Applications of Probability to Problems in Analysis

Banuelos 9700585首席调查员将在不同的分析领域调查几个具体的未决问题，在这些领域，概率论的思想和技术已经取得了相当大的成功，他认为有可能取得进一步的进展。这些内容包括：1)用鞅不等式研究二维和多维Beurling-Ahlfors算子的范数；2)用布朗运动研究欧几里得区域中拉普拉斯算子的基本频率和基本间隙的锐界；3)用Wiener泛函的渐近展开式研究薛定谔算子迹的渐近展开式，以及这种展开式中系数的符号如何依赖于势的符号。概率论植根于应用科学的各个领域，因此，现代随机分析的许多技术工具来自几个不同的数学领域也就不足为奇了。通常并不明显的是，概率论的思想和技术可以有效地用于研究表面上似乎与概率完全无关的问题和应用。上面讨论的问题属于这一类。Beurling-Ahlfors算子是(奇异的)积分算子，它描述了由弹性等引起的各种非线性方程的解的正则性(光滑性)性质。它们的算子范数的计算是一个有着许多应用的基本问题。从表面上看，这个运算符似乎离概率很远。首席调查员与G.Wang和A.J.Lindeman合作，成功地使用了鞅(公平博弈)理论来研究这个问题。该项目的第一部分介绍了在这一方向上进一步工作的各种新的概率方法。拉普拉斯算符的基频和基隙的界限是振膜理论中的基本量。薛定谔算符的迹在数学物理中的各种问题中扮演着基本的角色，其中包括散射理论。在这里，与概率的联系也是不透明的。在项目的第二部分，PI建议使用布朗运动理论和随机微积分来研究这一领域的几个公开问题。预计这些研究将导致概率在分析、几何、偏微分方程式中的新的和令人惊讶的应用，就像过去多次发生的那样，以及这些学科在数学物理中的应用。几名学生很可能会参与这些调查。

Banuelos 9700585 The Principle Investigator will investigate several concrete open problems in different areas of analysis where probabilistic ideas and techniques have already had considerable success and where he believes further progress is possible. These include 1) applications of martingale inequalities to investigate the operator norm of the Beurling- Ahlfors operators in two and several dimensions, 2) the use of Brownian motions to investigate sharp bounds for the fundamental frequency and fundamental gap of the Laplacian in Euclidean domains, and 3) the use of asymptotic expansions for Wiener functionals to investigate the asymptotic expansion in the trace of Schrodinger operators and how the signs of the coefficients in such expansion depend on the sign of the potential. Probability Theory has its roots in various fields of applied sciences and hence it should not be surprising that modern stochastic analysis draws many of its technical tools from several distinct areas of mathematics. What is often far from obvious is that probabilistic ideas and techniques can be effectively used to investigate problems and applications which on the surface do not seem to be related to probability at all. The problems discussed above fall in this category. The Beurling-Ahlfors operators are (singular) integral operators which describe regularity (smoothness) properties of solutions to various nonlinear equations arising from, among other topics, elasticity. The computation of their operator norms is a fundamental problem with many applications. On the surface this operator appears to be very far from probability. The Principal Investigator, in collaboration with G. Wang and A. J. Lindeman, has successfully used the theory of martingales (fair games) to study this problem. The first part of the project describes various new probabilistic approaches for further work in this direction. Bounds on the fundamental frequency and fundament al gap of the Laplacian operator are basic quantities in the theory of vibrating membranes. The trace of Schrodinger operators plays a fundamental role in various problems in mathematical physics related to, among other things, scattering theory. Here, too, the connection to probability is not transparent. In the second part of the project the PI proposes to use the theory of Brownian motion and stochastic calculus to investigate several open problems in this areas. It is expected that these investigations will lead, as it has happened many times in the past, to new and surprising applications of probability to analysis, geometry, partial differential equations, and the application of these subjects to mathematical physics. Several students will most likely participate in these investigations.