Degenerate Diffusions on Manifolds with Corners

带角流形上的简并扩散

• 批准号: 1205851
• 负责人: Charles Epstein
• 金额: $ 34.31万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-06-15 至 2016-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1205851&HistoricalAwards=false
• 关键词: Degenerate; Diffusions; Manifolds; Corners

Dr. Epstein's work concerns the analysis of Markov Processes that arise as infinite population limits of Wright-Fisher-like Markov chains. Such models also arise in Epidemiology and Mathematical Finance. These models are diffusion-like processes, but with variables that are constrained to lie in certain regions of space, e.g. the non-negative orthant for a Finance problem, or a simplex for a Genetics problem. Hence the paths of these Markov processes must also remain within the feasible part of space. This forces the generator to be an elliptic operator whose leading part degenerates along the boundary. A distinguishing feature of these processes is that, in the absence of an outside force, like mutation, a typical path reaches the boundary of the feasible region in finite time, but cannot cross it. This implies that the degeneracies are rather different from any that have heretofore been successfully analyzed. While a great deal is known about some classes of degenerate operators, very little is known either about the class of operators that arise here, or about degenerate PDEs on domains with boundaries as singular as that of a simplex or other polyhedra. A principal focus of Dr. Epstein's research is to understand the detailed analytic properties of the solutions to equations of this type. His recent work, with Rafe Mazzeo, establishes the existence and uniqueness of regular solutions for this natural class of equations, on a natural class of domains, and lays the foundation for a detailed study of the qualitative properties of these models. In applications one needs to solve equations that these prior results show cannot have regular solutions. Thus Dr. Epstein will now turn his attention to the analysis of the singular solutions that arise in applications to Probability, Mathematical Finance and Population Biology, etc. Amongst other things, this will entail an elaboration of the functional analytic framework used to analyze regular solutions, to address the singularities that arise in these applications. Combining these analytic techniques with methods used in Probability Theory, he hopes to precisely describe the structure of the heat kernel itself near to time zero, and along the boundaries of the feasible region, where it can exhibit various types of singularities.Mathematical models for many problems in Population Biology, Epidemiology and Finance involve the time evolution of a collection of discrete variables under both random and deterministic forces. These variables are frequently constrained: for example a population with a given genotype, or the number of people infected with a pathogen must be a non-negative integer, and the value of a stock is usually assumed to be non-negative. The dynamical behavior of such models is often described by a Markov chain. These are discrete models with "no history," meaning that the statistical properties of the current generation determine those of the next. Discrete models of this type are difficult to directly analyze, and so they are often replaced by continuum limits that are described by partial differential equations, which is also the language of classical and quantum physics. While much work has been done to study such models when there is a single variable, the purpose of this project is to develop analytic and computational tools to study the qualitative behavior of solutions to these equations when there are many, possibly interacting, variables. In applications to Population Genetics, such tools can be used to understand when random aspects of reproduction dominate the evolution of a population, and when deterministic forces like natural selection dominate. The investigator hopes to unravel the effects of fitness interactions among a small group mutations on the evolution of simple organisms. Similar methods could be applied to study the early stages of an epidemic, when it might be possible to make a small intervention that could control the growth of the infected population.

爱泼斯坦博士的工作涉及到对马尔可夫过程的分析，马尔可夫过程是作为Wright-Fisher类马尔可夫链的无限总体极限而出现的。此类模型也出现在《流行病学》和《数学金融学》中。这些模型是类似扩散的过程，但变量被约束在特定的空间区域，例如，金融问题的非负正交，或遗传问题的单纯形。因此，这些马尔可夫过程的路径也必须保持在空间的可行部分内。这迫使生成器是一个椭圆运算符，它的前导部分沿着边界退化。这些过程的一个显著特征是，在没有外力(如突变)的情况下，一条典型的路径在有限时间内到达可行域的边界，但不能越过它。这意味着，这些简并现象与迄今已成功分析的任何一种简并现象都有很大不同。虽然关于某些退化算子类的了解很多，但对于这里出现的这类算子，或者关于具有像单纯形或其他多面体那样奇异边界的区域上的退化偏微分方程类，我们知之甚少。爱泼斯坦博士研究的一个主要重点是了解这类方程的解的详细分析性质。他最近与Rafe Mazzeo合作的工作，建立了这类自然方程在一类自然区域上正则解的存在唯一性，并为详细研究这些模型的定性性质奠定了基础。在应用中，人们需要解这些先前的结果表明不可能有正则解的方程。因此，爱泼斯坦博士现在将把他的注意力转向对概率、数学金融和人口生物学等应用中出现的奇异解的分析。其中，这将需要详细阐述用于分析常规解的函数分析框架，以解决这些应用中出现的奇异性。将这些分析技术与概率论中使用的方法相结合，他希望精确地描述接近时间零的热核本身的结构，并沿着可行域的边界，在可行域的边界上，它可以显示出各种类型的奇异性。人口生物学、流行病学和金融学中许多问题的数学模型涉及一组离散变量在随机和确定性作用力下的时间演化。这些变量经常受到约束：例如，具有给定基因型的种群或感染病原体的人数必须是非负整数，并且通常假设股票的值是非负的。这类模型的动力学行为通常用马尔可夫链来描述。这些是“没有历史”的离散模型，这意味着当前一代的统计特性决定了下一代的统计特性。这种类型的离散模型很难直接分析，因此它们经常被用偏微分方程组描述的连续统极限所取代，偏微分方程组也是经典和量子物理学的语言。虽然已经做了很多工作来研究这类模型当有一个单一变量时，这个项目的目的是开发分析和计算工具来研究当有许多变量时这些方程的解的定性行为，可能是相互作用的。在人口遗传学的应用中，这样的工具可以用来理解何时繁殖的随机方面主导了种群的进化，以及何时自然选择等确定性力量主导了种群的进化。这位研究人员希望揭示一小群突变之间的适应性相互作用对简单有机体进化的影响。类似的方法也可以用来研究流行病的早期阶段，那时可能会进行一些小的干预，以控制受感染人口的增长。