流形上的Bakry-Emery曲率，泛函不等式和热核分析

项目名称： 流形上的Bakry-Emery曲率，泛函不等式和热核分析

项目编号： No.11201040

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 钱斌

作者单位： 常熟理工学院

项目金额： 23万元

中文摘要： 流形上的Bakry-Emery 曲率, 泛函不等式和热核分析是随机分析和马氏过程理论的一个重要研究分支，也是当前国内外研究的热点之一。泛函不等式（如B-E热核不等式）刻画了热半群的重要性质且与Ricci曲率（B-E曲率）下有界是等价的。为此我们在该项目中要研究Bakry-Emery曲率与传输信息不等式之间的等价关系。注意到对于亚椭圆算子（如Heisenberg群上的次Laplace算子）其对应的B-E曲率不可能下有界(即曲率维数条件不满足)，为此我们还将研究满足一定曲率维数条件（如Baudoin-Garofalo提出的一般曲率维数条件）的亚椭圆算子对应的泛函不等式和热核分析，主要内容包括：满足一般曲率维数条件的亚椭圆算子对应的热核对数的梯度估计；n个布朗运动模型上的B-E热核不等式和研究带有势能的亚椭圆算子（Schrodinger算子）对应热方程正解的梯度估计及其热核估计。

中文关键词： 泛函不等式；Bakry-Emery曲率；熵；梯度估计；热核估计

英文摘要： The Bakry-Emery curvature of Riemannian manifolds, functional inequalities and heat kernel ananlysis is one important branch of stochastic analysis and theory of Markov processes. Functional inequalities, such as Bakry-Emery inequalities, describe important properties of heat semigroups and are equivalent to the lower bounds of Ricci curvature(Bakry-Emery curvature). So one main object in this project is to study the equivalent relationship between the Bakry-Emery curvature and transportation-information inequalities. Notice that the Ricci curvature can not be bounded below for the hypoellitic operators(e.g. the sublaplace operator on Heisenberg group), i.e. the curvature dimensional condition does not hold. Thus we will study the functional inequalities and heat kernel analysis for the hypoelliptic operators satisfying the generalized curvature dimensional condition introduced by Baudoin and Garofalo. It mainly contains:study the gradient estimates of the logarithm of the heat kernel associated the hypoelliptic operators satisfying generalized curvature dimensional conditions; study the Barky-Emery inequalities on the n-Brownian motion model and study the gradient estimates for the positive solutions to the heat equations associated the hypoelliptic operator with potential (Schrodinger operators) and the co

英文关键词： functional inequalities；Bakry-Emery curvature；entropy；gradient estimates；heat kernel estimates