Generalized Gamma $z$ calculus via sub-Riemannian density manifold

We generalize the Gamma $z$ calculus to study degenerate drift-diffusion processes, where $z$ stands for the extra directions introduced into the degenerate system. Based on the new calculus, we establish the curvature dimension bound for general sub-Riemannian manifolds, which does not require the commutative iteration of Gamma and Gamma z operator and goes beyond the step two condition. It allows us to analyze the convergence properties of degenerate drift-diffusion processes and prove the entropy dissipation rate and several functional inequalities in sub-Riemannian manifolds. Several examples are provided. The new Gamma $z$ calculus is motivated by optimal transport and density manifold. We embed the probability density space over sub-Riemannian manifold with the $L^2$ sub-Riemannian Wasserstein metric. We call it sub-Riemannian density manifold (SDM). We study the dynamical behavior of the degenerate Fokker-Planck equation as gradient flows in SDM. Our derivation builds an equivalence relation between Gamma z calculus and second-order calculus in SDM.