Curvature-dimension inequalities for non-local operators in the discrete setting

Abstract

We study Bakry-Émery curvature-dimension inequalities for non-local operators on the one-dimensional lattice and prove that operators with finite second moment have finite dimension. Moreover, we show that a class of operators related to the fractional Laplacian fails to have finite dimension and establish both positive and negative results for operators with sparsely supported kernels. Moreover, a large class of operators is shown to have no positive curvature. The results correspond to CD inequalities on locally infinite graphs.

27 pages, 4 Figures. Accepted for publication in Calc. Var. PDE