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arXiv · 1209.6161

Equivalent Harnack and Gradient Inequalities for Pointwise Curvature Lower Bound

Marc Arnaudon·Anton Thalmaier·Feng-Yu Wang
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Abstract

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.

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Marc Arnaudon, Anton Thalmaier, Feng-Yu Wang. 2012-09-27. Equivalent Harnack and Gradient Inequalities for Pointwise Curvature Lower Bound. https://arxiv.org/abs/1209.6161

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