Superdiffusions and Partial Differential Equations

超扩散和偏微分方程

• 批准号: 0503977
• 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: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0503977&HistoricalAwards=false
• 关键词: Superdiffusions; Partial; Differential; Equations

The subject of the project is a class of measure-valued Markov processes called superdiffusions and the correponding class of semilinear elliptic equations. An intensive study of the connections between these two classes during the past 15 years resulted in developing a nonlinear analog of the classical probabilistic potential theory related to the Brownian motion. Some fundamental problems in this field were solved during the last three years which opens new directions of research. These directions will be explored. In particular, new probabilistic tools will be applied to problems on nonlinear differential equations. Harmonic functions in spaces of measures associated with superprocesses will be investigated which promises to open a new chapter in infinite dimensional analysis. In this connection, exit boundaries for superdiffusions will be explored - an attempt to extend Martin boundary theory to nonlinear PDEs.The goal of the proposal is to contribute to probabilistic analysis, which is an important branch of modern mathematics. Interactions between the theory of stochastic processes and the theory of partial differential equations are beneficial for both fields. At the beginning, mostly analytic results were used by probabilists. More recently, analysts (and physicists) took inspiration from the probabilistic approach. Of course, the development of analysis in general, and of theory of differential equations in particular, was motivated to a great extent by problems in physics. A difference between physics and probability is that the latter provides not only an intuition but also rigorous tools for proving theorems.

该项目的主题是一类测量值的马尔可夫过程，称为超级植物和半线性椭圆方程的CORREPONDING类。 在过去15年中，对这两个类别之间的联系的深入研究导致开发了与布朗运动有关的经典概率潜在理论的非线性类似物。 Some fundamental problems in this field were solved during the last three years which opens new directions of research. These directions will be explored. In particular, new probabilistic tools will be applied to problems on nonlinear differential equations. 将研究与超级过程相关的度量空间中的谐波功能，该措施将有望在无限维度分析中打开新章节。 在这方面，将探索超级潜水界的退出边界 - 将马丁边界理论扩展到非线性PDE的尝试。该提案的目的是为概率分析做出贡献，这是现代数学的重要分支。 Interactions between the theory of stochastic processes and the theory of partial differential equations are beneficial for both fields. At the beginning, mostly analytic results were used by probabilists. More recently, analysts (and physicists) took inspiration from the probabilistic approach. 当然，总体而言，分析的发展，尤其是微分方程的理论是由于物理学问题在很大程度上激发的。 A difference between physics and probability is that the latter provides not only an intuition but also rigorous tools for proving theorems.