矩阵值扩散过程、排它过程和随机增长模型的大偏差

该项目研究与 Airy 函数、Tracy-Widom 分布和 Airy 过程相关联的矩阵值扩散过程、排它过程、随机增长模型的大偏差和中偏差。我们主要研究以下几个方面：Airy 过程的性质；矩阵值扩散过程的特征值、排它过程和随机增长模型的大偏差和中偏差；与Airy 函数相关的区间删失模型的非参数极大似然估计的中偏差。由于Tracy-Widom 分布和 Airy 过程在随机矩阵、排它过程、随机增长模型中具有普适性，研究相应的大偏差是一个基本的问题。Tracy-Widom 分布和 Airy 过程比正态分布和 Brown 运动复杂得多，相应的大偏差问题也要困难得多， 需要探索新的方法和技术。

The project studies large deviations and moderate deviations for matrix-valued diffusion processes, exclusion processes and random growth models which have a closed relationship with airy function, Tracy-Widom distribution and Airy processes. We concentrate our efforts on: properties of Airy processes; large deviations and moderate deviations for the eigenvalue processes of matrix-valued diffusion processes, exclusion processes and random growth models; moderate deviations for nonparametric maximum likelihood estimators under interval censoring. Since Tracy-Widom distribution and Airy processes are universal in random matrices, exclusion processes and random growth models, it is a fundamental problem to study the large deviations associated with them. Tracy-Widom distribution and Airy processes are more complex than normal distribution and Brownian motion, and so the large deviation problems associated with them are more difficulties, and some new approaches and techniques are needed.