White noise driven quasilinear SPDEs with reflection
Facultat de MatemàtiquesUniversitat de Barcelona
Mathématiques, URA 225Université de Provence
Cite this article as:
Nualart, D. & Pardoux, E. Probab. Th. Rel. Fields (1992) 93: 77. doi:10.1007/BF01195389
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.