White noise driven SPDEs with reflection
type: Item
instance of: scholarly article
title: White noise driven SPDEs with reflection
zbMATH Open document ID: 0794.60059
DOI: 10.1007/BF01197335
author: Etienne Pardoux
author: Catherine Donati-Martin
publication date: 1994-08-15T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 35R45
zbMATH DE Number: 569036
zbMATH Keywords: nonlinear heat equation
zbMATH Keywords: Dirichlet boundary conditions
zbMATH Keywords: variational inequalities
zbMATH Keywords: existence theorem
zbMATH Keywords: comparison theorem
MaRDI profile type: Publication
cites work: Q4199298
cites work: Q4053162
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published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 569036