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

Abstract

Consider the linear stochastic fractional heat equation with vanishing initial condition: $$\frac{\partial u(t,x)}{\partial t}=-(-\Delta {)}^{\frac{\alpha }{2}}u(t,x)+\dot{W}(t,x),t>0,x\in \mathbb{R},$$ where $-(-\Delta {)}^{\frac{\alpha }{2}}$ 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 $H\in \left(\frac{2-\alpha }{2},1\right)$. We establish Chung's law of the iterated logarithm for the solution at $t=0$.

10 pages