A non-homogeneous wave equation driven by a Poisson process

León, Jorge A.; Sarrà, Mònica

doi:10.1090/conm/336/06035

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Introduction1

Converge In Law To Those Of the Brownian Sheet1

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2010

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Cited by 2 publications

(2 citation statements)

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“…Notice also that an effort has been made in order to cover the case of Lévy noises e.g. in [24,14,12,18].…”

Section: Introductionmentioning

confidence: 99%

The 1-d stochastic wave equation driven by a fractional Brownian sheet

Quer-Sardanyons

Tindel

2007

Stochastic Processes and their Applications

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“…Notice also that an effort has been made in order to cover the case of Lévy noises e.g. in [24,14,12,18].…”

Section: Introductionmentioning

confidence: 99%

The 1-d stochastic wave equation driven by a fractional Brownian sheet

Quer-Sardanyons

Tindel

2007

Stochastic Processes and their Applications

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“…Eventually, it is worth commenting that other type of problems concerning SPDEs driven by Poisson-type noises have been considered e.g. in [18,20,23,28,30].…”

Section: Converge In Law To Those Of the Brownian Sheetmentioning

confidence: 99%

Weak Convergence for the Stochastic Heat Equation Driven by Gaussian White Noise

Bardina

María

Quer-Sardanyons

2010

Electron. J. Probab.

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In this paper, we consider a quasi-linear stochastic heat equation on [0, 1], with Dirichlet boundary conditions and controlled by the space-time white noise. We formally replace the random perturbation by a family of noisy inputs depending on a parameter n ∈ that approximate the white noise in some sense. Then, we provide sufficient conditions ensuring that the realvalued mild solution of the SPDE perturbed by this family of noises converges in law, in the space ([0, T ] × [0, 1]) of continuous functions, to the solution of the white noise driven SPDE. Making use of a suitable continuous functional of the stochastic convolution term, we show that it suffices to tackle the linear problem. For this, we prove that the corresponding family of laws is tight and we identify the limit law by showing the convergence of the finite dimensional distributions. We have also considered two particular families of noises to that our result applies. The first one involves a Poisson process in the plane and has been motivated by a one-dimensional result of Stroock, which states that the family of processes n t 0 (−1) N (n 2 s) ds, where N is a standard Poisson process, converges in law to a Brownian motion. The second one is constructed in terms of the kernels associated to the extension of Donsker's theorem to the plane.

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A non-homogeneous wave equation driven by a Poisson process

Cited by 2 publications

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The 1-d stochastic wave equation driven by a fractional Brownian sheet

The 1-d stochastic wave equation driven by a fractional Brownian sheet

Weak Convergence for the Stochastic Heat Equation Driven by Gaussian White Noise

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