Exponential Asymptotics in Sample Path Intersection and Related Problems

[405188]陈：PI将研究由样本路径的交叉点引起的指数收敛。这个项目的重要性来自于以下三个原因：首先，样本交叉点的区域充满了数学的优雅和惊喜。它也引起了从事物理和生物科学领域工作的人们的极大兴趣。其次，本研究是理解多参数过程的加性泛函和局部时间的渐近模式（在指数水平上）的关键一步。作为多参数过程运行地方时的一个具体例子，交叉地方时的长期行为局部类似于布朗片地方时的长期行为。第三，所涉及的问题通常在数学上具有挑战性。交点局部时间是记忆性强、奇异性强的过程。在这一领域所证明的许多结果显示出严重的维度依赖性，这是这两个因素共同作用的结果。在已有的文献中，在多维的情况下，自交和径交也有很大的区别。与弱定律等其他方面的研究相比，对大偏差、小球概率等极限行为以及相关的强极限定律的了解要少得多。近年来，作者及其合作者在布朗运动、稳定过程和随机游走的交叉局部时间的指数渐近性和迭代对数律方面取得了一些实质性进展。这一进展在更广泛的情况下提出了进一步的问题和制度。该项目涉及三种类型的模型，将研究指数渐近性：随机漫步和一些更一般的过程的交集局部时间；随机游走的范围和交点；以及多参数过程的局部时间和加性泛函。交叉地方时的概念是研究聚合物和极化子模型的有效工具，这在生物和物理学界是一个众所周知的事实。这个项目与这些模型的长期行为有关。这项研究将提高我们对系统如何在不同尺度上进化的认识。因此，该项目的成功也将对生物和物理科学的一些领域产生影响。

0405188Chen The PI will study the exponential convergence arising from the intersections of sample path(s). The importance of this project comes from the following three reasons: First, the area of sample intersections is full of mathematical elegance and surprise. It is also of great interest to the people working in the fields of physical and biological sciences. Second, this study is a crucial step toward understanding the asymptotic patterns (at the exponential level) of additive functionals and local times of the multi-parameter processes. As a concrete example of the local times run by multi-parameter processes, the long term behaviors of intersection local times locally resemble those of the local times of Brownian sheets. Third, the problems involved are often mathematically challenging. The intersection local times are the processes with strong memory and sharp singularity. Many of the results proved in this field show a heavy dimension dependence, which results from a combination of these two factors. In existing literature, a substantial difference between self-intersection and inter-path intersection has also been observed in the multi-dimensional cases. Compared with other aspects of the study, such as the weak laws, much less has been known in the limit behaviors such as large deviations, small ball probabilities and related strong limit laws. Recently, the proposer and his collaborators have made some substantial progress on the exponential asymptotics and the law of the iterated logarithm for the intersection local times run by Brownian motions, stable processes and random walks. The progress suggests further questions and establishments in broader situations. The project concerns three types of models, to which the exponential asymptotics will be investigated: The intersection local times of random walks and some more general processes; the range and intersection of ranges of the random walks; and the local times and additive functionals of multi-parameter processes. It is a well known fact in the biological and physical communities that the notion of intersection local times is an effective tool in the study of polymer and polaron models. This project is related to the long term behaviors of these models. The study will advance our knowledge on how the systems evolve in various scales. Therefore, the success of this project will also have impact on some areas in the biological and physical sciences.