Wavelet analysis of the Besov regularity of Lévy white noise

Aziznejad, Shayan; Fageot, Julien

doi:10.1214/20-ejp554

Cited by 12 publications

(10 citation statements)

References 49 publications

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“…In the restricted case of independently scattered Lévy processes which are stationary in space, we can combine our results with the work Aziznejad and Fageot [4] to determine the Besov space, in which a cylindrical Lévy process attains its values. This indicates a potential reasoning for the often observed phenomena of irregular trajectories of solutions of heat equations driven by cylindrical Lévy processes, for example, in Brzeźniak and Zabczyk [10] and Priola and Zabczyk [35].…”

Section: Introductionmentioning

confidence: 76%

“…These embedding results enable us to compare the Lévy space-time white noise with a model of Lévy-type noise in the space of distributions, investigated in a series of papers by Dalang, Humeau, Unser and co-authors in, for example, [4,14,18,19]. Their model of noise is initiated from research on developing sparse statistical models for signal and image processing.…”

Section: Introductionmentioning

confidence: 99%

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Modelling Lévy space‐time white noises

Griffiths

Riedle

2021

J. London Math. Soc.

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In this work, we compare Lévy space-time white noises and cylindrical Lévy processes. Lévy space-time white noises are defined as infinitely divisible independently scattered random measures and cylindrical Lévy processes are defined by means of the theory of cylindrical processes. It is shown that Lévy space-time white noises correspond to an entire sub-class of cylindrical Lévy processes, which is completely characterised by the characteristic functions of its members. We embed the Lévy space-time white noise, or the corresponding cylindrical Lévy process, in the space of general and tempered distributions. For the latter case, we show that this embedding is possible if and only if a certain integrability condition is satisfied. We establish that both embedded cylindrical processes are induced by genuine Lévy processes in the space of general or tempered distributions. We complete the picture by establishing Lévy space-time white noise as the weak derivative of Lévy additive sheets.

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Section: Introductionmentioning

confidence: 76%

Section: Introductionmentioning

confidence: 99%

Modelling Lévy space‐time white noises

Griffiths

Riedle

2021

J. London Math. Soc.

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“…In [15], similar results have been obtained for Lévy white noise. Global smoothness properties of Lévy white noise in weighted spaces have been established in [4,14]. Results as in the latter, two references will be important for the derivation of mixed smoothness properties.…”

Section: Remark 225mentioning

confidence: 98%

“…Proof This is a collection of Proposition 6, 9 and 12 from [4]. Note that the authors of [4] use a different convention concerning the notation of weighted Besov spaces so that the weight parameters in our formulation are multiplied by p compared to the formulation in [4]. ◻…”

Section: Then It Holds Thatmentioning

confidence: 99%

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Sample paths of white noise in spaces with dominating mixed smoothness

Hummel

2021

Banach J. Math. Anal.

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The sample paths of white noise are proved to be elements of certain Besov spaces with dominating mixed smoothness. Unlike in isotropic spaces, here the regularity does not get worse with increasing space dimension. Consequently, white noise is actually much smoother than the known sharp regularity results in isotropic spaces suggest. An application of our techniques yields new results for the regularity of solutions of Poisson and heat equation on the half space with boundary noise. The main novelty is the flexible treatment of the interplay between the singularity at the boundary and the smoothness in tangential, normal and time direction.

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Boundary value problems of elliptic and parabolic type with boundary data of negative regularity

Hummel

2021

J. Evol. Equ.

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We study elliptic and parabolic boundary value problems in spaces of mixed scales with mixed smoothness on the half-space. The aim is to solve boundary value problems with boundary data of negative regularity and to describe the singularities of solutions at the boundary. To this end, we derive mapping properties of Poisson operators in mixed scales with mixed smoothness. We also derive $$\mathcal {R}$$ R -sectoriality results for homogeneous boundary data in the case that the smoothness in normal direction is not too large.

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Wavelet analysis of the Besov regularity of Lévy white noise

Cited by 12 publications

References 49 publications

Modelling Lévy space‐time white noises

Modelling Lévy space‐time white noises

Sample paths of white noise in spaces with dominating mixed smoothness

Boundary value problems of elliptic and parabolic type with boundary data of negative regularity

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