Mathematical Sciences: Degenerate Stochastic Differential Equations and Partial Differential Equations

在他们之前的工作中，提出者已经使用概率方法建立了二阶微分算子的Hormander定理的最优扩展。它们的推广在允许通常的李代数条件在超曲面集合上以指数速率失效的假设下断言了亚椭圆性。作者利用这些技术进一步研究了偏微分方程和随机微分方程领域的一系列问题。特别地，他们研究了欧氏空间光滑域上指数退化算子的Dirichlet问题。他们还研究了一类高度退化的随机泛函方程的光滑密度的存在性。本研究涉及物理学和工程学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，它在热传导、电势和流体流动的研究中起着重要作用。第二个研究领域是专门用于物理学、工程学和生物学的一类模型，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、经济和劳动模型、飞机动力学、具有记忆的材料和人口动力学等各种不同领域都非常重要。研究人员使用概率演算的技术，以便对上述模型有更深入的了解。

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.