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

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.

Entropy-information inequalities under curvature-dimension conditions for continuous-time Markov chains

“…The fact that the Bakry-Émery condition is necessary for CD Υ with a quadratic CD-function also motivates to study the former condition. In particular, we refer to [28], where CD(0, n) has been studied for a large class of operators with long range jumps and state space Z.…”

The fractional Laplacian has infinite dimension

“…The key idea, which is motivated by [SWZ19, Theorem 3.1], is to consider functions which are smooth enough near x = 0 and equal to |x| β−ε for |x| ≥ 1 with small ε > 0. In the very recent work [SWZ19], which deals with CD-inequalities for nonlocal discrete operators on the lattice Z, the described family of functions is used to prove that CD(0, N ) fails for all finite N > 0 for a certain class of operators with power type kernel, in particular for all powers −(−∆ disc ) β 2 of the discrete Laplacian ∆ disc on Z with β ∈ (0, 2). By considering functions which depend only on one real variable, the one-dimensional counterexample can be extended to the multi-dimensional case.…”

Li-Yau and Harnack inequalities via curvature-dimension conditions for discrete long-range jump operators including the fractional discrete Laplacian

“…What they also share is that they work on locally finite, if not finite, state spaces. In [16] the authors have studied the classical condition CD(0, d) for a special case of locally infinite graphs. Although Li-Yau estimates have not been addressed in [16] this work serves as the main inspiration for the present paper alongside [10,18,20].…”

“…As explained before, the purpose of [16] was to study the validity of the Bakry-Émery condition CD(0, d), which, by definition, is satisfied if…”

“…where u, v are elements of a suitable algebra of real-valued functions. The main positive result of [16] is that whenever the kernel k is non-increasing (on N) and has finite second moment (i.e.…”