Equivalent Harnack and Gradient Inequalities for Pointwise Curvature Lower Bound

Marc Arnaudon (1) , Anton Thalmaier (2) , Feng-Yu Wang (3)

1 IMB - Institut de Mathématiques de Bordeaux
2 Institute of Mathematics
3 SMS - School of Mathematical Sciences [Beijing]

By using a coupling method, an explicit log-Harnack inequality with local geometry quantities is established for (sub-Markovian) di usion 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.

Mots clés: Log-Harnack inequality; Riemannian manifold; diffusion process

Domaines: Probabilités [math.PR]

Marc Arnaudon, Anton Thalmaier, Feng-Yu Wang. Equivalent Harnack and Gradient Inequalities for Pointwise Curvature Lower Bound. Bulletin des Sciences Mathématiques, 2014, 138 (5), pp.643-655. ⟨hal-01057529⟩