SPDEs with affine multiplicative fractional noise in space with index 1/4 < H < 1/2

© 2015 University of Washington. All right reserved. In this article, we consider the stochastic wave and heat equations on R with nonvanishing 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 σ(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 L2(Ω)-continuous and its p-th moments are uniformly bounded, for any p ≥ 2.