A stochastic approach to quasi-everywhere boundary convergence of harmonic functions
Let X be a continuous strong Markov process living inside the unit ball, B, in \(R^ n\), \(n\geq 2\). Assume that X dies at the boundary, S, of B or \(\lim_{t\to \zeta}X_ t\in S\) exists a.s. Using Doob's h-transform X can be conditioned to exit B at a given point \(z\in S\). Denote this conditioned processes \(X^ z.\) In this paper it is proved - under some regularity assumptions on the \({\mathbb{P}}_ x\)-distribution of \(X_{\zeta}\) and an assumption of ``rotational invariance'' - that \[ \lim_{t\to \zeta}\tilde u(X^ z_ t)=\check u(z)\quad {\mathbb{P}}^ z-a.s. \] for quasi-all \(z\in S\). Here, ``quasi-all'' is with respect to a capacity induced by an appropriate Dirichlet form, \({\mathcal E}\), on S. The functions \(\tilde u\) and \(\check u\) are defined in terms of a real-valued function u on S such that \({\mathcal E}(u,u)<\infty\), \(\tilde u(x)={\mathbb{E}}_ x(u(X_{\zeta}))\) and \(\check u\) is a quasi-continuous version of u. A classical result of \textit{A. Beurling} [Acta Math. 72, 1-13 (1940; Zbl 0023.14204)] and its extensions by \textit{A. Broman} [On two classes of trigonometrical series, Thesis, Uppsala (1947)] and \textit{L. Carleson} [Selected problems on exceptional sets (1967; Zbl 0189.109)] are also discussed in this stochastic framework.
boundary convergence of harmonic functions
scientific article; zbMATH DE number 4036860
scientific article