Applications of Probability to Problems in Analysis

巴纽艾洛斯 9700585 首席研究员将调查几个具体的开放问题， 不同的分析领域，概率思想和技术已经有了 在这方面取得了很大的成功，他认为有可能取得进一步的进展。其中包括1） 应用鞅不等式研究二维和多维Beurling-Ahlfors算子的算子范数，2）利用布朗运动， 研究的基本频率和基本差距的尖锐界限， 拉普拉斯在欧几里德域，和3）使用的渐近展开的维纳 研究薛定谔算子迹的渐近展开的泛函 以及在这种展开中系数的符号如何取决于 潜力 概率论植根于应用科学的各个领域，因此它应该 这并不奇怪，现代随机分析的许多技术工具都来自于 几个不同的数学领域 通常不太明显的是， 思想和技术可以有效地用于研究问题和应用 从表面上看，这似乎与概率无关。的问题 上面讨论的都属于这一类。 Beurling-Ahlfors算子是（奇异的） 积分算子描述解的正则性（光滑性） 各种非线性方程产生的，除其他事项外，弹性。 计算 其算子范数的确定是一个具有许多应用的基本问题。 表面上 这个操作符看起来离概率很远。 首席研究员，在 与G。Wang和A.林德曼成功地运用了 鞅（公平博弈）来研究这个问题。 该项目的第一部分描述了 各种新的概率方法在这个方向上进一步的工作。 的界 拉普拉斯算子的基频和基隙是基本量 振动膜理论中最重要的一点薛定谔算子的迹起着 在数学物理的各种问题中的基本作用，除其他外， 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， 随机微积分，探讨在这方面的几个开放的问题。预计在 正如过去多次发生的那样，这些调查将导致新的， 概率在分析、几何、偏微分方程 以及这些学科在数学物理中的应用。大多数学生将 可能参与了这些调查。

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.