The heat equation with time-independent multiplicative stable Lévy noise

Mueller, Carl; Mytnik, Leonid; Stan, Aurel I.

doi:10.1016/j.spa.2005.08.001

Cited by 12 publications

(17 citation statements)

References 13 publications

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“…These arguments rely on the structure of G and cannot be used when O is a bounded domain.) Example 6.3 (Parabolic equations) Let L = ∂ ∂t −L where L is given by (17). Assuming (18), we see that (52) holds if p < 1 + 2/d.…”

Section: The Case α >mentioning

confidence: 99%

SPDEs with α-Stable Lévy Noise: A Random Field Approach

Balan

2014

International Journal of Stochastic Analysis

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This article is dedicated to the study of an SPDE of the form Lu(t, x) = σ(u(t, x))Ż(t, x) t > 0, x ∈ O with zero initial conditions and Dirichlet boundary conditions, where σ is a Lipschitz function, L is a second-order pseudo-differential operator, O is a bounded domain in R d , andŻ is an α-stable Lévy noise with α ∈ (0, 2), α = 1 and possibly non-symmetric tails. To give a meaning to the concept of solution, we develop a theory of stochastic integration with respect to Z, by generalizing the method of [11] to higher dimensions and non-symmetric tails. The idea is to first solve the equation with "truncated" noiseŻ K (obtained by removing from Z the jumps which exceed a fixed value K), yielding a solution u K , and then show that the solutions u L , L > K coincide on the event t ≤ τ K , for some stopping times τ K ↑ ∞ a.s. A similar idea was used in [22] in the setting of Hilbert-space valued processes. A major step is to show that the stochastic integral with respect to Z K satisfies a p-th moment inequality, for p ∈ (α, 1) if α < 1, and p ∈ (α, 2) if α > 1. This inequality plays the same role as the Burkholder-Davis-Gundy inequality in the theory of integration with respect to continuous martingales.MSC 2000 subject classification: Primary 60H15; secondary 60H05, 60G60

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Section: The Case α >mentioning

confidence: 99%

SPDEs with α-Stable Lévy Noise: A Random Field Approach

Balan

2014

International Journal of Stochastic Analysis

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“…where K ϕ is defined in (17). If t = T we will write |||w||| q,q instead of |||w||| q,q,[0,t] , w ∈ V q,q (T ).…”

Section: Lemma 41 Let K Be the Operator Defined Bymentioning

confidence: 99%

“…where K ϕ is defined in (17). If δ g − ρ < 1 p − r g q and δ f − ρ < 1 − r f q , then the following holds:…”

Section: Lemma 42 Let K Be the Operator Defined Bymentioning

confidence: 99%

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SPDEs driven by Poisson random measure with non Lipschitz coefficients: existence results

Hausenblas

2006

Probab. Theory Relat. Fields

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The article deals with SPDEs driven by Poisson random measure with non Lipschitz coefficients. Let A : E → E be a generator of an analytic semigroup on E, E being a certain Banach space. Let , F, (F t ) t≥0 , P be a stochastic basis carrying an E-valued Poisson random measure η with characteristic measure ν and compensator γ . Let 1 ≤ p ≤ 2. Our point of interest is the existence of solutions to SPDE's of e.g. the following typewherewhere 0 < r < 1 satisfy certain condition specified later and E 0 ⊂ E is again a certain Banach space.

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“…However, in the heavy-tailed situation, there are only few results. Articles [4,11,12,14,23] are devoted to heat equations with stable noise. To the best of our knowledge, there are no articles studying the wave equation with a heavy-tailed noise, neither there are attempts to consider equations with coloured stable noise.…”

Section: Introductionmentioning

confidence: 99%

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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The heat equation with time-independent multiplicative stable Lévy noise

Cited by 12 publications

References 13 publications

SPDEs with α-Stable Lévy Noise: A Random Field Approach

SPDEs with α-Stable Lévy Noise: A Random Field Approach

SPDEs driven by Poisson random measure with non Lipschitz coefficients: existence results

Wave equation with a coloured stable noise

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