Sample path properties and small ball probabilities for stochastic fractional diffusion equations
報告人簡介
宋健，山東大學教授、博士生導師。2010年在美國堪薩斯大學博士畢業，2010-2012年在美國Rutgers大學任訪問助理教授，2013-2018在香港大學任助理教授，2018年至今任山東大學數學與交叉科學研究中心教授。主要研究方向為隨機偏微分方程、統計物理模型、隨機矩陣、隨機控制、隨機分析及其應用等。
內容簡介
We consider the following stochastic space-time fractional diffusion equation with vanishing initial condition:\begin{equation*}\partial^{\beta}u(t, x)=-\left(-\Delta\right)^{\alpha/2}u(t, x)+I_{0+}^{\gamma}\left[\dot{W}(t, x)\right],\quad t\in[0,T],\: x \in \mathbb{R}^d, \end{equation*} where $\alpha>0$, $\beta\in(0,2)$, $\gamma\in[0,1)$, $\left(-\Delta\right)^{\alpha/2}$ is the fractional Laplacian and $\W$ is a fractional space-time Gaussian noise. We prove the existence and uniqueness of the solution and then focus on various sample path properties of the solution. More specifically, we establish the exact uniform and local moduli of continuity and Chung’s laws of the iterated logarithm. The small ball probability is also studied. This is joint work with Yuhui Guo, Ran Wang, and Yimin Xiao.
