Wiener sausage的重对数率研究

Wiener sausage is an important functional of Brownian motion. It is widely used in various stochastic phenomena, such as heat conduction, trapping in random media, and the analysis of spectral properties of random Schrodinger operators. In this program, we plan to study the laws of the iterated logarithm for Wiener sausage, which is the precise version of the large number law and also a very intresting topic to probability scholars. . In this research, the main tool is the large deviations and the key technique is the triangle decomposition of Wiener sausage. We want to switch appropriatly and try to construct a suitable Feynman-Kac semigroup. Combining high moment approximation method put forward recently with Feynman-Kac semigroup technique, we obtain the large deviations for the volume of Wiener sausage. Finally, using the tail estimates provided by large deviiations and the Borel-Cantelli lemma, we explore the corrsponding laws of the iterated logarithm.

Wiener sausage是Brown运动的一个重要泛函，它在热传导、随机介质的传播以及随机Schrodinger算子的谱分析等众多随机现象中有着广泛的应用。本课题拟研究Wiener sausage的重对数率，这是大数定律的精确化，也是概率论学者非常感兴趣的一个课题。. 该研究的主要工具是大偏差理论，其中很关键的一个技巧是Wiener sausage的三角分解性质。我们拟进行适当变换，通过构造Feynman-Kac半群，结合学界新近提出的高阶矩逼近方法以及经典的Fenyman-Kac方法，得到Wiener sausage体积的大偏差结果。最后，我们利用大偏差提供的尾估计和Borel-Cantelli引理方法，研究其相应的重对数率。

Wiener sausage是Brown运动的一类重要泛函，本项目主要研究Wiener sausage的重对数律。研究结果包括以下三个方面的内容：（1）单个Wiener sausage体积的重对数律；（2）多个相互独立的Wiener sausage相交部分体积的重对数律；（3）多个相互独立的Wiener sausage相交部分时间的重对数律。在第一方面，我们完整地得到了在各种维数空间下单个Wiener sausage体积的重对数律。在第二方面，我们主要得到了在下临界维数情形下Wiener sausage相交体积的中偏差和重对数律。在第三方面，我们借鉴了处理第二方面的一些技巧，得到了下临界维数下Wiener sausage相交时间的中偏差和重对数律。