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 with the power $\alpha/2\in (1/2,1]$, 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 $u_t(\mathbf{ x})-u_t(\mathbf{ x}-\varepsilon \mathbf e)$ at any fixed time $t>0$, as $\varepsilon\downarrow 0$, 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.
Categories: math.PR