宋健，山东大学教授、博士生导师。2010年在美国堪萨斯大学博士毕业，2010-2012年在美国Rutgers大学任访问助理教授，2013-2018在香港大学任助理教授，2018年至今任山东大学数学与交叉科学研究中心教授。主要研究方向为随机偏微分方程、统计物理模型、随机矩阵、随机控制、随机分析及其应用等。
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.
