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Stochastic partial differential equations with reflection and Malliavin calculus. (EPDS réfléchies et calcul de Malliavin.) (French) Zbl 0884.60049

This paper deals with the following nonlinear one-dimensional heat equation with reflection perturbed by a space-time white noise: \[ {\partial u\over\partial t}={\partial^2u\over\partial x^2}+ f(u(t, x))+ \sigma(u(t, x))\dot W(t, x)+ \eta(t, x), \] \(t>0\), \(x\in[0,1]\), with Dirichlet boundary conditions \(u(t, 0)= u(t, 1)= 0\) and initial condition \(u(0, x)= u_0(x)\). Here \(\eta\) is a random measure that forces the solution \(u\) to be nonnegative. The coefficients \(\sigma\) and \(f\) are supposed to be continuously differentiable with bounded derivatives. These equations were studied by D. Nualart and E. Pardoux [Probab. Theory Relat. Fields 93, No. 1, 77-89 (1992; Zbl 0767.60055)] in the case where \(\sigma\) is constant, and by C. Donati-Martin and E. Pardoux [ibid. 95, No. 1, 1-24 (1993; Zbl 0794.60059)] in the general case. In this paper, using the techniques of the Malliavin calculus, the authors show that if \(\sigma>0\), then the law of the solution \(u(t_0, x_0)\) restricted to \((0,+\infty)\) is absolutely continuous, for any \(t_0>0\) and \(0<x_0<1\).

MSC:

60H07 Stochastic calculus of variations and the Malliavin calculus
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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