Mathematical Sciences: Degenerate Stochastic Differential Equations and Partial Differential Equations

在他们以前的工作中，提出者使用概率方法建立了二阶微分算子的Hormander定理的最优推广。它们的推广断言了假设下的椭圆性，这些假设允许通常的李代数条件在一组超曲面上以指数速度失效。作者利用这些技巧研究了偏微分方程和随机微分方程领域中的一系列进一步的问题。特别地，他们研究了欧氏空间中光滑区域上指数退化算子的Dirichlet问题。他们还研究了一大类高度退化的随机函数方程的光滑密度的存在性。这项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，它们在热传导、电势和流体流动的研究中发挥着基础作用。第二个研究领域致力于一类用于物理、工程和生物学的模型，以分析其演化受随机波动和过去历史影响的动力系统。这些模型在各种不同的领域都非常重要，从信号处理、股市波动、经济和劳动力模型、飞机动力学、具有记忆的材料和人口动力学。研究人员使用概率演算中的技术，以加深对上述模型的理解。

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.