Partial Differential Equations and Statistical Mechanics

偏微分方程和统计力学

• 批准号: 0500608
• 负责人: Joseph Conlon
• 金额: $ 9.6万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-15 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0500608&HistoricalAwards=false
• 关键词: Partial; Differential; Equations; Statistical; Mechanics

Partial Differential Equations and Statistical Mechanics.Joseph G. Conlon University of Michigan AbstractThis research project is concerned with partial differential equations of elliptic and parabolic type, especially equations with random coefficients. There are deep connections between equations with random coefficients and statistical mechanics, in particular particle systems. In recent work the principal investigator and co-authors have investigated connections between the one dimensional nonlinear stochastic Fisher-Kolmogorov-Petrowski-Piscunov (FKPP) equation and certain particle systems. Their work introduced new techniques which, if more fully developed, should yield the proof of some conjectures of physicists concerning the wave speed of the stochastic FKPP equation. A major goal of the project is to obtain the proof of some of these conjectures. In another direction the PI plans to continue his research on linear stochastic elliptic and parabolic equations, in particular with the equations governing random walk in random environment. He has already established some unexpected connections between this problem and certain results in combinatorics related to graph connectivity. These arose in the study of the formal perturbation theory for the problem. The main goal in the current project is to prove an inequality for the effective diffusivity constant. He expects to uncover some new connections with other areas in the study of this problem also. The final part of the project is concerned with problems in partial differential equations which occur in finance. The principal investigator plans to continue working with his graduate student on a model for efficient management of an insurance company. Mathematically the problem is a problem of stochastic control theory. A satisfactory analysis of the problem will include existence and uniqueness proofs and a detailed study of the free boundary which occurs.The purpose of this research project is the rigorous mathematical analysis of certain partial differential equations. These equations govern the behavior of many processes which occur in physics and engineering. During the last thirty years they have also been used in finance in the systematic study and pricing of financial instruments which depend on market volatility, such as options. The principal investigator plans to do an in depth study of some problems related to physics-engineering applications and also some problems with applications in finance. The financial application is concerned with a simple model of an insurance company which holds a risky portfolio and wants to maximize payout to investors. The physics-engineering application is concerned with understanding the dynamics of phase boundaries which occur for example in chemical reactions. Both the financial and the physics-engineering models are the simplest possible prototypes. Nevertheless, sophisticated mathematical techniques are needed to understand their basic properties.

偏微分方程与统计力学。密歇根大学Conlon摘要本研究项目主要研究椭圆型和抛物型偏微分方程，特别是具有随机系数的方程。具有随机系数的方程与统计力学，特别是粒子系统之间有着深刻的联系。在最近的工作中，主要研究者和合著者研究了一维非线性随机Fisher-Kolmogorov-Petrowski-Piscunov（FKPP）方程和某些粒子系统之间的联系。他们的工作介绍了新的技术，如果更充分的发展，应产生的证明物理学家关于波速的随机FKPP方程。该项目的一个主要目标是获得其中一些假设的证明。在另一个方向上，PI计划继续他对线性随机椭圆和抛物方程的研究，特别是随机环境中的随机行走方程。他已经建立了一些意想不到的连接之间的问题和某些结果的组合有关的图形连接。这些都出现在研究的正式扰动理论的问题。本课题的主要目的是证明有效扩散系数的一个不等式。他希望在这个问题的研究中也能发现一些与其他领域的新联系。该项目的最后一部分是有关问题的偏微分方程发生在金融。首席研究员计划继续与他的研究生一起研究一个保险公司有效管理的模型。从数学上讲，这个问题是一个随机控制理论的问题。一个令人满意的分析问题将包括存在性和唯一性的证明和自由边界的详细研究发生。本研究项目的目的是某些偏微分方程的严格的数学分析。这些方程支配着物理学和工程学中许多过程的行为。在过去的三十年里，它们也被用于金融领域，对依赖于市场波动的金融工具（如期权）进行系统研究和定价。主要研究者计划深入研究与物理工程应用相关的一些问题，以及在金融应用中的一些问题。金融应用程序关注的是一个简单的保险公司模型，该公司持有一个风险投资组合，并希望最大限度地提高对投资者的支付。物理工程应用涉及理解例如在化学反应中发生的相边界的动力学。金融模型和物理工程模型都是最简单的原型。然而，需要复杂的数学技术来理解它们的基本性质。