Osaka Journal of Mathematics

Ricci curvature of Markov chains on Polish spaces revisited

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

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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

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

First available in Project Euclid: 21 June 2013

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Zentralblatt MATH identifier

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


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.

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