Analysis of the gradient of the solution to a stochastic heat equation via fractional Brownian motion

Abstract

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 $α/2\in(\nicefrac12\,,1]$. We study the detailed behavior of the approximate spatial gradient $u_t(x)-u_t(x-\epsilon )$ at fixed times $t>0$, as $\epsilon \downarrow 0$. We discuss a few applications of this work to the study of the sample functions of the solution to the KPZ equation as well.

25 pages