Analysis of the gradient for the stochastic fractional heat equation with spatially-colored noise in ${\mathbb{R}}^d$

Consider the stochastic partial differential equation $$\frac{\mathrm{\partial }}{\mathrm{\partial }t}{u}_t(\mathbf{x})=-(-\mathrm{\Delta }{)}^{\frac{\alpha }{2}}{u}_t(\mathbf{x})+b(u_t(\mathbf{x}))+\sigma (u_t(\mathbf{x}))\dot{F}(t,\mathbf{x}),t\ge 0,\mathbf{x}\in {\mathbb{R}}^d,$$ where $-(-\mathrm{\Delta }{)}^{\frac{\alpha }{2}}$ denotes the fractional Laplacian with the power $\alpha \mathrm{/}2\in (1\mathrm{/}2,1]$, and the driving noise $\dot{F}$ is a centered Gaussian field which is white in time and with a spatial homogeneous covariance given by the Riesz kernel. We study the detailed behavior of the approximation spatial gradient $u_t(\mathbf{x})-u_t(\mathbf{x}-\epsilon \mathbf{e})$ at any fixed time $t>0$, as $\epsilon \downarrow 0$, where $\mathbf{e}$ is the unit vector in ${\mathbb{R}}^d$. As applications, we deduce the law of iterated logarithm and the behavior of the $q$-variations of the solution in space.