معادلات انتشار غیرخطی و شرایط انحنا در فضاهای اندازه‌گیری متریک (خاطرات انجمن ریاضی آمریکا)

The aim of this paper is to provide new characterizations of the curvature dimension condition in the context of metric measure spaces $(X,\mathsf d,\mathfrak m)$. On the geometric side, the authors'new approach takes into account suitable weighted action functionals which provide the natural modulus of $K$-convexity when one investigates the convexity properties of $N$-dimensional entropies. On the side of diffusion semigroups and evolution variational inequalities, the authors'new approach uses the nonlinear diffusion semigroup induced by the $N$-dimensional entropy, in place of the heat flow. Under suitable assumptions (most notably the quadraticity of Cheeger's energy relative to the metric measure structure) both approaches are shown to be equivalent to the strong $\mathrm {CD}^{•}(K,N)$ condition of Bacher-Sturm. Cover Title page Chapter 1. Introduction Chapter 2. Contraction and Convexity via Hamiltonian Estimates: an Heuristic Argument Part I . Nonlinear Diffusion Equations and Their Linearization in Dirichlet Spaces Chapter 3. Dirichlet Forms, Homogeneous Spaces and Nonlinear Diffusion 3.1. Dirichlet forms 3.2. Completion of quotient spaces w.r.t. a seminorm 3.3. Nonlinear diffusion Chapter 4. Backward and Forward Linearizations of Nonlinear Diffusion Part II . Continuity Equation and Curvature Conditions in Metric Measure Spaces Chapter 5. Preliminaries 5.1. Absolutely continuous curves, Lipschitz functions and slopes 5.2. The Hopf-Lax evolution formula 5.3. Measures, couplings, Wasserstein distance 5.4. W_{p}-absolutely continuous curves and dynamic plans 5.5. Metric measure spaces and the Cheeger energy 5.6. Entropy estimates of the quadratic moment and of the Fisher information along nonlinear diffusion equations 5.7. Weighted Γ-calculus Chapter 6. Absolutely Continuous Curves in Wasserstein Spaces and Continuity Inequalities in a Metric Setting Chapter 7. Weighted Energy Functionals along Absolutely Continuous Curves Chapter 8. Dynamic Kantorovich Potentials, Continuity Equation and Dual Weighted Cheeger Energies Chapter 9. The \RCDSKN Condition and Its Characterizations through Weighted Convexity and Evolution Variational Inequalities 9.1. Green functions on intervals 9.2. Entropies and their regularizations 9.3. The \CDSKN condition and its characterization via weighted action convexity 9.4. \RCDK∞ spaces and a criterium for \CDSKN via EVI Part III . Bakry-Émery Condition and Nonlinear Diffusion Chapter 10. The Bakry-Émery Condition 10.1. The Bakry-Émery condition for local Dirichlet forms and interpolation estimates 10.2. Local and “nonlinear” characterization of the metric \BEKN condition in locally compact spaces Chapter 11. Nonlinear Diffusion Equations and Action Estimates Chapter 12. The Equivalence Between \BEKN and \RCDSKN 12.1. Regular curves and regularized entropies 12.2. \BEKN yields EVI for regular entropy functionals in \MCN 12.3. \RCDSKN implies \BEKN Bibliography Back Cover Provides new characterizations of the curvature dimension condition in the context of metric measure spaces $(X,\mathsf d,\mathfrak m)$. The authors' approach takes into account suitable weighted action functionals, and uses the nonlinear diffusion semigroup induced by the $N$-dimensional entropy, in place of the heat flow