Nontangential and probabilistic boundary behavior of pluriharmonic functions

Abstract

Let $u$ be a pluriharmonic function on the unit ball in ${\mathbb{C}}^n$. I consider the relationship between the set of points $L_u$ on the boundary of the ball at which $u$ converges nontangentially and the set of points ${\mathcal{L}}_u$ at which $u$ converges along conditioned Brownian paths. For harmonic functions $u$ of two variables, the result $L_u\stackrel{a.e.}{=}{\mathcal{L}}_u$ has been known for some time, as has a counterexample to the same equality for three variable harmonic functions. I extend the $L_u\stackrel{a.e.}{=}{\mathcal{L}}_u$ result to pluriharmonic functions in arbitrary dimensions.

arXiv commentPublished at http://dx.doi.org/10.1214/009117906000000188 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)