Mathematical Sciences: Degenerate Stochastic Differential Equations and Partial Differential Equations

数学科学：简并随机微分方程和偏微分方程

• 批准号: 9505039
• 负责人: Denis Bell
• 金额: $ 4万
• 依托单位: University of North Florida
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-06-15 至 1997-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9505039&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Degenerate; Stochastic; Differential

In their previous work, the proposers have used probabilistic methods to establish an optimal extension of Hormander's theorem for second order differential operators. Their extension asserts hypoellipticity under hypotheses that allow the usual Lie algebra condition to fail at an exponential rate on a collection of hypersurfaces. The proposers use these techniques to investigate a series of further problems in the areas of partial and stochastic differential equations. In particular, they study the Dirichlet problem for exponentially degenerate operators on smooth domains in Euclidean space. They also investigate the existence of smooth densities for a large class of highly degenerate stochastic functional equations. The research deals with two important problem areas that arise in physics and engineering. The first area concerns an important class of mathematical models, called partial differential equations, that play a fundamental role in the study of heat conduction, electric potential and fluid flow. The second area of investigation is devoted to a class of models that are used in physics, engineering and biology in order to analyze dynamical systems whose evolution is influenced by random fluctuations and past history. These models are very important in a variety of diverse areas ranging from signal processing, stock market fluctuations, economic and labor models, aircraft dynamics, materials with memory and population dynamics. The investigators use techniques from the calculus of probability in order to develop a deeper understanding of the above-mentioned models.

在他们以前的工作中，提议者使用概率方法建立了二阶微分算子的Hormander定理的最佳扩展。他们的扩展断言hypoellipticity的假设下，允许通常的李代数条件，以指数率失败的超曲面的集合。提议者使用这些技术来研究偏微分方程和随机微分方程领域的一系列进一步问题。特别是，他们研究了欧氏空间中光滑域上指数退化算子的狄利克雷问题。他们还研究了一大类高度退化的随机泛函方程的光滑密度的存在性。 该研究涉及物理和工程中出现的两个重要问题领域。第一个领域涉及一类重要的数学模型，称为偏微分方程，在热传导，电势和流体流动的研究中发挥着重要作用。 研究的第二个领域是专门用于物理，工程和生物学中使用的一类模型，以分析其演化受随机波动和过去历史影响的动力系统。这些模型在信号处理、股票市场波动、经济和劳动力模型、飞机动力学、记忆材料和人口动力学等各个领域都非常重要。 研究人员使用概率演算的技术，以加深对上述模型的理解。