Exponential Asymptotics in Sample Path Intersection and Related Problems

陈教授将研究样本路径交点的指数收敛问题(S)。这个项目的重要性来自于以下三个原因：第一，样本交叉口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的人们的极大兴趣。其次，这项研究是了解多参数过程的加性泛函和局部时的渐近模式(在指数水平)的关键一步。作为多参数过程局部时的一个具体例子，交局部时的长期行为类似于布朗单的局部时。第三，涉及的问题在数学上往往具有挑战性。相交局部时是具有强记忆性和尖锐奇异性的过程。在这一领域中证明的许多结果都显示出严重的维度相关性，这是这两个因素共同作用的结果。在现有的文献中，在多维情况下也观察到了自相交和路径间相交之间的本质区别。与弱律等其他方面的研究相比，大偏差、小球概率等极限行为以及相关的强极限律所知甚少。最近，作者和他的合作者在布朗运动、稳定过程和随机游动运行的相交局部时的指数渐近性和重对数律方面取得了一些实质性的进展。这一进展表明，在更广泛的情况下，还有更多的问题和机构。研究了三类模型的指数渐近性：随机游动的局部时和一些更一般的过程的交集；随机游动的值域和交集；多参数过程的局部时和可加泛函。在生物界和物理界，相交当地时间的概念是研究聚合物和极化子模型的有效工具，这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。这项研究将增进我们对这些系统如何在不同尺度上进化的了解。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.