Wave equation with a stable noise

Pryhara, Larysa; Shevchenko, Georgiy

doi:10.1090/tpms/1040

Cited by 10 publications

(9 citation statements)

References 13 publications

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“…In particular, the integral I H (f) = R 3 f(x)Z H (dx) may be equivalently defined as a limit in probability of integrals of simple functions approximating f in L 1 (R 3 ) ∩ L 2 (R 2 ). Another important observation is that our definition agrees with that given in [16].…”

Section: Preliminariessupporting

confidence: 81%

“…This equation was already studied in [16], wherein we proved that its candidate solution given by Kirchhoff's formula…”

Section: Wave Equation With Coloured Stable Noisementioning

confidence: 81%

“…In this article we continue our research started in [15], where a planar wave equation was studied, and [16], which, as the present work, was devoted to wave equation in R 3 . The main object of our study is the wave equation…”

Section: Introductionmentioning

confidence: 84%

“…It was proved in [16] that there exists a weak derivative ∂ 3 ∂x 1 ∂x 2 ∂x 3 Z H,ε (x) and, in view of this, the following definition was proposed:…”

Section: Preliminariesmentioning

confidence: 99%

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Wave equation with a coloured stable noise

Pryhara

Shevchenko

2017

Random Operators and Stochastic Equations

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Abstract. We define a random measure generated by a real anisotropic harmonizable fractional stable field Z H with stability parameter α ∈ (1, 2) and Hurst index H ∈ (1/2, 1) and prove that the measure is σ-additive in probability. An integral with respect to this measure is constructed, which enables us to consider a wave equation in R 3 with a random source generated by Z H . We show that the solution to this equation, given by Kirchhoff's formula, has a modification, which is Hölder continuous of any order up to (3H − 1) ∧ 1. In the case where H ∈ (2/3, 1), we show further that the modification is absolutely continuous.

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Section: Preliminariessupporting

confidence: 81%

“…This equation was already studied in [16], wherein we proved that its candidate solution given by Kirchhoff's formula…”

Section: Wave Equation With Coloured Stable Noisementioning

confidence: 81%

Section: Introductionmentioning

confidence: 84%

“…It was proved in [16] that there exists a weak derivative ∂ 3 ∂x 1 ∂x 2 ∂x 3 Z H,ε (x) and, in view of this, the following definition was proposed:…”

Section: Preliminariesmentioning

confidence: 99%

See 2 more Smart Citations

Wave equation with a coloured stable noise

Pryhara

Shevchenko

2017

Random Operators and Stochastic Equations

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show abstract

“…The wave equation driven by stochastic measure defined on subsets of the spatial variable was considered in [1]. Similar equations driven by random stable noises are studied in [9,10], where the properties of generalized solutions are investigated.…”

Section: Introductionmentioning

confidence: 99%