Partial Differential Equations and Statistical Mechanics

偏微分方程组和统计力学。密歇根大学约瑟夫·G·康伦摘要本研究项目涉及椭圆型和抛物型偏微分方程组，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)方程与某些粒子系统之间的联系。他们的工作引入了新的技术，如果得到更充分的发展，应该会证明物理学家关于随机FKPP方程的波速的一些猜想的证据。该项目的一个主要目标是获得其中一些猜想的证明。在另一个方向，PI计划继续他对线性随机椭圆型和抛物型方程的研究，特别是关于随机环境中控制随机游动的方程。他已经在这个问题和与图连通性相关的组合学中的某些结果之间建立了一些意想不到的联系。这些都是在研究该问题的形式摄动理论时出现的。当前项目的主要目标是证明有效扩散系数常数的一个不等式。他希望在这个问题的研究中也能发现与其他领域的一些新的联系。该项目的最后部分涉及金融中出现的偏微分方程组的问题。首席调查员计划继续与他的研究生合作，建立一个有效管理保险公司的模型。从数学上讲，这是一个随机控制理论问题。一个令人满意的分析将包括存在唯一性证明和对自由边界的详细研究。本研究项目的目的是对某些偏微分方程作严格的数学分析。这些方程支配着物理和工程中发生的许多过程的行为。在过去的三十年里，它们也被用于金融学中，对依赖于市场波动的金融工具进行系统研究和定价，例如期权。首席研究员计划深入研究一些与物理工程应用有关的问题，以及一些在金融方面的应用问题。金融应用程序涉及一家保险公司的简单模型，该公司持有高风险的投资组合，并希望最大限度地向投资者赔付。物理工程应用涉及理解相界的动力学，例如在化学反应中发生的动力学。金融模型和物理工程模型都可能是最简单的原型。然而，需要复杂的数学技术来理解它们的基本性质。

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.