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

2014/01/03 by Karl‐Theodor Sturm, Sturm, Karl-Theodor
Mathematics · Physics and Astronomy · #31Dxx #31Exx #35Jxx #51Fxx #53xxx #58Jxx #60Jxx #Advanced Differential Geometry Research #Black Holes and Theoretical Physics #FOS: Mathematics #Geometric Analysis and Curvature Flows #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.1401.0687

openalex publication_date 2014/01/03 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28

Abstract

Within the \Γ₂-calculus of Bakry and Ledoux, we define the Ricci\ntensor induced by a diffusion operator, we deduce precise formulas for its\nbehavior under drift transformation, time change and conformal transformation,\nand we derive new transformation results for the curvature-dimension conditions\nof Bakry-Emery as well as for those of Lott-Sturm-Villani. Our results are\nbased on new identities and sharp estimates for the N-Ricci tensor and for\nthe Hessian. In particular, we obtain Bochner's formula in the general setting.\n

Related

• Curvature-dimension conditions for diffusions under time change
• Lagrangian calculus for nonsymmetric diffusion operators
• Sub-Riemannian Ricci curvature via generalized Gamma z calculus
• Curvature tensor under the Ricci-Harmonic flow
• Self-improvement of the Bakry-Emery criterion for Poincaré inequalities and Wasserstein contraction using variable curvature bounds