English

Nontangential and probabilistic boundary behavior of pluriharmonic functions

Probability 2007-05-23 v2 Complex Variables

Abstract

Let uu be a pluriharmonic function on the unit ball in Cn\mathbb{C}^n. I consider the relationship between the set of points LuL_u on the boundary of the ball at which uu converges nontangentially and the set of points Lu\mathcal{L}_u at which uu converges along conditioned Brownian paths. For harmonic functions uu of two variables, the result Lu=a.e.LuL_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 Lu=a.e.LuL_u\stackrel{\mathrm{a.e.}}{=}\mathcal{L}_u result to pluriharmonic functions in arbitrary dimensions.

Keywords

Cite

@article{arxiv.math/0511368,
  title  = {Nontangential and probabilistic boundary behavior of pluriharmonic functions},
  author = {Steve Tanner},
  journal= {arXiv preprint arXiv:math/0511368},
  year   = {2007}
}

Comments

Published 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)