Log-Sobolev inequalities: Different roles of Ric and Hess

Details Log-Sobolev inequalities: Different roles of Ric and Hess Log-Sobolev inequalities: Different roles of Ric and Hess

2007-12-19

arxiv.org/abs/0712.3143v2

Annals of Probability 2009, Vol. 37, No. 4, 1587-1604

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/08-AOP444 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/08-AOP444

Let $P_t$ be the diffusion semigroup generated by $L:=\Delta +\nabla V$ on a complete connected Riemannian manifold with $\operatorname {Ric}\ge-(\sigma ^2\rho_o^2+c)$ for some constants $\sigma, c>0$ and $\rho_o$ the Riemannian distance to a fixed point. It is shown that $P_t$ is hypercontractive, or the log-Sobolev inequality holds for the associated Dirichlet form, provided $-\operatorname {Hess}_V\ge\delta$ holds outside of a compact set for some constant $\delta >(1+\sqrt{2})\sigma \sqrt{d-1}.$ This indicates, at least in finite dimensions, that $\operatorname {Ric}$ and $-\operatorname {Hess}_V$ play quite different roles for the log-Sobolev inequality to hold. The supercontractivity and the ultracontractivity are also studied.

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