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

TL;DR

This paper investigates the behavior of spatial gradients of solutions to a stochastic heat equation with fractional Laplacian and white noise, revealing detailed properties and applications to KPZ equation analysis.

Contribution

It provides a detailed analysis of the spatial gradient behavior of solutions to a fractional stochastic heat equation, connecting to KPZ sample function studies.

Findings

Characterizes the asymptotic behavior of spatial gradients as epsilon approaches zero.

Links the gradient analysis to properties of KPZ equation solutions.

Provides new insights into the regularity and sample path properties of stochastic PDE solutions.

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 $\alpha/2\in(\nicefrac12\,,1]$. 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.