Degenerate Diffusions on Manifolds with Corners

生物学和经济学中许多问题的模型涉及变量集合的演化，这些变量集合被约束在由超曲面集合限定的欧几里得空间的区域内。例如，证券的价值或其方差可能被约束为非负的，而基因的频率必须在0到1之间。特定证券的价值或等位基因在单个群体中的流行所遵循的实际路径非常复杂，实际上从根本上是不可预测的。这些路径共同构成了所谓的随机过程。虽然单一路径的时间进程是不可预测的，但整个可能路径族的统计特性通常可以被证明为满足偏微分方程式，从而允许对其进行详细分析。爱泼斯坦博士研究的主要目标是了解在这些背景下出现的各类方程的解。这很有挑战性，因为方程显示了与以下事实相关的简并性：路径被约束位于空间的某些区域，如三角形和四面体，这些区域本身具有非光滑的边界。爱泼斯坦博士正试图对这些方程有足够详细的理解，以开发精辟的数值工具，供生物学家和种群遗传学家使用。爱泼斯坦博士研究的一个主要重点是了解木村扩散方程解的详细分析性质，这些方程是赖特-费舍尔马尔可夫模型的极限情况。在过去的七年里，他与合作者一起工作，为这类自然的方程建立了分析基础，在一类自然的领域上。在他们最近的工作中，很明显，算子的类别需要扩展到包括具有某种奇异系数的方程，这是分析在概率、数学金融和人口生物学的更现实应用中出现的问题所必需的。这里提出的大部分工作涉及开发分析工具来解决这些类型的现实世界问题。这将包括对热核、稳定度量、注视概率和注视时间等量的分析。除了抽象的分析工作，爱泼斯坦博士还将开发数值算法，精确地解决木村扩散方程和相关的椭圆问题，这些问题是在研究这类过程的统计性质时出现的，在真正具有应用价值的情况下。爱泼斯坦博士还将致力于解决电磁学中依赖时间的问题的稳定、准确的数值方法的分析方面。这涉及到找到波动方程和全麦克斯韦方程解的新表示法，这反过来又导致具有比现有方法更好的精度和稳定性的数值方法。

Models for many problems in Biology and Economics involve the evolution of a collection of variables that are constrained to lie in a domain of Euclidean space bounded by a collection of hypersurfaces. For example the value of a security, or its variance might be constrained to be non-negative, whereas the frequency of a gene must lie between 0 and 1. The actual path that the value of a particular security, or the prevalence of an allele in a single population follow is very complicated and, indeed, fundamentally unpredictable. Collectively these paths constitute what is called a stochastic process. While the time course of a single path is unpredictable, the statistical properties of the whole family of possible paths can often be shown to satisfy partial differential equations, which allow for their detailed analysis. The main goal of Dr. Epstein's research is to understand the solutions of the types of equations that arise in these contexts. This is challenging because the equations display degeneracies connected to the fact that the paths are constrained to lie in certain regions of space, like triangles and tetrahedra, which themselves have non-smooth boundaries. Dr. Epstein is trying to develop a detailed enough understanding of these equations to develop incisive numerical tools for use by biologists and population geneticists.A principal focus of Dr. Epstein's research is to understand the detailed analytic properties of the solutions to Kimura diffusion equations, which arise as limiting cases of Wright-Fisher Markov models. Working jointly with collaborators for the past seven years, he has established analytic foundations for this natural class of equations, on a natural class of domains. In their recent work it became clear that the class of operators needed to be expanded to include equations with somewhat singular coefficients, which are needed for the analysis of problems that arise in more realistic applications to Probability, Mathematical Finance and Population Biology. Much of the work proposed herein deals with developing analytic tools to address these sorts of real world problems. This will include the analysis of such quantities as the heat kernel, stationary measures, probabilities of fixation, and times to fixation. Beyond the abstract analytic work, Dr. Epstein will also develop numerical algorithms to accurately solve Kimura diffusion equations and the associated elliptic problems that arise in the study of statistical property of such processes in cases of genuine applied interest. Dr. Epstein will also pursue the analytic aspects of stable, accurate numerical methods for solving time-dependent problems in electromagnetics. This involves finding novel representations of solutions to the wave equation and full Maxwell equations, which in turn lead to numerical methods with better accuracy and stability properties than pre-existing approaches.