Boundary Behaviour of Harmonic Functions in a Half-Space and Brownian Motion

  • Burgess Davis
  • Renming Song
Part of the Selected Works in Probability and Statistics book series (SWPS)


The behaviour of harmonic functions in the half-space \(R_ + ^{n + 1}\) has been discussed from two points of view: geometrical and probabilistic. In this paper, we compare these two view points with respect to (1) local convergence at the boundary and (2) the Hp-spaces. The results are as follows: (1) The existence of a nontangential limit for almost all points in a set E of positive Lebesgue measure in \({R^n}\left( { = \partial \,R_ + ^{n + 1}} \right)\) is more restrictive than the existence of a « fine » or probability limit almost everywhere in E when \(n \geqslant 2\). When \(n=1\), the existence of a nontangential limit almost everywhere in E implies the existence of a « fine » limit almost everywhere in E and conversely.


Brownian Motion Harmonic Function Maximal Function Local Convergence Area Function 
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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Mathematics and Department of StatisticsPurdue UniversityWest LafayetteUSA
  2. 2.Department of MathematicsUniversity of IllinoisUrbanaUSA

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