Partial Differential Equations and Statistical Mechanics

PI：Joseph G.康伦，密歇根大学DMS-0138519 这个项目是关于椭圆和抛物偏微分方程及其在统计力学中的应用。第一部分的建议是关于椭圆和抛物方程的散度形式的系数是随机变量。本文研究了绿色函数期望值的正则性和方程解收敛到齐次方程解的速度。建议的第二部分是关于从随机系数抛物方程的发散理论到欧氏场论的应用。Helffer和Sjostrand最近发现了随机系数椭圆和抛物方程与欧氏场论之间的关系。后来由Naddaf-Spencer进一步发展。作者打算把他和Naddaf关于随机椭圆方程的一些思想应用到欧几里德场论的情形中，在第三部分中，作者提出研究具有随机系数的非发散型椭圆方程，特别是具有随机漂移的布朗运动所对应的方程。Sinai已经证明，在一维中，随机漂移扩散在长时间内是强亚扩散的。证明了当维数大于2时，随机漂移扩散的标度极限是布朗运动。作者发现了这个猜想与某些关于图中圈存在性的组合问题之间的联系。他计划继续他的程序来解决这些组合问题。该方案的最后一部分是关于具有确定系数的一致椭圆型方程，但系数可以任意快速振荡。作者计划继续他的工作，以获得估计的潜在的扩散是独立的程度的振荡系数。这一部分的建议是与前几部分，因为随机方程具有系数是迅速振荡. 该提案的目的是了解的性质的解决方案，以偏微分方程时，唯一的信息之一是，该方程的系数是有界的某种方式。在这个目标中有两个子主题：（a）理解最坏的可能行为，（B）理解“平均”行为--考虑到我们对系数的了解。子主题（a）是提案最后部分的主题。它与随机控制理论问题密切相关。随机控制理论的例子在工程和金融领域比比皆是。举一个金融例子，考虑股票期权的估值问题。Black和Scholes的经典研究表明，期权的价值仅取决于股票的波动率。对于一只波动率恒定的股票，他们有一个公式来计算波动率下的期权价值。该公式是一个波动率为系数的偏微分方程的解。(a)因此，当一个人只能假设股票波动率的一些界限时，与估计期权价值的最坏可能情况的问题有关。分主题（B）与（a）有类似的应用。对作者来说，最令人兴奋的是它提供了一种开始理解流体湍流问题的方法。湍流大致上是流体速度随机行为的开始。 众所周知，当流体受到足够大的扰动时，它将经历湍流行为。理解湍流的数学问题是明确的。人们只需要理解一个偏微分方程的解，即纳维-斯托克斯方程。到目前为止，甚至还没有开始了解如何从Navier-Stokes方程中以严格的方式导出湍流。原因是流体速度是方程的系数。在湍流区，Navier-Stokes方程类似于一个具有随机系数的偏微分方程。子主题（B）则涉及在这种情况下流体速度的“典型“行为。

PI: Joseph G. Conlon, University of MichiganDMS-0138519ABSTRACT This project is concerned with elliptic and parabolic partial differential equations and their applications in statistical mechanics. The first part of theproposal is concerned with elliptic and parabolic equations in divergence form where the coefficients are random variables. The author intends to study the regularity properties of the expectation value of the Green's function, and the rate of convergence of the solution of the equation to the solution of the homogenized equation. The second part of the proposal is concerned with the application of ideas from the theory of divergence form parabolic equations withrandom coefficients to Euclidean field theory. The relationship between ellipticand parabolic equations with random coefficients and Euclidean field theory wasdiscovered recently by Helffer and Sjostrand. It was then more fully developed by Naddaf-Spencer. The author intends to apply some ideas he and Naddaf have developed for random elliptic equations to the Euclidean field theory situation.In the third part of the proposal the author proposes to study nondivergence form elliptic equations with random coefficients, in particular anequation corresponding to Brownian motion with a random drift. It has beenshown by Sinai that in one dimension diffusion with random drift is stronglysubdiffusive for large time. It has been conjectured that in dimension larger than 2 the scaling limit of diffusion with random drift is Brownian motion. The author has found a connection between this conjecture and certain combinatorialproblems concerning the existence of cycles in graphs. He plans to continue hisprogram to solve these combinatorial problems. The final part of the proposal isconcerned with uniformly elliptic equations with deterministic coefficients.The coefficients can oscillate arbitrarily rapidly however. The author plans tocontinue his work to obtain estimates on the underlying diffusion which areindependent of the degree of oscillation of the coefficients. This part of theproposal is related to the previous parts since random equations havecoefficients which are rapidly oscillating. The goal of the proposal is to understand properties of the solution to apartial differential equation when the only information one has is that thecoefficients of the equation are bounded in some way. Within this goal thereare two sub-themes: (a) understanding worst possible behavior, (b)understanding "on average" behavior -given our knowledge of the coefficients.The sub-theme (a) is the subject of the final part of the proposal. It isintimately related to problems of stochastic control theory. Examples ofstochastic control theory abound in the world of engineering and finance. To take a financial example, consider the problem of valuing a stock option. Theclassic work of Black and Scholes shows that the value of the option dependsonly on the stock volatility. For a stock with constant volatility they have aformula for the value of the option in terms of the volatility. The formula isthe solution to a partial differential equation in which the volatility is acoefficient. (a) is therefore related to the problem of estimating worstpossible scenarios for option values when one can only assume some bounds on stock volatility. The sub-theme (b) has similar applications to (a). The mostexciting of these to this author is that it offers a way of beginning tounderstand the problem of turbulence in fluids. Turbulence is roughly speakingthe onset of random behavior in the velocity of a fluid. It is well known that a fluid will undergo turbulent behavior when subject to a sufficiently largedisturbance. The mathematical problem of understanding turbulence is welldefined. One simply needs to understand the solutions of a partial differentialequation known as the Navier-Stokes equation. To date there is not even thebeginnings of an understanding how to derive turbulence in a mathematicallyrigorous way out of the Navier-Stokes equation. The reason is that the fluidvelocity is a coefficient of the equation. In the turbulent regime thereforethe Navier-Stokes equation is like a partial differential equation with arandom coefficient. The sub-theme (b) is then concerned with the "typical"behavior of the fluid velocity in this situation.