Li-yau and harnack inequalities via curvature-dimension conditions for discrete long-range jump operators including the fractional discrete laplacian

We consider operators of the form Lu(x) = E-y is an element of Z k(x - y)(u(y) - u(x)) on the one-dimensional lattice with symmetric, integrable kernel k. We prove several results stating that under certain conditions on the kernel the operator L satisfies the curvature-dimension condition CD Upsilon(0, F) (recently introduced by the last two authors) with some CD-function F, where attention is also paid to the asymptotic properties of F (exponential growth at infinity and power-type behaviour near zero). We show that CD Upsilon(0, F) implies a LiYau inequality for positive solutions of the heat equation associated with the operator L. The Li-Yau estimate in turn leads to a Harnack inequality, from which we also derive heat kernel bounds. Our results apply to a wide class of operators including the fractional discrete Laplacian.