Partial Differential Equations and Statistical Mechanics

摘要本课题研究椭圆型和抛物型偏微分方程及其在统计力学中的应用。该建议的第一部分涉及散度形式的椭圆和抛物方程，其中系数是随机变量。研究了格林函数期望值的正则性，以及方程的解收敛到齐次化方程解的速度。第二部分讨论了随机系数抛物型方程散度理论的思想在欧几里得场论中的应用。带随机系数的椭圆型和抛物型方程与欧几里得场论之间的关系是Helffer和Sjostrand最近发现的。后来，纳达夫-斯宾塞把它发展得更加充分。作者打算将他和Naddaf关于随机椭圆方程的一些思想应用到欧几里得场论中。在论文的第三部分，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方程。西奈已经证明，在一维中随随机漂移的扩散在大时间内是强次扩散的。在大于2维的情况下，随机漂移扩散的标度极限为布朗运动。作者发现了这个猜想与若干关于图中环的存在性的组合问题之间的联系。他计划继续他的项目来解决这些组合问题。最后讨论了具有确定性系数的一致椭圆方程。然而，系数可以任意快速地振荡。作者计划继续他的工作，以获得与系数的振荡程度无关的潜在扩散的估计。该建议的这一部分与前面的部分有关，因为随机方程具有快速振荡的系数。这个建议的目标是理解当一个人唯一的信息是方程的系数在某种程度上是有界的时候，偏微分方程解的性质。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——给定我们对系数的知识。分主题(a)是提案最后部分的主题。它与随机控制理论的问题密切相关。随机控制理论的例子在工程和金融领域比比皆是。举一个金融方面的例子，考虑给股票期权估值的问题。Black和Scholes的经典研究表明，期权的价值只取决于股票的波动率。对于波动率恒定的股票，他们有一个根据波动率计算期权价值的公式。该公式是一个偏微分方程的解，其中波动性是协效的。因此，(a)与当人们只能假设股票波动的某些界限时，估计期权价值的最坏可能情况的问题有关。分主题(b)与(a)具有类似的用途。其中最令笔者兴奋的是，它提供了一种开始理解流体湍流问题的方法。湍流大致是流体速度随机行为的开始。众所周知，当受到足够大的扰动时，流体会发生湍流行为。理解湍流的数学问题已经有了很好的定义。人们只需要了解一个偏微分方程的解，即纳维-斯托克斯方程。迄今为止，人们甚至还没有开始理解如何用数学上严格的方法从纳维-斯托克斯方程中推导出湍流。原因是流体速度是方程的一个系数。因此，在湍流状态下，Navier-Stokes方程就像一个带有随机系数的偏微分方程。子主题(b)则涉及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.