Temporal properties of the stochastic fractional heat equation with rough dependence in space

Abstract

This paper investigates the nonlinear stochastic fractional heat equation driven by a Gaussian noise that is white in time and fractional in space with a Hurst parameter $H\in (\frac{3-\alpha }{4},\frac{1}{2})$. Specifically, the driving operator is the fractional Laplacian of order $\alpha /2\in (1/2,1)$. We characterize the asymptotic behavior of the temporal increment $u(t+\epsilon ,x)-u(t,x)$ for fixed $t\ge 0$ and $x\in \mathbb{R}$ as $\epsilon \downarrow 0$. Utilizing these precise asymptotic estimates, we establish Khinchin's and Chung's laws of the iterated logarithm for the temporal process $t\mapsto u(t,x)$.