Temporal properties of the stochastic fractional heat equation with spatially-colored noise

Consider the stochastic partial differential equation \begin{equation*} \frac {\partial }{\partial t}u_t(\boldsymbol {x})= -(-\Delta )^{\frac {\alpha }{2}}u_t(\boldsymbol {x}) +b\left (u_t(\boldsymbol {x})\right )+\sigma \left (u_t(\boldsymbol {x})\right ) \dot F(t, \boldsymbol {x}), \quad t\ge 0,\: \boldsymbol {x}\in \mathbb R^d, \end{equation*} where $-(-\Delta )^{\frac {\alpha }{2}}$ denotes the fractional Laplacian with power $\frac {\alpha }{2}\in (\frac 12,1]$, and the driving noise $\dot F$ is a centered Gaussian field which is white in time and has a spatial homogeneous covariance given by the Riesz kernel. We study the detailed behavior of the approximation of the temporal gradient $u_{t+{\varepsilon }}(\boldsymbol {x})-u_t(\boldsymbol {x})$ at any fixed $t > 0$ and $\boldsymbol {x}\in \mathbb R^d$, as ${\varepsilon }\downarrow 0$. As applications, we deduce Khintchin’s law of iterated logarithm, Chung’s law of iterated logarithm, and a result on the $q$-variations of the temporal process $\{u_t(\boldsymbol {x})\}_{ t \ge 0}$ of the solution, where $\boldsymbol {x}\in \mathbb R^d$ is fixed.