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

Abstract

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 H ∈ ( 1 4 , 1). 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. MSC 2010: 60B10; 60H07; 60H15At this point, we aim to extend the random fieldW H defined in (9) to an isonormal Gaussian process in H H . We need the following corollary of [5, Thm. 4.3]:Proposition 2.6. The space of finite linear combinations of functions of the form f (r, z) = 1 (s,t]×(x,y] (r, z), with 0 ≤ s < t and x < y, is dense in the Hilbert space H H .Proof. The result is a direct consequence of [5, Thm. 4.3]. Indeed, in the latter paper it is proved that any predictable process {X(t, x), (t, x) ∈ R + × R} belonging to L 2 (Ω; H H ) can be approximated by finite linear combinations of processes of the form (r, z, ω) → 1 G (ω)1 (s,t] (r)1 (x,y] (z), for some G ∈ F. To prove our result, it suffices to observe that, if we choose a deterministic element ϕ in their proof, also its approximating sequence ϕ n is deterministic, and the norm in the space L 2 (Ω; H H ) coincides with the norm in H H for deterministic elements.

Continuity with respect to the Hurst parameter of solutions to stochastic evolution equations driven by H-valued fractional Brownian motion

“…Jolis and Viles in [13][14][15][16] have given a series of results on the stability in law with respect to the Hurst index. Some other works in this aspect can be found in [9,11,12,25]. Another impressive work in a recent year is the study of Koch and Neuenkirch in [18], where the infinite differentiability of fractional Brownian motion has been investigated.…”

A note on the continuity in the Hurst index of the solution of rough differential equations driven by a fractional Brownian motion

“…The estimation of H is very important, since it determines the magnitude of the self-correlation of the noise in the models. As emphasized in [8], not only one has to deal with the problem of the estimation of the Hurst parameter H of the noise, as in [9,11,12], but one needs to check that the model does not exhibit a large sensitivity with respect to the values of H. Hence, the study of the continuity problem is important in the case of both time (SDE) and time-space (SPDE) stochastic differential equations driven by fractional noises, and it is a very interesting problem not only from a theoretical point of view, but also in the modeling applications ( [1,2,4,13,14]).…”