Intermittency for the Hyperbolic Anderson Model with rough noise in space

“…Note that, in this latter case, though the noise is more regular in space than a white noise, the corresponding Hilbert space H H may be rather big, and indeed contains genuine distributions. This makes our proof different compared to the one of [4,Thm. 4.2], in which H H is a space of functions (because H < 1 2 ).…”

Section: Itô and Skorohod Stochastic Integralsmentioning

confidence: 86%

“…We point out that we restrict to Hurst indices greater than 1 4 . This is due to the fact that, as proved in [4,Prop. 3.7], H > 1 4 is also a necessary condition in order to have a solution to (SWE) and (SHE).…”

Section: Introductionmentioning

confidence: 95%

“…Here, we split the proof taking into account that the sequence of Hurst indices is contained in ( 1 4 , 1 2 ] or [ 1 2 , 1), for the definition and properties of the stochastic integral in (2) differ significantly between those two cases. Indeed, the main difficulty here is concentrated in the rough case, where we carefully extend some moment estimates appearing in [4] in order to make them uniform with respect to H ∈ ( 1 4 , 1 2 ). Secondly, in order to identify the limit law, we prove the convergence of the corresponding finite dimensional distributions (see Section 4).…”

Section: Introductionmentioning

confidence: 99%

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

Giordano¹,

María²,

Quer-Sardanyons³

2020

Bernoulli

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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.

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“…This phenomenon is called full intermittency, which means that the random field solution develops high peaks concentrated on small sets for large time values. For further details on this topic, see [72,115,11,10]. Results regarding Lyapunov-exponents for parabolic SPDEs on bounded domains are available within the RDS approach using Oseledets' multiplicative ergodic theorem for compact operators in [70,142,37,129,131].…”

Section: Stabilitymentioning

confidence: 99%