Abstract
In this paper,we investigate the coarse Ricci curvature on metric spaces with random walks. There exists no canonical random walk on metric space with a reference measure. However, we prove that a Bishop–Gromov inequality gives a lower bound of coarse Ricci curvature with respect to a random walk called an \(r\)-step random walk. The lower bound does not coincide with the constant corresponding to curvature in Bishop–Gromov inequality. As a corollary, we obtain a lower bound of coarse Ricci curvature with respect to an \(r\)-step random walk for a metric measure space satisfying the curvature-dimension condition. Moreover we give an important example, Heisenberg group, which does not satisfy the curvature-dimension condition for any constant but has a lower bound of coarse Ricci curvature. We also have an estimate of the eigenvalues of the Laplacian by a lower bound of coarse Ricci curvature.
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References
Ambrosio, L., Gigli, N., Savaré, G.: Metric measure spaces with Riemannian Ricci curvature bounded from below. arXiv:1109.0222
Bauer, J.J.F., Shiping, L: Ollivier-Ricci curvature and the spectrum of the normalized graph Laplace operator. arXiv: 1105.3803v1
Erbar, M.: The heat equation on manifolds as a gradient flow in the Wasserstein space. Ann. Inst. Henri Poincar’e Probab. Stat. 46(1), 1–23 (2010) (English, with English and French summaries)
Gigli, N., Kuwada, K., Ohta, S.-i: Heat flow on Alexandrov spaces. arXiv:1008.1319
Jordan, R., Kinderlehrer, D., Otto, F.: The variational formulation of the Fokker–Planck equation. SIAM J. Math. Anal. 29(1), 1–17 (1998)
Juillet, N.: Geometric inequalities and generalized Ricci bounds in the Heisenberg group. Int. Math. Res. Not. IMRN 13, 2347–2373 (2009)
Jost, J., Liu, S.: Ollivier’s Ricci curvature, local clustering and curvature dimension inequalities on graphs. arXiv: 1103.4037v2
Lin, Y., Yau, S.-T.: Ricci curvature and eigenvalue estimate on locally finite graphs. Math. Res. Lett 17(2), 343–356 (2010)
Lott, J., Villani, C.: Ricci curvature for metric-measure spaces via optimal transport. Ann. Math. (2) 169(3), 903–991 (2009). doi:10.4007/annals.2009.169.903
Ohta, S-i: On the measure contraction property of metric measure spaces. Comment. Math. Helv. 82(4), 805–828 (2007). doi:10.4171/CMH/110
Ollivier, Y.: Ricci curvature of Markov chains on metric spaces. J. Funct. Anal. 256(3), 810–864 (2009). doi:10.1016/j.jfa.2008.11.001
Savaré, G.: Gradient flows and diffusion semigroups in metric spaces under lower. C. R. Math. Acad. Sci. Paris 345(3), 151–154 (2007). doi:10.1016/j.crma.2007.06.018 (English, with English and French summaries)
Sturm, K.T.: On the geometry of metric measure spaces. I. Acta Math. 196(1), 65–131 (2006). doi:10.1007/s11511-006-0002-8
Sturm, K.-T.: On the geometry of metric measure spaces. II. Acta Math. 196(1), 133–177 (2006). doi:10.1007/s11511-006-0003-7
Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics. Mathematical Society, Providence, RI (2003)
Villani, C.: Optimal Transport: Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (2009)
von Renesse, M.-K., Sturm, K.-T.: Transport inequalities. gradient estimates,en- tropy, and Ricci curvature. Comm. Pure Appl. Math. 58(7), 923–940 (2005)
Acknowledgments
The author is grateful to Professor Nicola Gigli for pointing out Proposition 3.3,Professor Shin-ichi Ohta for helpful comments, Professor Kazuhiro Kuwae for valuable comments and Professor Takashi Shioya for reading this paper and giving useful advices. The author is partly supported by the Grant-in-Aid for JSPS Fellows, The Ministry of Education, Culture, Sports, Science and Technology, Japan.
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Kitabeppu, Y. Lower bound of coarse Ricci curvature on metric measure spaces and eigenvalues of Laplacian. Geom Dedicata 169, 99–107 (2014). https://doi.org/10.1007/s10711-013-9844-3
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DOI: https://doi.org/10.1007/s10711-013-9844-3