Degenerate Stochastic Systems and Related Problems in Analysis

小行星9703852 调查人员将继续与萨拉赫·穆罕默德合作， 退化随机微分方程及其相关问题的研究， 拟线性二阶偏微分方程拟议的研究福尔斯属于 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。 微分方程在他们最近的工作中，调查人员已经证明了一个非常普遍的 二阶线性偏微分的Hormander型亚椭圆性定理 运营商这个定理的假设允许Hormander的一般李代数 欧氏空间中光滑超曲面以最优指数速率失效的条件 空间这种算子被称为超简并算子。调查人员将确定 光滑解的存在性Dirichlet和Neumann问题与 超退化算子在第二部分中，研究者将研究光滑 密度的一类广泛的退化随机遗传方程。他们将寻求 使用他们的方法来建立相应算子的亚椭圆性。此外 解决无限维亚椭圆问题（显然是第一个）， 这里得到的估计应该导致存在一个李雅普诺夫谱， 奇异遗传系统的概率在第三部分，他们将使用他们的方法来研究 具有超退化主部的拟线性二阶偏微分算子 这些算子与超扩散密切相关。本部分的目的是 研究拟线性Cauchy，Dirichlet， 以及与这些算子相关的Neumann问题。 这项研究涉及物理学中出现的两个重要问题领域， 工程.第一个领域涉及一类重要的数学模型，称为 偏微分方程，这是基本的目标，在现代纯粹和应用 数学这些方程源于对热传导、电 潜力和流体流动。偏微分方程与 数学的几个领域，特别是概率论和几何。第二 该领域致力于一类模型，用于物理学，工程学和生物学， 为了分析其演化受随机波动影响的动力系统， 和过去的历史。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、记忆材料和种群动力学。调查人员将使用 最新的概率技术，以便更深入地了解这些 模型

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.