Journal of Evolution Equations

, Volume 15, Issue 1, pp 1–26 | Cite as

Second order PDEs with Dirichlet white noise boundary conditions

  • Zdzisław Brzeźniak
  • Ben Goldys
  • Szymon Peszat
  • Francesco Russo
Article

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.

Mathematical Subject Classification

60H15 35J25 35K10 35K51 60G20 

Keywords

Heat equation Poisson equation Dirichlet problem White noise Boundary conditions Fractional Brownian motion 

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Copyright information

© Springer Basel 2014

Authors and Affiliations

  • Zdzisław Brzeźniak
    • 1
  • Ben Goldys
    • 2
  • Szymon Peszat
    • 3
  • Francesco Russo
    • 4
  1. 1.Department of MathematicsUniversity of YorkYorkUK
  2. 2.School of Mathematics and StatisticsThe University of SydneySydneyAustralia
  3. 3.Institute of MathematicsJagiellonian UniversityKrakówPoland
  4. 4.Ecole Nationale Supérieure des Techniques Avancées, ENSTA-ParisTech, Unité de Mathématiques appliquéesPalaiseauFrance

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