Martingale solutions for Stochastic Equation of Reaction Diffusion Type driven by Lévy noise or Poisson random measure

Zdzislaw Brzezniak; Erika Hausenblas

Martingale solutions for Stochastic Equation of Reaction Diffusion Type driven by Lévy noise or Poisson random measure

Mathematics

Research output: Working paper › Preprint

Abstract

In this paper we are interested in a reaction diffusion equation driven by Poissonian noise respective L\'evy noise. For this aim we first show existence of a martingale solution for an SPDE of parabolic type driven by a Poisson random measure with only continuous and bounded coefficients. This result is transferred to an parabolic SPDE driven by L\'evy noise. In a second step, we show existence of a martingale solution of reaction diffusion type, also driven by Poissonian noise respective L\'evy noise. Our results answer positively a long standing open question about existence of martingale solutions driven by genuine L\'evy processes.

Original language	Undefined/Unknown
Number of pages	79
Publication status	Published - Oct 2010

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http://arxiv.org/abs/1010.5933

Cite this

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title = "Martingale solutions for Stochastic Equation of Reaction Diffusion Type driven by L{\'e}vy noise or Poisson random measure",

abstract = "In this paper we are interested in a reaction diffusion equation driven by Poissonian noise respective L\'evy noise. For this aim we first show existence of a martingale solution for an SPDE of parabolic type driven by a Poisson random measure with only continuous and bounded coefficients. This result is transferred to an parabolic SPDE driven by L\'evy noise. In a second step, we show existence of a martingale solution of reaction diffusion type, also driven by Poissonian noise respective L\'evy noise. Our results answer positively a long standing open question about existence of martingale solutions driven by genuine L\'evy processes. ",

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T1 - Martingale solutions for Stochastic Equation of Reaction Diffusion Type driven by Lévy noise or Poisson random measure

AU - Brzezniak, Zdzislaw

AU - Hausenblas, Erika

PY - 2010/10

Y1 - 2010/10

N2 - In this paper we are interested in a reaction diffusion equation driven by Poissonian noise respective L\'evy noise. For this aim we first show existence of a martingale solution for an SPDE of parabolic type driven by a Poisson random measure with only continuous and bounded coefficients. This result is transferred to an parabolic SPDE driven by L\'evy noise. In a second step, we show existence of a martingale solution of reaction diffusion type, also driven by Poissonian noise respective L\'evy noise. Our results answer positively a long standing open question about existence of martingale solutions driven by genuine L\'evy processes.

AB - In this paper we are interested in a reaction diffusion equation driven by Poissonian noise respective L\'evy noise. For this aim we first show existence of a martingale solution for an SPDE of parabolic type driven by a Poisson random measure with only continuous and bounded coefficients. This result is transferred to an parabolic SPDE driven by L\'evy noise. In a second step, we show existence of a martingale solution of reaction diffusion type, also driven by Poissonian noise respective L\'evy noise. Our results answer positively a long standing open question about existence of martingale solutions driven by genuine L\'evy processes.

M3 - Preprint

BT - Martingale solutions for Stochastic Equation of Reaction Diffusion Type driven by Lévy noise or Poisson random measure

ER -