Continuity in Law with Respect to the Spatial Hurst Index of the Solutions to Some Linear SPDEs

Abstract

In this thesis, we study the linear stochastic heat and wave equations with zero initial conditions, driven by a Gaussian noise, which is fractional in space with Hurst index H ∈ (0, 1), and is either white in time (i.e. fractional in time with index H_0 = 1/2) or fractional in time with index H_0 > 1/2. We prove that the solution of each of these equations is continuous in law in the space C([0,T] × R) of continuous functions, with respect to the index H. This result has already been proved in the recent article [15] for the case H_0 = 1/2, and we extend it here to the case H_0 > 1/2.