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Curvature of Nonlocal Markov Generators

Bakry’s curvature-dimension condition will be extended to certain nonlocal Markov generators. In particular this gives rise to a possible notion of curvature for graphs. 1. Definition of Curvature Let (Ω, μ) be a probability space and L a self-adjoint negative but not necessarily bounded operator on L2(μ) given by Lf(x) := ∫ (f(y)− f(x))K(x, y)μ(dy) (1) where K is a non negative symmetric kernel. Obviously L remains unchanged if we change K on the diagonal. By Pt = e we denote the continuous contraction semigroup on L2(μ) with generator L. We will assume that Pt is ergodic and that there exists an algebra A ⊆ n domL of bounded functions which is a form core of L. Then the Beurling–Deny condition implies that Pt is a symmetric Markov semigroup, i.e., Pt preserves positivity and extends to a continuous contraction semigroup on Lp(μ) for all 1 ≤ p 0. Let us say a a word about the motivation for this definition. Assume L is the Laplacian on a Riemannian manifold, then