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.
