SPDEs with fractional noise in space with index $H<1/2$

TL;DR

This paper studies stochastic wave and heat equations driven by fractional noise in space with index H between 1/4 and 1/2, establishing existence, uniqueness, and regularity of solutions.

Contribution

It proves the existence and uniqueness of mild solutions for these SPDEs with fractional spatial noise and bounded H"older continuous initial conditions.

Findings

Solutions are $L^{2}( ext{Omega})$-continuous.

$p$-th moments are uniformly bounded.

Existence and uniqueness are established for the equations.

Abstract

In this article, we consider the stochastic wave and heat equations on $\mathbb{R}$ with non-vanishing initial conditions, driven by a Gaussian noise which is white in time and behaves in space like a fractional Brownian motion of index $H$, with $1/4<H<1/2$. We assume that the diffusion coefficient is given by an affine function $\sigma(x)=ax+b$, and the initial value functions are bounded and H\"older continuous of order $H$. We prove the existence and uniqueness of the mild solution for both equations. We show that the solution is $L^{2}(\Omega)$-continuous and its $p$-th moments are uniformly bounded, for any $p \geq 2$.