Probability Theory and Related Fields

, Volume 93, Issue 1, pp 77–89

White noise driven quasilinear SPDEs with reflection

  • D. Nualart
  • E. Pardoux

DOI: 10.1007/BF01195389

Cite this article as:
Nualart, D. & Pardoux, E. Probab. Th. Rel. Fields (1992) 93: 77. doi:10.1007/BF01195389


We study reflected solutions of the heat equation on the spatial interval [0, 1] with Dirichlet boundary conditions, driven by an additive space-time white noise. Roughly speaking, at any point (x, t) where the solutionu(x, t) is strictly positive it obeys the equation, and at a point (x, t) whereu(x, t) 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. An existence and uniqueness result is proved, which relies heavily on new results for a deterministic variational inequality.

Mathematics Subject Classification


Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • D. Nualart
    • 1
  • E. Pardoux
    • 2
  1. 1.Facultat de MatemàtiquesUniversitat de BarcelonaBarcelonaSpain
  2. 2.Mathématiques, URA 225Université de ProvenceMarseille Cedex 3France