The Concavity Of Entropy Power And Displacement Convexity In Nonlinear Diffusion

As two basic concepts in information theory and optimal transport,the concavity of entropy power and displacement convexity have gained significant development in stochastic analysis and geometric analysis,and formed close relationship with the functional geometric inequalities in recent years.This thesis mainly discusses the concavity of entropy power and displacement convexity relating to nonlinear diffusion on Riemannian manifolds.More specifically,the main contents consist of five chapters as follows:The first chapter is an introduction,which introduces the research background and progress on the concavity of entropy power and displacement convexity,and lists the research contents and main conclusions of this thesis.In chapter two,we research the concavity of p-entropy power along solutions of the p-heat equation and give three applications in Euclidean space:generalization of isoperimetric entropy power inequality,new proofs of the Lp-Nash inequality and Lp-logarithmic Sobolev inequality,and moreover an improvement Lp-logarithmic Sobolev inequality.In chapter three,the concavity property is extended to weighted Riemannian manifolds satisfying the Bakry-Emery Ricci curvature dimensional condition.In chapter four,we study the monotonicity of logarithmic p-entropy functional along solutions of the weighted p-heat equation and give its decreasing formulae.And we establish a relation with the concavity of p-entropy power.In chapter five,we research the convexity of general entropy functional along solutions of the generalized geodesic equation and get its displacement convexity and applications under the generalized McCann condition.At last,we connect it with the concavity property.