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

本计画旨在发展一种称为高阶矩渐近法的强大工具，并将其应用于交叉口当地时间及相关模型的大偏差研究。矩量法在研究交点地方时和交点距方面已经探索了大约二十年。经典的应用包括交局部时的存在性和弱收敛性。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，需要理解当幂增加到无穷大时矩的渐近行为。近年来，这方面有了新的发展。沿着这条线发展起来的方法叫做高矩渐近法。高矩渐近方法在其早期阶段就已经在相交局部时和可加Levy过程的局部时方面显示出了强大的能力，理解相交局部时和相关模型的大偏差对于应用于欧几里德量子场论、聚合物的生长和极化子问题是至关重要的。开展这一项目将导致解决这一领域中遗留的一些难题，并发展新的方法，所处理的问题可以清楚地说明，没有太多的技术术语，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，特别是有助于吸引更多的学生学习数学和概率。

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.