Partial Differential Equations and Statistical Mechanics

偏微分方程与统计力学。摘要本课题研究椭圆型和抛物型偏微分方程，特别是带随机系数的偏微分方程。随机系数方程和统计力学之间有着深刻的联系，特别是粒子系统。在最近的工作中，首席研究员和合作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，如果这些技术得到更充分的发展，将会证明物理学家关于随机FKPP方程的波速的一些猜想。该项目的一个主要目标是获得其中一些猜想的证明。在另一个方向上，PI计划继续他对线性随机椭圆方程和抛物方程的研究，特别是在随机环境中控制随机行走的方程。他已经在这个问题和与图连通性相关的组合学的某些结果之间建立了一些意想不到的联系。这些问题是在研究形式摄动理论时产生的。本课题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一些与其他领域的新联系。项目的最后一部分涉及金融中出现的偏微分方程问题。首席研究员计划继续与他的研究生一起研究一个有效管理保险公司的模型。从数学上讲，这个问题是一个随机控制理论问题。一个令人满意的问题的分析将包括存在性和唯一性的证明，以及对所发生的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进行严格的数学分析。这些方程支配着物理学和工程学中发生的许多过程的行为。在过去的三十年中，它们也被用于金融领域，用于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。项目负责人计划对物理工程应用方面的一些问题以及金融应用方面的一些问题进行深入研究。金融应用程序关注的是一个简单的保险公司模型，该保险公司持有一个有风险的投资组合，并希望最大化对投资者的支付。物理工程应用关注于理解相边界的动力学，例如发生在化学反应中。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了解它们的基本性质。

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.