Probability Theory and Related Fields

, Volume 95, Issue 1, pp 1–24

White noise driven SPDEs with reflection


  • C. Donati-Martin
    • Mathématiques, URA 225Université de Provence
  • E. Pardoux
    • Mathématiques, URA 225Université de Provence

DOI: 10.1007/BF01197335

Cite this article as:
Donati-Martin, C. & Pardoux, E. Probab. Th. Rel. Fields (1993) 95: 1. doi:10.1007/BF01197335


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.

Mathematics Subject Classification

60 H 1535 R 6035 R 45

Copyright information

© Springer-Verlag 1993