Degenerate Diffusions on Manifolds with Corners

爱泼斯坦博士的工作涉及马尔可夫过程的分析，这些过程是赖特-费舍尔式马尔可夫链的无限种群极限。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，但变量被限制在某些空间区域，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。 这迫使生成元成为一个椭圆算子，其前导部分沿边界沿着退化。这些过程的一个显著特征是，在没有外力的情况下，如突变，一条典型的路径在有限时间内到达可行域的边界，但不能穿过它，这意味着退化与迄今为止成功分析的任何退化都有很大的不同。虽然我们对退化算子的某些类有很多了解，但对这里出现的一类算子，或者对边界像单形或其他多面体那样奇异的域上的退化偏微分方程，我们知之甚少。爱泼斯坦博士的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。他最近的工作，与雷夫Mazzeo，建立了存在性和唯一性的正规解决方案，这类自然方程，对自然类域，并奠定了基础，详细研究这些模型的定性性质。在应用中，需要求解这些先前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分析的奇异的解决方案，出现在应用概率，数学金融和人口生物学等，除其他事项外，这将需要一个功能的分析框架，用于分析定期解决方案的阐述，以解决这些应用中出现的奇异性。结合这些分析技术和概率论中使用的方法，他希望精确地描述热核本身在时间零点附近的结构，以及沿着可行域的边界，在那里它可以表现出各种类型的奇点。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随机和确定性力量下离散变量集合的时间演化。这些变量经常受到约束：例如，具有给定基因型的人群，或感染病原体的人数必须是非负整数，并且股票的价值通常被假设为非负。这种模型的动力学行为通常由马尔可夫链描述。这些都是“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，因此它们通常被用偏微分方程描述的连续极限所取代，这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种模型时，有一个单一的变量，这个项目的目的是开发分析和计算工具，研究这些方程的解决方案时，有很多，可能相互作用的定性行为，变量。在群体遗传学的应用中，这些工具可以用来了解繁殖的随机方面何时主导群体的进化，以及自然选择等确定性力量何时占主导地位。研究人员希望解开一小群突变之间的适应性相互作用对简单生物进化的影响。类似的方法也可以应用于研究流行病的早期阶段，那时也许可以进行小规模的干预，控制感染人口的增长。

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.