No abstract

bstract. 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. Furthermore, a large class of operators is shown to have no positive curvature. The results correspond to CD inequalities on locally infinite graphs.

We show that the fractional Laplacian on R d fails to satisfy the Bakry-Émery curvaturedimension inequality CD(κ, N ) for all curvature bounds κ ∈ R and all finite dimensions N > 0.