Equivalent Harnack and Gradient Inequalities for Pointwise Curvature   Lower Bound
  By using a coupling method, an explicit log-Harnack inequality with local geometry quantities is established for (sub-Markovian) diffusion semigroups on a Riemannian manifold (possibly with boundary). This inequality as well as the consequent $L^2$-gradient inequality, are proved to be equivalent to the pointwise curvature lower bound condition together with the convexity or absence of the boundary. Some applications of the log-Harnack inequality are also introduced. 
Marc Arnaudon, Anton Thalmaier, Feng-Yu Wang
math.DG math.PR