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White noise driven SPDEs with reflection. (English) Zbl 0794.60059

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 solution \(u(t,x)\) is strictly positive it obeys the equation, and at a point \((t,x)\) where \(u(t,x)\) 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.
Reviewer: C.Donati-Martin

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
35R45 Partial differential inequalities and systems of partial differential inequalities
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