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

Abstract

Consider the stochastic partial differential equation $$\frac{\partial }{\partial t}{u}_t(\mathbf{x})=-(-\hat{I}”{)}^{\frac{\hat{I}}{\pm }2}{u}_t(\mathbf{x})+b\left(u_t(\mathbf{x})\right)+\ddot{I}ƒ\left(u_t(\mathbf{x})\right)\dot{F}(t,\mathbf{x}),t\ge 0,\mathbf{x}\in {\mathbb{R}}^d,$$ where $-(-\hat{I}”{)}^{\frac{\hat{I}}{\pm }2}$ denotes the fractional Laplacian with the power $\hat{I}\pm /2\in (1/2,1]$, 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 $u_t(\mathbf{x})-u_t(\mathbf{x}-\epsilon \mathbf{e})$ 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.

18 pages. All comments are welcome