High Moment Asymptotics and its Applications in Intersection Local Times and Related Models

这个项目的目的是开发一个强大的工具，称为高阶矩渐近，并将其应用于研究本地时间交叉口的大偏差及其相关模型。矩方法用于研究交叉口的当地时间和范围已有近二十年的历史。经典的应用包括相交局部时的存在性和弱收敛。在这种情况下，瞬间的力量是固定的。然而，为了研究这些模型的大偏差，我们需要了解随着功率增加到无穷大，矩的渐近行为。近年来，在这个方向上有了新的发展。沿着这条线发展的方法称为高阶矩渐近法。即使在其早期阶段，高阶矩渐近方法已经在相交局部时和可加Levy过程的局部时方面显示出其强大的能力，理解相交局部时和相关模型的大偏差对于欧几里德量子场论、聚合物生长和极化子问题的应用是至关重要的。开展这一项目将解决这一领域遗留下来的一些难题，并开发新的方法。所解决的问题可以清楚地陈述，而不需要太多的技术术语，因此可以很容易地向非专家解释。在这方面，该项目将有利于数学教育，并有助于吸引更多的学生学习数学，特别是概率学。

This project aims to develop a powerful tool known as high moment asymptotics and to apply it to the study of the large deviations for intersection local times and for related models. The moment method has been explored in studying the intersection local times and ranges for about two decades. The classic applications include the existence and weak convergence of intersection local times. In this setting, the power of the moment is fixed. To study the large deviations for these models, however, one needs to understand the asymptotic behaviors of the moments as the power increases to infinity. In recent years, there have been new development in this direction. The method developed along this line is called high moment asymptotics. Even in its early stage, the method of high moment asymptotics has shown its power in the areas of intersection local times and of the local times of additive Levy processes.Understanding large deviations for intersection local times and related models is crucial for the applications to Euclidean quantum field theory, the growth of polymers, and the polaron problem. Carrying out this project will lead to the solution of some hard problems remaining in this area and to the development of new methodology.The problems addressed can be clearly stated without too much technical jargon and therefore can be easily explained to the non-experts. In this regard, the project will benefit mathematical education and help attracting more students to mathematics and probability in particular.