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.

9623190 Dynkin 摘要 提出了对分支测值过程及其与非线性偏微分方程的联系的研究。目标之一是可微流形上的超扩散边界理论以及方程Lu=F(u)的相关问题，其中L是二阶椭圆微分算子，F是非线性函数。这些问题包括所有正解及其在马丁边界处的行为的描述，以及根据马丁容量可移除边界奇点的标准。 该项目的这一部分是通过对欧几里得空间中具有平滑边界的有界域的情况进行的深入研究而准备的。另一个方向是研究最一般的分支测值过程的结构。先前的结果是在总质量的二阶矩有限的假设下获得的。人们希望利用线性加性泛函理论的最新进展来消除或削弱这种限制。 在过去的十年中，分支测值过程（也称为超级过程）吸引了世界各地许多研究人员的大量努力。 对这些过程的兴趣源于它们在生物学，特别是群体遗传学中的应用。另一方面，此类过程为研究具有无限多个自由度的复杂物理系统提供了新工具。