Ricci Tensor for Diffusion Operators and Curvature-Dimension Inequalities under Conformal Transformations and Time Changes

Within the $\Gamma_2$-calculus of Bakry and Ledoux, we define the Ricci tensor induced by a diffusion operator, we deduce precise formulas for its behavior under drift transformation, time change and conformal transformation, and we derive new transformation results for the curvature-dimension conditions of Bakry-Emery as well as for those of Lott-Sturm-Villani. Our results are based on new identities and sharp estimates for the $N$-Ricci tensor and for the Hessian. In particular, we obtain Bochner's formula in the general setting.