Probability Theory and Related Fields

, Volume 93, Issue 1, pp 77-89

First online:

White noise driven quasilinear SPDEs with reflection

  • D. NualartAffiliated withFacultat de Matemàtiques, Universitat de Barcelona
  • , E. PardouxAffiliated withMathématiques, URA 225, Université de Provence

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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

60H15 35H60 35R45