Diffusions, Superdiffusions and Partial Differential Equations

扩散、超扩散和偏微分方程

• 批准号: 0204237
• 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: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0204237&HistoricalAwards=false
• 关键词: Diffusions; Superdiffusions; Partial; Differential; Equations

A close relationship between linear elliptic and parabolic partial differential equations and Markov diffusion processes played an important role in the development of both fields during the last century. A new chapter in this direction -- an interplay between a class of semilinear equations and superdiffusions -- has evolved over the past 15 or so years. The theory has reached a certain stage of maturity. However some key problems remain open (for instance, the uniqueness of the solution with a given fine trace and the structure of exit boundaries of superdiffusions). The work on these problems is the subject of the proposal. Superdiffusions is a special class of branching measure-valued processes also known as superprocesses. Outside of pure mathematics, they play a significant role in population genetics and they provide new tools for the study of complex physical systems with infinitely many degrees of freedom. For about two decades these processes have attracted the efforts of many investigators around the world.

线性椭圆形和抛物线偏微分方程与马尔可夫扩散过程之间的密切关系在上个世纪的两个领域的发展中起着重要作用。在过去的15年左右的时间里，朝这个方向上的新章节（一类半线性方程式和超级潜水事件之间的相互作用）已经发展。该理论已经达到了成熟的一定阶段。然而，一些关键问题仍然是打开的（例如，具有给定的良好痕迹的解决方案的独特性以及超级停留的出口边界的结构）。这些问题的工作是该提案的主题。 SuperDiffusions是一类特殊的分支测量值过程，也称为超级过程。在纯数学之外，它们在人群遗传学中​​发挥了重要作用，并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。大约二十年来，这些过程吸引了世界上许多调查人员的努力。