Probabilistic approach to semilinear and generalized mixed boundary value problems
In the late eighties the authors have written several papers on probabilistic approach to various types of the Schrödinger equation. A survey of methods and results is given in [\textit{Z. Ma} and \textit{R. Song}, Stochastic processes, Semin., San Diego/CA (USA) 1989, Prog. Probab. 18, 135-164 (1990; Zbl 0691.60050)]. One of the main steps in their approach is the analysis of the gauge function for the corresponding problem. They succeeded to extend the method to the case of infinite gauge which corresponds to nonnegative eigenvalues of the infinitesimal generator. In this paper the authors present a probabilistic solution to the following semilinear and generalized mixed boundary value problem: \[ \begin{aligned} -(\Delta/2+\mu I_ D)u +f_ 1(u) & =\nu I_ D\quad\text{in } D, \\ u|_{I_ 1} & =g, \\ \partial u/\partial n|_{I_ 2}- 2u\mu I_{I_ 2}+2f_ 2(u) & =2\nu I_{I_ 2},\end{aligned} \] where \(D=G-B\), \(G\) a bounded \(C^ 3\) domain in \(\mathbb{R}^ d\) and \(B\) a closed subset of \(\overline G\), \(I_ 1=\partial D\cap B\), \(I_ 2=\partial D- B\), \(\mu\) and \(\nu\) signed measures on \(\tilde D=D\cup I_ 2\) belonging to the generalized Kato class, \(f_ 1\) and \(f_ 2\) continuously differentiable functions on \(\mathbb{R}\), \(g\) a bounded measurable function on \(I_ 1\), and \(\partial/\partial n\) is the outward normal derivative. The authors first solve the linear mixed boundary value problem (for the finite gauge), and based on the obtained results they give the probabilistic approach to the above semilinear problem.
nonnegative eigenvalues of the infinitesimal generator
generalized mixed boundary value problem
scientific article; zbMATH DE number 177970