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.
Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows