SPDEs with linear multiplicative fractional noise: Continuity in law with respect to the Hurst index

In this article, we consider the one-dimensional stochastic wave and heat equations driven by a linear multiplicative Gaussian noise which is white in time and behaves in space like a fractional Brownian motion with Hurst index [Formula presented]. We prove that the solution of each of the above equations is continuous in terms of the index H, with respect to the convergence in law in the space of continuous functions. The proof is based on a tightness criterion on the plane and Malliavin calculus techniques in order to identify the limit law.