Equivalent Harnack and Gradient Inequalities for Pointwise Curvature Lower Bound
by Marc Arnaudon, Anton Thalmaier and Feng-Yu Wang
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 L²-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.
Bull. Sci. Math. 138 (2014) 643-655
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http://dx.doi.org/10.1016/j.bulsci.2013.11.001
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