具有测度初值的分数阶抛物Anderson模型精确间歇性的研究

Anderson模型描述了粒子在随机媒介中的扩散现象，其研究是以概率论、偏微分方程和位势论等学科为基础的交叉领域。一个重要的数学问题是考察该模型的间歇性，主要包含解的空间渐近、长时矩渐近、几乎必然长时渐近、高阶矩渐近和尾概率渐近等。. 本项目拟使用概率方法，研究由空间粗高斯场驱动的连续分数阶抛物Anderson模型的精确间歇性问题，有助于深入理解分数阶抛物Anderson模型解的复杂结构。为了由克服空间粗高斯场、分数阶扩散结构和测度初值带来的困难，该研究拟对测度初值进行细致分类，结合基于Levy桥的Feynman-Kac公式、随机场极值渐近以及非负随机变量的大偏差等技术证明空间渐近和长时矩渐近。此项目的研究有利于丰富和发展Anderson模型及其相关模型间歇性的数学理论。

Anderson model describes the diffusion of particles in random media, the studies on which have become an interdisciplinary field based on probability theory, partial differential equation and potential theory. One of the most important mathematical questions is to consider intermittency of the model, which mainly includes the spatial asymptotics, long time moment asymptotics, almost surely long time asymptotics, high moment asymptotics, tailed probability asymptotics and so on.. This project intends to apply the probability method to studying the precise intermittency of continuous fractional parabolic Anderson model driven by space-rough Gaussian field, which contributes to a deeper understanding of the complex structure of solutions to the fractional parabolic Anderson model. In order to overcome some mathematical difficulties coming from the space-rough Gaussian field, fractional diffusion structure and measure-valued initial data, we intend to classify the initial value of measure in detail, and combining the Feynman-Kac formula based on Levy bridge, the asymptotics of extreme value of random field and large deviation technology for non-negative random variable to prove the precise spatial asymptotics and long time moment asymptotics. This study will enriches and develops the mathematical theory of the intermittency of Anderson models and its related models.