Abstract
We establish the existence of weak martingale solutions to a class of second order parabolic stochastic partial differential equations. The equations are driven by multiplicative jump type noise, with a non-Lipschitz multiplicative functional. The drift in the equations contains a dissipative nonlinearity of polynomial growth.
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Acknowledgments
Open access funding provided by Montanuniversität Leoben. The research by E. Hausenblas and P. A. Razafimandimby has been funded by the FWF-Austrian Science Fund through the project P21622. The research on this paper was initiated during the visit of Hausenblas to the University of York in October 2008. She would like to thank the Mathematics Department at York for hospitality. A major part of this paper was written when Razafimandimby was an FWF Lise Meitner fellow (with project number M1487) at the Montanuniversität Leoben. He is very grateful to the FWF and the Montanuniversität Leoben for their support. Razafimandimby’s current research is partially National supported by the National Research Foundation South Africa (Grant number 109355). The authors would like to thank Jerzy Zabczyk for discussion related to the dual predictable projection of a Poisson random measure and to Szymon Peszat for discussion related to an example from his paper [64].
We also would like thank the anonymous referees for their insightful comments and help to clarify issues from previous version of the paper; in particular for their help in clarifying the construction of stochastic integral with respect to Poisson random measure (PRM) and progressively measurable integrands.
Last but not least, the authors would like to thank Carl Chalk, Pani Fernando, Ela Motyl, Markus Riedle, Akash Panda and Nimit Rana for a careful reading of the manuscript. Earlier versions of this paper can be found on arXiv:1010.5933.
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This work was supported by the FWF-Project P17273-N12
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Brzeźniak, Z., Hausenblas, E. & Razafimandimby, P.A. Stochastic Reaction-diffusion Equations Driven by Jump Processes. Potential Anal 49, 131–201 (2018). https://doi.org/10.1007/s11118-017-9651-9
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DOI: https://doi.org/10.1007/s11118-017-9651-9
Keywords
- Itô integral driven by a Poisson random measure
- Stochastic partial differential equations
- Lévy processes
- Reaction diffusion equations