## Osaka Journal of Mathematics

### Ricci curvature of Markov chains on Polish spaces revisited

#### 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

https://projecteuclid.org/euclid.ojm/1371833497

Mathematical Reviews number (MathSciNet)
MR3080812

Zentralblatt MATH identifier
1283.60102

#### 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

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