Multidimensional Lévy white noise in weighted Besov spaces

Fageot, Julien; Fallah, Alireza; Unser, Michaël

doi:10.1016/j.spa.2016.08.011

Cited by 28 publications

(40 citation statements)

References 39 publications

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“…The knowledge that β ∞ = 0 is enough to deduce that Ψ(ξ) is also dominated by |ξ| for |ξ| ≥ 1. Thus, there exists C > 0 such that |Ψ(ξ)| ≤ C |ξ| for every ξ ∈ R. Restarting from (20), we obtain that log P a H s(·/a) (ϕ) ≤ C…”

Section: Discussion and Converse Resultsmentioning

confidence: 93%

Scaling Limits of Solutions of Linear Stochastic Differential Equations Driven by Lévy White Noises

Fageot

Unser

2018

J Theor Probab

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Correspondence to be sent to: julien.fageot@epfl.ch Consider a random process s solution of the stochastic differential equation Ls = w with L a homogeneous operator and w a multidimensional Lévy white noise. In this paper, we study the asymptotic effect of zooming in or zooming out of the process s. More precisely, we give sufficient conditions on L and w such that a H s(·/a) converges in law to a non-trivial self-similar process for some H, when a → 0 (coarse-scale behavior) and a → ∞ (fine-scale behavior). The parameter H depends on the homogeneity order of the operator L and the Blumenthal-Getoor and Pruitt indices associated to the Lévy white noise w. Finally, we apply our general results to several notorious classes of random processes and random fields and illustrate our results on simulations of Lévy processes. 2 homogeneous of order 1. Our aim is to study the statistical behavior of the rescaling x → s(x/a) of a solution of (2) when a > 0 is varying. Our two main questions are:• What is the asymptotic behavior of s(·/a) when we zoom out the process (i.e., when a → 0)?• What is the local behavior when we zoom in (i.e., when a → ∞)?Our main contribution is to identify sufficient conditions such that the rescaling a H s(·/a) of a solution of (2) has a non-trivial self-similar asymptotic limit as a goes to 0 or ∞. When this limit exists, the parameter H is unique and depends essentially on the degree of homogeneity γ of L and on the Blumenthal-Getoor and Pruitt indices β ∞ and β 0 of w [8,44]. The indices β 0 and β ∞ are used in the literature to characterize the asymptotic and local behaviors of Lévy processes, and more generally Lévy-type processes [9]. We summarize the main results of our paper in Theorem 1.1. Precise definitions and rigorous statements are given later.Theorem 1.1. Let L be a γ-homogeneous operator and w a Lévy process with indices β ∞ and β 0 . Under some technical conditions, the solution s to the equation Ls = w has the following properties.

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Section: Discussion and Converse Resultsmentioning

confidence: 93%

Scaling Limits of Solutions of Linear Stochastic Differential Equations Driven by Lévy White Noises

Fageot

Unser

2018

J Theor Probab

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“…Thus w f = 0 in Q. By (19) and the finite speed of wave propagation, it holds that w f = 0 in (t 0 , ∞) × R n . Using the finite speed of wave propagation once more, we see that w f (t, x) = 0 when t ∈ R and d g (x, B 0 (x 0 , ǫ)) ≥ ǫ.…”

Section: Reduction To the Deterministic Inverse Problemmentioning

confidence: 94%

“…where the weight function is defined by x = (1 + |x| 2 ) 1/2 . Moreover, we have [19,Prop. 7]) and therefore we can consider W as a random variable restricted to H −d/2−ǫ (R d ; x −d/2−ǫ ) assigned with the cylindrical σ-algebra.…”

Section: Definition 1 a Generalized Random Variable Is A Measurable mentioning

confidence: 99%

Correlation based passive imaging with a white noise source

Helin

Lassas

Oksanen

et al. 2018

Journal de Mathématiques Pures et Appliquées

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Passive imaging refers to problems where waves generated by unknown sources are recorded and used to image the medium through which they travel. The sources are typically modelled as a random variable and it is assumed that some statistical information is available. In this paper we study the stochastic wave equation ∂ 2 t u − ∆ g u = χW , where W is a random variable with the white noise statistics on R 1+n , n ≥ 3, χ is a smooth function vanishing for negative times and outside a compact set in space, and ∆ g is the Laplace-Beltrami operator associated to a smooth non-trapping Riemannian metric tensor g on R n . The metric tensor g models the medium to be imaged, and we assume that it coincides with the Euclidean metric outside a compact set. We consider the empirical correlations on an open set X ⊂ R n ,for T > 0. Supposing that χ is non-zero on X and constant in time after t > 1, we show that in the limit T → ∞, the data C T becomes statistically stable, that is, independent of the realization of W . Our main result is that, with probability one, this limit determines the Riemannian manifold (R n , g) up to an isometry. To our knowledge, this is the first result showing that a medium can be determined in a passive imaging setting, without assuming a separation of scales.Date: May 7, 2019.

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“…Recently, the Besov space regularity of white noise was studied, e.g., in [13,54]. For a Gaussian white noise ξ : Ω → D (R n ), it was shown in [13,Corollary 3] that P-almost surely ξ locally belongs to the Besov space B s 2,r (R n ) for all r ∈ [1, ∞] and s < −n/2. In [54], white noise on the n-dimensional torus was studied.…”

Section: Interpolation Properties Of Related Spaces Consider the Hilmentioning

confidence: 99%

“…Based on an embedding result (Proposition 8.1), we show that the solution belongs pathwise to the space H α (Ω) under some condition on α. The investigation of such boundary-value problems is motivated by recent results on boundary noise (see, e.g., [50]) and on the Besov smoothness of white noise [13,54].…”

mentioning

confidence: 99%