Degenerate Stochastic Systems and Related Problems in Analysis

9703852 Bell研究员将继续与Salah Mohammed合作研究退化随机微分方程以及线性和拟线性二阶偏微分方程中的相关问题。本文的研究分为三个部分。第一部分讨论退化扩散及其对线性偏微分方程的影响。在他们最近的工作中，研究人员证明了二阶线性偏微分算子的一个非常一般的hormander型半椭圆性定理。该定理的假设允许Hormander的一般李代数条件在欧氏空间的光滑超曲面上以最优指数速率失效。这样的算子被称为超简并算子。研究者将建立与超简算子相关的狄利克雷和诺伊曼问题的光滑解的存在性。在第二部分中，研究者将研究一类广泛的退化随机遗传方程的光滑密度的存在性。他们将寻求用他们的方法来建立相应算子的亚椭圆性。除了解决了一个无限维的半椭圆性问题（显然是第一个此类问题），这里得到的估计应该导致奇异遗传系统的李雅普诺夫谱在概率上的存在。在第三部分中，他们将用他们的方法研究具有超退化主部的拟线性二阶偏微分算子。这些算符与超扩散密切相关。本部分研究的目的是寻求与此类算子相关的拟线性Cauchy， Dirichlet和Neumann问题的经典光滑正解。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，它是现代纯数学和应用数学的基本对象。这些方程源于对热传导、电势和流体流动的研究。偏微分方程与数学的几个领域有重要的联系，特别是概率论和几何。第二个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。研究人员将使用最新的概率技术，以便对这些模型有更深入的了解。

9703852 Bell The investigator will continue his collaborative work with Salah Mohammed on the study of degenerate stochastic differential equations and related problems in linear and quasilinear second-order partial differential equations. The proposed research falls into three parts. Part I deals with degenerate diffusions and their impact on linear partial differential equations. In their recent work, the investigators have proved a very general Hormander-type hypoellipticity theorem for second-order linear partial differential operators. The hypotheses of this theorem allow Hormander's general Lie algebra condition to fail at an optimal exponential rate on smooth hypersurfaces in Euclidean space. Such operators have been termed superdegenerate. The investigators will establish the existence of smooth solutions to the Dirichlet and Neumann problems associated with superdegenerate operators. In Part II, the investigators will study the existence of smooth densities for a wide class of degenerate stochastic hereditary equations. They will seek to use their methods to establish hypoellipticity of the corresponding operators. In addition to solving an infinite-dimensional hypoellipticity problem (apparently the first of its kind), the estimates obtained here should lead to the existence of a Lyapunov spectrum in probability for singular hereditary systems. In Part III, they will use their methods to study quasilinear second-order partial differential operators with superdegenerate principal parts. These operators are closely related to superdiffusions. The objective of this part of the research is to seek classical smooth positive solutions of the quasilinear Cauchy, Dirichlet, and Neumann problems associated with such operators. This 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 are fundamental obje cts in modern day pure and applied mathematics. These equations arose from the study of heat conduction, electrical potential, and fluid flow. Partial differential equations have important connections with several areas of mathematics, in particular probability theory and geometry. The second area 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 will use the most current probabilistic techniques in order to develop a deeper understanding of these models.