Applications of Probability to Problems in Analysis

首席研究员将研究不同分析领域的几个具体的开放问题，在这些领域，概率思想和技术已经取得了相当大的成功，并且他认为进一步的进展是可能的。这包括：1)利用鞅不等式研究二维和多维的Beurling- Ahlfors算子的算子范数；2)利用布朗运动研究欧几里得域拉普拉斯算子的基频和基隙的尖锐界限；3)利用维纳泛函的渐近展开式来研究薛定谔算子迹中的渐近展开式，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，现代随机分析从几个不同的数学领域汲取许多技术工具也就不足为奇了。通常不明显的是，概率论的思想和技术可以有效地用于调查表面上似乎与概率论完全无关的问题和应用。上面讨论的问题就属于这一类。Beurling-Ahlfors算子是一种（奇异）积分算子，它描述了各种非线性方程的解的正则性（光滑性）。它们的算符范数的计算是许多应用中的一个基本问题。从表面上看，这个算符似乎与概率相差甚远。首席研究员与G. Wang和A. J. Lindeman合作，成功地利用鞅理论（公平博弈）研究了这个问题。项目的第一部分描述了在这个方向上进一步工作的各种新的概率方法。拉普拉斯算子的基频和基间隙的边界是振动膜理论中的基本量。薛定谔算符的迹在数学物理中与散射理论等有关的各种问题中起着基本作用。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个领域的几个开放问题。正如过去多次发生的那样，预计这些研究将导致概率论在分析、几何、偏微分方程中的新的和令人惊讶的应用，以及这些学科在数学物理中的应用。一些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.