Mathematical Sciences: Branching Measure-Valued Processes and Related Nonlinear Partial Differential Equations

数学科学：分支测值过程及相关非线性偏微分方程

• 批准号: 9623190
• 负责人: Eugene Dynkin
• 金额: --
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-06-01 至 1999-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9623190&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Branching; Measure; Valued

9623190 Dynkin ABSTRACT Investigations on branching measure-valued processes and their connections with nonlinear PDEs are proposed. One of the targets is the boundary theory of superdiffusion on a differentiable manifold and related problems on the equation Lu=F(u) where L is the second order elliptic differential operator and F is a nonlinear function. Among the problems are the description of all positive solutions and their behavior at the Martin boundary, and criteria for removable boundary singularities in terms of the Martin capacity. This part of the project has been prepared for by an intensive study, done in the last years, of the case of a bounded domain with a smooth boundary in a Euclidean space. Another direction iq the investigation of the structure of the most general branching measure-valued process. Previous results have been obtained under an assumption that the second moment of the total mass is finite. There is a hope to remove or to weaken this restriction by using recent progress in the theory of linear additive functionals. During the last decade branching measure-valued processes also known as superprocesses have attracted a great deal of effort from many investigators around the world. Interest in these processes is motivated by their applications to biology, especially, to population genetics. On the other hand, this class of processes provides new tools for the study of complex physical systems with infinitely many degrees of freedom.

提出了9623190 Dynkin关于分支测量值过程及其与非线性PDE的连接的抽象研究。目标之一是在方程式LU = f（u）上的可区分流形和相关问题上的超级扩散的边界理论，其中l是二阶椭圆差算子，f是非线性函数。问题之一是对马丁边界上所有积极解决方案及其行为的描述，以及马丁能力上可移动边界奇点的标准。 该项目的这一部分是由过去几年进行的一项密集研究为欧几里得空间中具有平稳边界的有界域而进行的。智商的另一个方向研究了最通用的分支测量过程的结构。在假设总质量的第二刻是有限的假设下，已经获得了先前的结果。希望通过使用线性添加功能理论中的最新进展来消除或削弱这一限制。 在过去的十年中，分支措施价值的过程也称为超级过程，吸引了世界各地许多调查人员的大量努力。 对这些过程的兴趣是由于它们在生物学上的应用，尤其是对人群遗传学的兴趣。另一方面，此类过程为研究具有无限多种自由度的复杂物理系统提供了新的工具。