Osaka Journal of Mathematics

Ricci curvature of Markov chains on Polish spaces revisited

Fu-Zhou Gong, Yuan Liu, and Zhi-Ying Wen

Full-text: Open access

Abstract

Recently, Y. Ollivier defined the Ricci curvature of Markov chains on Polish spaces via the contractivity of transition kernels under the $L^{1}$ Wasserstein metric. In this paper, we will discuss further the spectral gap, entropy decay, and logarithmic Sobolev inequality for the $\lambda$-range gradient operator. As an application, given resistance forms (i.e. symmetric Dirichlet forms with finite effective resistance) on fractals, we can construct Markov chains with positive Ricci curvature, which yields the Gaussian-then-exponential concentration of invariant distributions for Lipschitz test functions.

Article information

Source
Osaka J. Math., Volume 50, Number 2 (2013), 491-502.

Dates
First available in Project Euclid: 21 June 2013

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1371833497

Mathematical Reviews number (MathSciNet)
MR3080812

Zentralblatt MATH identifier
1283.60102

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 47D07: Markov semigroups and applications to diffusion processes {For Markov processes, see 60Jxx}
Secondary: 28A80: Fractals [See also 37Fxx] 31C25: Dirichlet spaces

Citation

Gong, Fu-Zhou; Liu, Yuan; Wen, Zhi-Ying. Ricci curvature of Markov chains on Polish spaces revisited. Osaka J. Math. 50 (2013), no. 2, 491--502. https://projecteuclid.org/euclid.ojm/1371833497


Export citation

References

  • S. Aida and D. Stroock: Moment estimates derived from Poincaré and logarithmic Sobolev inequalities, Math. Res. Lett. 1 (1994), 75–86.
  • M.T. Barlow and R.F. Bass: Brownian motion and harmonic analysis on Sierpinski carpets, Canad. J. Math. 51 (1999), 673–744.
  • F. Barthe and C. Roberto: Modified logarithmic Sobolev inequalities on $\mathbb{R}$, Potential Anal. 29 (2008), 167–193.
  • M. Fukushima, Y. Oshima and M. Takeda: Dirichlet Forms and Symmetric Markov Processes, de Gruyter, Berlin, 1994.
  • J. Kigami: Analysis on Fractals, Cambridge Univ. Press, Cambridge, 2001.
  • \begingroup S. Kusuoka and X.Y. Zhou: Dirichlet forms on fractals: Poincaré constant and resistance, Probab. Theory Related Fields 93 (1992), 169–196. \endgroup
  • M. Ledoux: The Concentration of Measure Phenomenon, Mathematical Surveys and Monographs 89, Amer. Math. Soc., Providence, RI, 2001.
  • Z.-M. Ma and M. Röckner: Introduction to the Theory of (Non-Symmetric) Dirchlet Forms, Springer-Verlag, Berlin, 1992.
  • Y. Ollivier: Ricci curvature of Markov chains on metric spaces, J. Funct. Anal. 256 (2009), 810–864.
  • K.T. Sturm: Diffusion processes and heat kernels on metric spaces, Ann. Probab. 26 (1998), 1–55.
  • T. Szarek: Feller processes on nonlocally compact spaces, Ann. Probab. 34 (2006), 1849–1863.
  • C. Villani: Topics in Optimal Transportation, Graduate Studies in Mathematics 58, Amer. Math. Soc., Providence, RI, 2003.