Survival Time Probabilities and Applications to Hot-Spots and Spectral Gaps

这个建议研究的问题在于概率和其他数学领域的接口，或在更现代的语言，问题在随机分析。这些问题包括：（1）与Jeff Rauch对Neumann特征函数的热点猜想有关的问题，以及PI最初对条件布朗运动提出的对应问题;（2）与Schrodinger算子的尖锐谱隙界猜想有关的问题。 将（1）和（2）中的问题用布朗运动的生存时间概率来表示，然后将其归结为对有限维分布函数的研究。概率公式比原始公式更一般，但从本建议中提出的随机分析思想和技术的角度来看，更自然。引发这些问题的原因是众所周知的，而且非常困难。这些问题的任何进展都可能导致其他应用，并将引起研究人员的极大兴趣，这些研究人员致力于数学的几个不同领域，包括调和分析，偏微分方程，几何和概率。这些问题的版本也将被调查以外的布朗运动过程，如对称稳定过程和更一般的列维过程。 自上个世纪初以来，人们已经知道许多物理和生物现象可以用随机过程来建模。 事实上，布朗运动的经典模型是上述问题的核心，是由英国植物学家罗伯特·布朗（Robert Brown）（甚至更早）通过对悬浮在液体中的花粉颗粒的物理观察发现的。最近，随机模型已被用于物理，生物和社会科学的许多其他设置与许多复杂的应用程序的经济模型，特别是金融市场。许多这些应用涉及随机过程，不像布朗运动，没有连续的轨迹。 在上述问题中出现的有限维分布技术是由于与这些更一般的随机过程的联系而产生的。

This proposal studies problems that lie at the interface of probability and other fields of mathematics, or in more contemporary language, problems in stochastic analysis. These include (1) problems related to the hot-spots conjecture of Jeff Rauch for Neumann eigenfunctions and counterparts originally raised by the PI for conditioned Brownian motion and (2) problems related to the sharp spectral gap bounds conjecture for Schrodinger operators. The problems in (1) and (2) are formulated in terms of survival time probabilities for Brownian motion which are then reduced to the study of finite dimensional distributions functions. The probabilistic formulations are more general than the original conjectures but much more natural from the point of view of the stochastic analysis ideas and techniques suggested in this proposal. The conjectures that motivate these problems are well known and very difficult. Any progress on these problems will likely lead to other applications and will be of considerable interest to researchers working on several different areas of mathematics, including harmonic analysis, partial differential equations, geometry and probability. Versions of these problems will also be investigated for processes other than Brownian motion such as symmetric stable processes and more general Levy processes. It has been known since the beginning of the last century that many physical and biological phenomena can be modeled by random stochastic processes. Indeed, the classical model of Brownian motion which is at the heart of the problems discussed above was discovered by the British botanist Robert Brown (even earlier) from physical observations of pollen particles suspended in fluids. More recently, stochastic models have been used in many other settings in the physical, biological and social sciences with many sophisticated applications to economic models, and particularly to financial markets. Many of these applications involve stochastic processes which, unlike Brownian motion, do not have continuous trajectories. The finite dimensional distribution techniques that arise in the problems above came about because of the connections to these more general stochastic processes.