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Second order PDEs with Dirichlet white noise boundary conditions

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Abstract

In this paper, we study inhomogeneous Dirichlet boundary problems associated to the Poisson and heat equations on bounded and unbounded domains with smooth boundary and random boundary data. The main novelty of this work is a convenient framework for the analysis of equations excited by the white in time and/or space noise on the boundary. Our approach allows us to show the existence and uniqueness of weak solutions in the space of distributions. We also prove that the solutions can be identified as smooth functions inside the domain, and finally, the rate of their blow up at the boundary is estimated. A large class of noises including Wiener and fractional Wiener space-time white noise, homogeneous noise and Lévy noise are considered.

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Authors and Affiliations

  1. Department of Mathematics, University of York, York, YO10 5DD, UK

    Zdzisław Brzeźniak

  2. School of Mathematics and Statistics, The University of Sydney, Sydney, 20006, Australia

    Ben Goldys

  3. Institute of Mathematics, Jagiellonian University, Łojasiewicza 6, 30-348, Kraków, Poland

    Szymon Peszat

  4. Ecole Nationale Supérieure des Techniques Avancées, ENSTA-ParisTech, Unité de Mathématiques appliquées, 828, boulevard des MarÉchaux, 91120, Palaiseau, France

    Francesco Russo

Authors
  1. Zdzisław Brzeźniak
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  2. Ben Goldys
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  3. Szymon Peszat
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  4. Francesco Russo
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Correspondence to Zdzisław Brzeźniak.

Additional information

Szymon Peszat’s Research was supported by Polish National Science Center Grant DEC2013/09/ B/ST1/03658. Ben Goldys Research was supported by the ARC Grant DP120101886. Francesco Russo was partially supported by the ANR Project MASTERIE 2010 BLAN 0121 01.

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Brzeźniak, Z., Goldys, B., Peszat, S. et al. Second order PDEs with Dirichlet white noise boundary conditions. J. Evol. Equ. 15, 1–26 (2015). https://doi.org/10.1007/s00028-014-0246-2

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  • DOI: https://doi.org/10.1007/s00028-014-0246-2

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