Khinchin's and Chung's Laws of the Iterated Logarithm at Time Zero for the Linear Stochastic Fractional Diffusion Equation

Abstract

We consider the linear stochastic fractional diffusion equation \begin{equation*} \partial^β u(t,x)=-\left(-Δ\right)^{α/2}u(t,x) +I_t^γ\bigl[\dot W(t,x)\bigr], \qquad t>0,\quad x\in\mathbb R^d, \end{equation*} with zero initial conditions, where $α>0$, $β\in(0,2)$, and $γ\ge0$. The driving noise $\dot W$ is a centered Gaussian generalized field that is fractional in time and has Riesz-type spatial covariance. For each fixed $x\in\mathbb R^d$, we establish a Khinchin-type law of the iterated logarithm at time zero for the temporal process $t\mapsto u(t,x)$. Under the additional conditions $0\leγ<1$ and $β+γ<2+H$, we also prove the corresponding Chung-type law. The proofs rely on a harmonizable representation, sharp frequency-truncation estimates, an exact small-ball asymptotic, and a localization argument. These results extend the initial-time laws of the iterated logarithm for stochastic heat equations to a broad class of time-fractional stochastic diffusion equations.