Stochastic partial differential equations with reflection and Malliavin calculus
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 \textit{D. Nualart} and \textit{E. Pardoux} [Probab. Theory Relat. Fields 93, No. 1, 77-89 (1992; Zbl 0767.60055)] in the case where \(\sigma\) is constant, and by \textit{C. Donati-Martin} and \textit{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 \(0<x_0<1\).
stochastic partial differential equations
scientific article; zbMATH DE number 1057339