Abstract
Recently, Y. Ollivier defined the Ricci curvature of Markov chains on Polish spaces. In this paper, we will discuss further about the spectral gap, entropy decay, and logarithmic Sobolev inequality for the -range gradient operator. As an application, given resistance forms (i.e. symmetric Dirichlet forms with finite effective resistance) on frac- tal sets, we can construct Markov chains with positive Ricci curvature, which yields the Gaussian-then-exponential concentration of invariant probability measures for Lipschitz test functions.
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