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

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