Chung's LIL for the linear stochastic fractional heat equation at origin

Consider the linear stochastic fractional heat equation with vanishing initial condition: $$\frac{\mathrm{\partial }u(t,x)}{\mathrm{\partial }t}=-(-\mathrm{\Delta }{)}^{\frac{\alpha }{2}}u(t,x)+\dot{W}(t,x),t>0,x\in \mathbb{R},$$ where $-(-\mathrm{\Delta }{)}^{\frac{\alpha }{2}}$ denotes the fractional Laplacian with power $\alpha \in (1,2]$, and the driving noise $\dot{W}$ is a centered Gaussian field which is white in time and has the covariance of a fractional Brownian motion with Hurst parameter $H\in (\frac{2-\alpha }{2},1)$. We establish Chung’s law of the iterated logarithm for the solution at $t=0$.