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Modified curvatures on manifolds with boundary and applications. (English) Zbl 1308.58022

Relations between curvature and heat semigroup, in the spirit of the celebrated Bakry-Emery criterion, were extensively studied on a closed Riemannian manifold. In order to extend such studies to a reflecting diffusion process and its semigroup \(P_t\), to the framework of a manifold with boundary, the author introduces new curvature operators taking the second fundamental form of the boundary into account.
Relying on this, the author obtains the equivalence of some estimates relating to \(|\nabla P_t|\) with the boundedness by below of his modified Ricci curvature operator.
As a consequence, under such assumption, the author derives functional inequalities relating to \(P_t\), successively of Poincaré, log-Harnack, log-Sobolev and entropic type.

MSC:

58J65 Diffusion processes and stochastic analysis on manifolds
58J32 Boundary value problems on manifolds
60J60 Diffusion processes
47D07 Markov semigroups and applications to diffusion processes
35K05 Heat equation
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