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White noise driven SPDEs with reflection
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  • Published: March 1993

White noise driven SPDEs with reflection

  • C. Donati-Martin1 &
  • E. Pardoux1 

Probability Theory and Related Fields volume 95, pages 1–24 (1993)Cite this article

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Summary

We study reflected solutions of a nonlinear heat equation on the spatial interval [0, 1] with Dirichlet boundary conditions, driven by space-time white noise. The nonlinearity appears both in the drift and in the diffusion coefficient. Roughly speaking, at any point (t, x) where the solutionu(t, x) is strictly positive it obeys the equation, and at a point (t, x) whereu(t, x) is zero we add a force in order to prevent it from becoming negative. This can be viewed as an extension both of one-dimensional SDEs reflected at 0, and of deterministic variational inequalities. Existence of a minimal solution is proved. The construction uses a penalization argument, a new existence theorem for SPDEs whose coefficients depend on the past of the solution, and a comparison theorem for solutions of white-noise driven SPDEs.

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Authors and Affiliations

  1. Mathématiques, URA 225, Université de Provence, 3 place Victor Hugo, F-13331, Marseille Cedex 3, France

    C. Donati-Martin & E. Pardoux

Authors
  1. C. Donati-Martin
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  2. E. Pardoux
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Partially supported by DRET under contract 901636/A000/DRET/DS/SR

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Cite this article

Donati-Martin, C., Pardoux, E. White noise driven SPDEs with reflection. Probab. Th. Rel. Fields 95, 1–24 (1993). https://doi.org/10.1007/BF01197335

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  • Received: 12 April 1991

  • Revised: 25 March 1992

  • Issue Date: March 1993

  • DOI: https://doi.org/10.1007/BF01197335

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Mathematics Subject Classification

  • 60 H 15
  • 35 R 60
  • 35 R 45
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