Any O-simple dual semigroup is completely O-simple

1971 ◽  
Vol 2 (1) ◽  
pp. 90-92 ◽  
Author(s):  
Štefan Schwarz
Keyword(s):  
Dual Semigroup

scholarly journals Lagrangian calculus for nonsymmetric diffusion operators

2020 ◽  
Vol 13 (4) ◽  
pp. 361-383
Author(s):  
Christian Ketterer
Keyword(s):  
Ricci Curvature ◽  
Vector Fields ◽  
Probabilistic Approach ◽  
Gradient Estimates ◽  
Metric Measure Spaces ◽  
Curvature Bounds ◽  
Maximal Diameter ◽  
Measure Spaces ◽  
Evolution Variational Inequality ◽  
Dual Semigroup

AbstractWe characterize lower bounds for the Bakry–Emery Ricci tensor of nonsymmetric diffusion operators by convexity of entropy and line integrals on the {L^{2}}-Wasserstein space, and define a curvature-dimension condition for general metric measure spaces together with a square integrable 1-form in the sense of [N. Gigli, Nonsmooth differential geometry—an approach tailored for spaces with Ricci curvature bounded from below, Mem. Amer. Math. Soc. 251 2018, 1196, 1–161]. This extends the Lott–Sturm–Villani approach for lower Ricci curvature bounds of metric measure spaces. In generalized smooth context, consequences are new Bishop–Gromov estimates, pre-compactness under measured Gromov–Hausdorff convergence, and a Bonnet–Myers theorem that generalizes previous results by Kuwada [K. Kuwada, A probabilistic approach to the maximal diameter theorem, Math. Nachr. 286 2013, 4, 374–378]. We show that N-warped products together with lifted vector fields satisfy the curvature-dimension condition. For smooth Riemannian manifolds, we derive an evolution variational inequality and contraction estimates for the dual semigroup of nonsymmetric diffusion operators. Another theorem of Kuwada [K. Kuwada, Duality on gradient estimates and Wasserstein controls, J. Funct. Anal. 258 2010, 11, 3758–3774], [K. Kuwada, Space-time Wasserstein controls and Bakry–Ledoux type gradient estimates, Calc. Var. Partial Differential Equations 54 2015, 1, 127–161] yields Bakry–Emery gradient estimates.

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