Ricci tensor for diffusion operators and curvature-dimension inequalities under conformal transformations and time changes

Abstract

Within the Γ2-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–Émery 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.