Wiener sausage相交轨道的大偏差理论及其在传感器网络中的应用研究

布朗运动常描述无体积的纯几何点运动，Wiener sausage是以布朗运动轨道为中心、以一定长度为半径的空间邻域。它是布朗运动的一个随机集合值泛函，在热传导、随机介质以及随机薛定谔算子的谱分析等领域有着广泛应用，也一直是学界关注的热点。本课题拟从大偏差角度研究Wiener sausage相交轨道的性质，尤其是它在临界和上临界维数下的情形。我们拟结合经典的Feynman-Kac方法和学界新近提出的高阶矩逼近方法，并借助Wiener sausage的三角分解性质、次可加性以及排列组合等一些分析技巧，研究其相交体积和相交时间的大偏差理论。其次，利用大偏差提供的尾估计进一步探索相应的收敛性质，包括重对数律、强逼近等。最后，我们拟将该理论应用于传感器网络节点最优移动策略、传输效率等问题的研究。通过该项目的工作，我们希望加深对Wiener sausage的理解，并为相关领域提供新的视角与分析工具。

Brownian motion is often used to describe the movement of pure geometric point without volume. Wiener sausage is a neighborhood centered at the trajectory of the Brownian motion with some given radius. It is a random set-valued functional of Brownian motion and widely used in the study of various stochastic phenomena, such as heat conduction and random media, as well as in the analysis of the spectral properties of random Schrödinger operators. Meanwhile, it is also the hot spot of related researches. In this project, we will approach the intersections of Wiener sausages from the view of large deviations, especially in the critical and supercritical dimensions. We want to combine the classical Feynman-Kac formula and the high moment approximation put forward recently, complemented by some techniques including triangle decomposition and subadditivity and permutations and combinations, and focus on large deviations for the volume and time of intersections of Wiener sausages. Secondly, based on the tail estimates provided by large deviations, we further explore the convergence properties such as laws of the iterated logarithm and strong approximation results. Finally, we want to apply the related theory to study the optimal moving strategy and transimission efficiency of sensor network nodes. We hope our research will understand Wiener sausage further, and provide a new perspective and analysis tool for related topics.