  Consider the stochastic partial differential equation $\partial_t u =
Lu+\sigma(u)\xi$, where $\xi$ denotes space-time white noise and
$L:=-(-\Delta)^{\alpha/2}$ denotes the fractional Laplace operator of index
$\alpha/2\in(\nicefrac12\,,1]$. We study the detailed behavior of the
approximate spatial gradient $u_t(x)-u_t(x-\varepsilon)$ at fixed times $t>0$,
as $\varepsilon\downarrow0$. We discuss a few applications of this work to the
study of the sample functions of the solution to the KPZ equation as well.
