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Consider the stochastic partial differential equation $$  \frac{\partial }{\partial t}u_t(\mathbf{x})=-(-\Delta)^{\frac{\alpha}{2}}u_t(\mathbf{x})+b\left(u_t(\mathbf{x})\right)+\sigma\left(u_t(\mathbf{x})\right) \dot F(t,\mathbf{x}), \ \ \ t\ge0, \mathbf{x}\in \mathbb R^d, $$  where $-(-\Delta)^{\frac{\alpha}{2}}$ denotes the fractional Laplacian withthe power $\alpha/2\in (1/2,1]$, and the driving noise $\dot F$ is a centeredGaussian field which is white in time and with a spatial homogeneous covariancegiven by the Riesz kernel. We study the detailed behavior of the approximationspatial gradient $u_t(\mathbf{x})-u_t(\mathbf{x}-\varepsilon \mathbf e)$ at anyfixed time $t>0$, as $\varepsilon\downarrow 0$, where $\mathbf e$ is the unitvector in $\mathbb R^d$. As applications, we deduce the law of iteratedlogarithm and the behavior of the $q$-variations of the solution in space.

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