SPDEs with affine multiplicative fractional noise in space with index $\frac{1}{4}\langle H\langle\frac{1}{2}$

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 Hölder 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 $p \geq 2$.