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 $H \in \big(\frac{3-α}{4}, \frac{1}{2}\big)$. Specifically, the driving operator is the fractional Laplacian of order $α/2 \in (1/2, 1)$. We characterize the asymptotic behavior of the temporal increment $u(t+\varepsilon,x)-u(t,x)$ for fixed $t\ge 0$ and $x\in\mathbb{R}$ as $\varepsilon\downarrow 0$. Utilizing these precise asymptotic estimates, we establish Khinchin's and Chung's laws of the iterated logarithm for the temporal process $t \mapsto u(t,x)$.