White noise driven quasilinear SPDEs with reflection

Summary 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.INRIAPartially supported by DRET under contract 901636/A000/DRET/DS/SR