Lagrangian calculus for nonsymmetric diffusion operators

We characterize lower bounds for the Bakry-Emery Ricci tensor of nonsymmetric diffusion operators by convexity of entropy 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 \cite{giglinonsmooth}. 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 \cite{kuwadamaximaldiameter}. 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 \cite{kuwadaduality, kuwadaspacetime} yields Bakry-Emery gradient estimates.