Non-negative Ollivier curvature on graphs, reverse Poincar\'e inequality, Buser inequality, Liouville property, Harnack inequality and eigenvalue estimates

DOI10.1016/J.MATPUR.2022.12.007arXiv1907.13514MaRDI QIDQ6322961

Florentin Münch

Publication date: 31 July 2019

Abstract: We prove that for combinatorial graphs with non-negative Ollivier curvature, one has [ |P_t mu - P_t

u|_1 leq frac{W_1(mu,

u)}{sqrt{t}} ] for all probability measures $mu,u$ where $P_t$ is the heat semigroup and $W_1$ is the $el{l}_1$-Wasserstein distance. This turns out to be an equivalent formulation of a version of reverse Poincar'e inequality. Furthermore, this estimate allows us to prove Buser inequality, Liouville property and the the eigenvalue estimate $lambd{a}_{1}geqlog(2)/operatorname{diam}^2$.

Mathematics Subject Classification ID

Heat equation (35K05) Methods of global Riemannian geometry, including PDE methods; curvature restrictions (53C21) Continuous-time Markov processes on discrete state spaces (60J27)

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