S. (2012). Condensation in randomly perturbed zero-range processes. J. Phys. A: Math. Theor. 45 205001.
  • [18] Diaconis, P. and Saloff-Coste, L. (1996). Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 695–750.
  • [19] Fathi, M. and Maas, J. (2016). Entropic Ricci curvature bounds for discrete interacting systems. Ann. Appl. Probab. 26 1774–1806.
  • [20] Fjordholm, U. S., Mishra, S. and Tadmor, E. (2012). Arbitrarily high-order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws. SIAM J. Numer. Anal. 50 544–573.
  • [21] Furihata, D. and Matsuo, T. (2011). The Discrete Variational Method. A Structure-Preserving Numerical Method for Partial Differential Equations. Chapman & Hall/CRC, Boca Raton, FL.
  • [22] Gao, F. and Quastel, J. (2003). Exponential decay of entropy in the random transposition and Bernoulli–Laplace models. Ann. Appl. Probab. 13 1591–1600.
  • [23] Guionnet, A. and Zegarlinski, B. (2003). Lectures on logarithmic Sobolev inequalities. In Séminaire de Probabilités 36 (J. Azéma et al., eds.). Lect. Notes Math. 1801 1–134. Springer, Berlin.
  • [24] Jerrum, M., Son, J.-B., Tetali, P. and Vigoda, E. (2004). Elementary bounds on Poincaré and log-Sobolev constants for decomposable Markov chains. Ann. Appl. Probab. 14 1741–1765.
  • [25] Johnson, O. (2016). A discrete log-Sobolev inequality under a Bakry–Emery type condition. Ann. Inst. Henri Poincaré Probab. Stat. To appear. Available at arXiv:1507.06268.
  • [26] Maas, J. (2016). Personal communication.
  • [27] Miclo, L. (1999). An example of application of discrete Hardy’s inequalities. Markov Process. Related Fields 5 319–330.
  • [28] Mielke, A. (2013). Geodesic convexity of the relative entropy in reversible Markov chains. Calc. Var. Part. Diff. Eqs. 48 1–31.
  • [29] Montenegro, R. and Tetali, P. (2006). Mathematical aspects of mixing times in Markov chains. Found. Trends Theor. Comput. Sci. 1 x+121.
  • [30] Morris, B. (2006). Spectral gap for the zero range process with constant rate. Ann. Probab. 34 1645–1664.
  • [31] Wang, F.-Y. (2005). Functional Inequalities, Markov Semigroups and Spectral Theory. Science Press, Beijing.