# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Bensoussan, A., Lions, J.L.: Applications des inéquations variationnelles en contrôle stochastique. Paris: Dunod 1978; English translation. Amsterdam: North-Holland 1982 [2] Buckdahn, R., Pardoux, E.: Monotonicity methods for white noise driven SPDEs. In: Pinsky, M. (ed.) Diffusion processes and related problems in Analysis, vol. I, pp. 219-233. Boston Basel Stuttgart: Birkhäuser 1990 · Zbl 0722.60061 [3] Haussmann, U.G., Pardoux, E.: Stochastic variational inequalities of parabolic type. Appl. Math. Optimization20, 163-192 (1989) · Zbl 0701.60059 [4] Lions, J.L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Paris: Dunod 1969 [5] Manthey, R.: On the Cauchy problem for reaction diffusion equations with white noise. Math. Nachr.136, 209-228 (1988) · Zbl 0658.60089 [6] Mignot, F., Puel, J.P.: Inéquations d’évolution paraboliques avec convexe dépendant du temps. Applications aux inéquations quasivariationnelles d’évolution. Arch. Ration. Mech. Anal.64, 59-91 (1977). · Zbl 0362.49011 [7] Walsh, J.: An introduction to stochastic partial differential equations. In: Hennequin, P.L. (ed.), Ecole d’été de Probabilité de St Flour. (Lect. Notes Math., vol. 1180) Berlin Heidelberg New York: Springer 1986
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.