The Annals of Applied Probability

Entropic Ricci curvature bounds for discrete interacting systems

Max Fathi and Jan Maas

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We develop a new and systematic method for proving entropic Ricci curvature lower bounds for Markov chains on discrete sets. Using different methods, such bounds have recently been obtained in several examples (e.g., 1-dimensional birth and death chains, product chains, Bernoulli–Laplace models, and random transposition models). However, a general method to obtain discrete Ricci bounds had been lacking. Our method covers all of the examples above. In addition, we obtain new Ricci curvature bounds for zero-range processes on the complete graph. The method is inspired by recent work of Caputo, Dai Pra and Posta on discrete functional inequalities.

Article information

Source
Ann. Appl. Probab., Volume 26, Number 3 (2016), 1774-1806.

Dates
Received: January 2015
Revised: July 2015
First available in Project Euclid: 14 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1465905019

Digital Object Identifier
doi:10.1214/15-AAP1133

Mathematical Reviews number (MathSciNet)
MR3513606

Zentralblatt MATH identifier
1345.60076

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Discrete Ricci curvature transport metrics functional inequalities birth-death processes zero-range processes Bernoulli–Laplace model random transposition model

Citation

Fathi, Max; Maas, Jan. Entropic Ricci curvature bounds for discrete interacting systems. Ann. Appl. Probab. 26 (2016), no. 3, 1774--1806. doi:10.1214/15-AAP1133. https://projecteuclid.org/euclid.aoap/1465905019


Export citation

References

  • [1] Bakry, D. and Émery, M. (1985). Diffusions hypercontractives. In Séminaire de Probabilités, XIX, 1983/84. Lecture Notes in Math. 1123 177–206. Springer, Berlin.
  • [2] Bobkov, S. G. and Tetali, P. (2006). Modified logarithmic Sobolev inequalities in discrete settings. J. Theoret. Probab. 19 289–336.
  • [3] Bonciocat, A.-I. and Sturm, K.-T. (2009). Mass transportation and rough curvature bounds for discrete spaces. J. Funct. Anal. 256 2944–2966.
  • [4] 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.
  • [5] 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.
  • [6] Caputo, P. and Posta, G. (2007). Entropy dissipation estimates in a zero-range dynamics. Probab. Theory Related Fields 139 65–87.
  • [7] Chow, S.-N., Huang, W., Li, Y. and Zhou, H. (2012). Fokker–Planck equations for a free energy functional or Markov process on a graph. Arch. Ration. Mech. Anal. 203 969–1008.
  • [8] Dai Pra, P. and Posta, G. (2013). Entropy decay for interacting systems via the Bochner–Bakry–Émery approach. Electron. J. Probab. 18 52.
  • [9] Erbar, M. and Maas, J. (2012). Ricci curvature of finite Markov chains via convexity of the entropy. Arch. Ration. Mech. Anal. 206 997–1038.
  • [10] Erbar, M. and Maas, J. (2014). Gradient flow structures for discrete porous medium equations. Discrete Contin. Dyn. Syst. 34 1355–1374.
  • [11] Erbar, M., Maas, J. and Tetali, P. (2015). Discrete Ricci curvature bounds for Bernoulli–Laplace and random transposition models. Annales Fac. Sci. Toulouse (6) 24 781–800.
  • [12] Gao, F. and Quastel, J. (2003). Exponential decay of entropy in the random transposition and Bernoulli–Laplace models. Ann. Appl. Probab. 13 1591–1600.
  • [13] Gigli, N. and Maas, J. (2013). Gromov–Hausdorff convergence of discrete transportation metrics. SIAM J. Math. Anal. 45 879–899.
  • [14] Goel, S. (2004). Modified logarithmic Sobolev inequalities for some models of random walk. Stochastic Process. Appl. 114 51–79.
  • [15] Jordan, R., Kinderlehrer, D. and Otto, F. (1998). The variational formulation of the Fokker–Planck equation. SIAM J. Math. Anal. 29 1–17.
  • [16] Lin, Y. and Yau, S.-T. (2010). Ricci curvature and eigenvalue estimate on locally finite graphs. Math. Res. Lett. 17 343–356.
  • [17] Lott, J. and Villani, C. (2009). Ricci curvature for metric-measure spaces via optimal transport. Ann. of Math. (2) 169 903–991.
  • [18] Maas, J. (2011). Gradient flows of the entropy for finite Markov chains. J. Funct. Anal. 261 2250–2292.
  • [19] Mielke, A. (2011). A gradient structure for reaction–diffusion systems and for energy-drift-diffusion systems. Nonlinearity 24 1329–1346.
  • [20] Mielke, A. (2013). Geodesic convexity of the relative entropy in reversible Markov chains. Calc. Var. Partial Differential Equations 48 1–31.
  • [21] Morris, B. (2006). Spectral gap for the zero range process with constant rate. Ann. Probab. 34 1645–1664.
  • [22] Ollivier, Y. (2009). Ricci curvature of Markov chains on metric spaces. J. Funct. Anal. 256 810–864.
  • [23] Ollivier, Y. (2013). A visual introduction to Riemannian curvatures and some discrete generalizations. In Analysis and Geometry of Metric Measure Spaces. CRM Proc. Lecture Notes 56 197–220. Amer. Math. Soc., Providence, RI.
  • [24] Sturm, K.-T. (2006). On the geometry of metric measure spaces. I. Acta Math. 196 65–131.