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White noise driven quasilinear SPDEs with reflection. (English) Zbl 0767.60055

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 solution \(u(x,t)\) is strictly positive it obeys the equation, and at a point \((x,t)\) where \(u(x,t)\) 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.

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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