Let u be a pluriharmonic function on the unit ball in ℂn. I consider the relationship between the set of points Lu on the boundary of the ball at which u converges nontangentially and the set of points ℒu 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.
@article{1158673331,
author = {Tanner, Steve},
title = {Nontangential and probabilistic boundary behavior of pluriharmonic functions},
journal = {Ann. Probab.},
volume = {34},
number = {1},
year = {2006},
pages = { 1623-1634},
language = {en},
url = {http://dml.mathdoc.fr/item/1158673331}
}
Tanner, Steve. Nontangential and probabilistic boundary behavior of pluriharmonic functions. Ann. Probab., Tome 34 (2006) no. 1, pp. 1623-1634. http://gdmltest.u-ga.fr/item/1158673331/