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 $L_u\stackrel{\mathrm{a.e.}}{=}\mathcal{L}_u$ has been known for some time, as has a counterexample to the same equality for three variable harmonic functions. I extend the $L_u\stackrel{\mathrm{a.e.}}{=}\mathcal{L}_u$ result to pluriharmonic functions in arbitrary dimensions.