Brownian motion with killing and reflection, stable processes and projections of martingales

[0303259]这个项目是研究一些处于概率论和其他数学领域交界的开放性问题。这些问题包括：(1)关于布朗运动的生存时间概率的“热点”性质的问题，布朗运动在一个区域的一部分边界上被消灭，而在其余部分上被反映；(2)关于对称稳定过程的“精细”谱理论性质的问题，如特征值的变分表征、节点线的性质、有限维分布作为起点函数的几何、解析和概率性质；(3)利用时空布朗运动研究Beurling-Ahlfors奇异积分算子所产生的最优问题。“热点”问题的解决将导致J. Rauch著名的“热点”猜想的进一步发展，该猜想断言基态诺伊曼特征函数的最大值和最小值仅在边界点处获得。许多关于稳定过程的问题都是由众所周知的布朗运动引起的。然而，稳定过程的一些问题反过来又会导致一些问题，这些问题甚至会为布朗运动提供新的信息。由于布朗运动结果的技术不适用，必须开发新的技术来解决这些问题。这种技术可能会在其他领域带来新的和意想不到的应用，特别是对于其他具有跳跃的随机过程，但其转移概率仍然保持一定的旋转不变性。鞅在一般奇异积分算子的研究中，特别是在Beurling-Ahlfors算子的研究中起着重要的作用。提出了一种新的基于热鞅的概率方法。这将使我们更好地理解T. Iwaniec关于该算子范数大小的著名猜想。当代数学的一个显著方面是在以前非常不同的领域之间意想不到而又深刻的联系。拟议项目中的问题本质上都是跨学科的，因此有可能影响数学、工程、科学和经济学的不同领域。对称稳定过程已被用于模拟许多物理和经济现象，特别是对于某些布朗运动模型不充分的股票。Beurling-Ahlfors算子的估计在非线性弹性问题中有许多应用。“热点”猜想，当用热传导理论表述时，断言如果一个人从一个在其边界周围绝缘的板上的初始热分布开始，并等待初始瞬态稳定下来，那么在板的边界上就会发现最热和最冷的区域。由于热的数学描述是由一个二阶偏微分算子给出的，这个算子是由皮埃尔-西蒙·德·拉普拉斯侯爵（1749-1827）在200多年前引入的，这个猜想引起了数学、物理和化学研究人员的兴趣。与概率的联系源于诺伯特·维纳（1894-1964）用布朗运动描述的热流。

0303259Banuelos This project is to study a number of open problems which lie at the interface of probability theory and other fields of mathematics. These include: (1) Problems concerning `hot-spots' properties for the survival time probabilities of Brownian motion which is killed on part of the boundary of a domain and reflected on the rest; (2) Problems concerning the `fine' spectral theoretic properties of symmetric stable processes, such as variational characterization for eigenvalues, properties of nodal lines, geometric, analytic and probabilistic properties of finite dimensional distributions as a function of the starting point; and (3) Optimal problems that arise from studying the Beurling-Ahlfors singular integral operator using space-time Brownian motion. Solutions to the 'hot-spots' problems will lead to further progress on the celebrated 'hot-spots' conjecture of J. Rauch which asserts that the maximum and minimum of ground state Neumann eigenfunctions are attained only at boundary points. Many of the proposed problems for stable processes are motivated by their well known counterparts for Brownian motion. However, some of the questions for stable processes in turn lead to problems which will provide new information even for Brownian motion. Since the techniques for the Brownian motion results do not apply, new techniques must be developed for these problems. Such techniques are likely to lead to new and unexpected applications in other areas particularly for other stochastic processes with jumps but whose transition probabilities still retain certain rotational invariance properties. Martingales have played an important role in the study of singular integral operators in general and in the study of Beurling-Ahlfors operator in particular. A new probabilistic approach, based on heat martingales, is proposed. This will give a better understanding of a celebrated conjecture of T. Iwaniec on the size of the norm of this operator. A remarkable aspect of contemporary mathematics is the unexpected yet deep connections between previously very different fields. The problems in the proposed project are all interdisciplinary in nature and as such have the potential to impact different fields in mathematics, engineering, science and economics. Symmetric stable processes have been used to model many physical and economic phenomena, particularly for certain stocks where the Brownian motion models are not adequate. Estimates on the Beurling-Ahlfors operator have many applications to nonlinear problems in elasticity. The 'hot-spots' conjecture, when formulated in terms of the theory of heat conduction, asserts that if one begins with an initial heat distribution on a plate which is insulated around its boundary and waits for the initial transients to settle down, then the hottest and coldest regions will be found on the boundary of the plate. Since the mathematical description of heat is given by a second order partial differential operator, introduced by the Marquis Pierre-Simon De Laplace (1749-1827) more than 200 years ago, the conjecture has been of interest to researchers in mathematics, physics and chemistry. The connection to probability arises from the description by Norbert Wiener (1894-1964) of heat flow in terms of Brownian motion.