Nontangential and probabilistic boundary behavior of pluriharmonic functions

Abstract

Let u be a pluriharmonic function on the unit ball in ℂⁿ. 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 ℒ_u at which u converges along conditioned Brownian paths. For harmonic functions u of two variables, the result $L_{u}\stackrel{\mathrm{a.e.}}{=}\mathscr{L}_{u}$ has been known for some time, as has a counterexample to the same equality for three variable harmonic functions. I extend the $L_{u}\stackrel{\mathrm{a.e.}}{=}\mathscr{L}_{u}$ result to pluriharmonic functions in arbitrary dimensions.