Gradient Estimate and Harnack Inequality on Non-Compact Riemannian   Manifolds
  A new type of gradient estimate is established for diffusion semigroups on non-compact complete Riemannian manifolds. As applications, a global Harnack inequality with power and a heat kernel estimate are derived for diffusion semigroups on arbitrary complete Riemannian manifolds. 
Marc Arnaudon (LMA), Anton Thalmaier, Feng-Yu Wang