A stochastic approach to quasi-everywhere boundary convergence of harmonic functions
type: Item
instance of: scholarly article
title: A stochastic approach to quasi-everywhere boundary convergence of harmonic functions
zbMATH Open document ID: 0636.60078
DOI: 10.1016/0022-1236(87)90077-2
author: Rodrigo Bañuelos
author: B. Øksendal
published in: Journal of Functional Analysis
publication date: 1987-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 60J45
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 31A20
Mathematics Subject Classification ID: 31A15
zbMATH DE Number: 4036860
zbMATH Keywords: capacity
zbMATH Keywords: symmetric Markov process
zbMATH Keywords: boundary convergence of harmonic functions
zbMATH Keywords: strong Markov process
zbMATH Keywords: Doob's h-transform
zbMATH Keywords: regularity assumptions
reviewed by: Paavo Salminen
MaRDI profile type: Publication
full work available at URL: 0022-1236(87)90077-2
OpenAlex ID: W2063422697
cites work: A Maximal Function Characterization of the Class H p
cites work: Boundary behavior of harmonic functions in a half-space and Brownian motion
cites work: Q5583306
cites work: Brownian motion and analytic functions
cites work: Conditional brownian motion and the boundary limits of harmonic functions
cites work: Q3342857
cites work: Q3217388
cites work: Q3855442
cites work: Q3672857
cites work: Stochastic differential equations. An introduction with applications
cites work: Q3862204
Recommended article: On Stochastic Boundary Problems for Harmonic Functions in Banach Spaces
Recommended article: Q3737763
Recommended article: Gradient estimates for positive harmonic functions by stochastic analysis
Recommended article: Probabilistic proof of boundary Harnack principle for \(\alpha\)-harmonic functions
Recommended article: A stochastic approach to a priori estimates and Liouville theorems for harmonic maps
Recommended article: Boundary behavior of harmonic functions for truncated stable processes
Recommended article: A stochastic characterization of harmonic morphisms
Recommended article: Approximation by harmonic functions, and stability of the Dirichlet problem
description: scientific article; zbMATH DE number 4036860