Curvature of Nonlocal Markov Generators

Summary

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

where K is a non negative symmetric kernel. Obviously L remains unchanged if we change K on the diagonal. By P ^t = e^tL we denote the continuous contraction semigroup on L₂(μ) with generator L. We will assume that P_t is ergodic and that there exists an algebra A⊆∩_n dom Lⁿ of bounded functions which is a form core of L. Then the Beurling-Deny condition implies that P_t is a symmetric Markov semigroup, i.e., P^t preserves positivity and extends to a continuous contraction semigroup on L_p(μ) for all 1 ≤p < ∞. We will also assume that A is stable under P_t.