Generalized Curvature Condition for Subelliptic Diffusion Processes

By using a general version of curvature condition, derivative inequalities are established for a large class of subelliptic diffusion semigroups. As applications, the Harnack/cost-entropy/cost-variance inequalities for the diffusion semigroups, and the Poincar\'e/log-Sobolev inequalities for the associated Dirichlet forms in the symmetric case, are derived. Our results largely generalize and partly improve the corresponding ones obtained recently in [BB].