Open Access
Translator Disclaimer
July 1997 Holomorphic diffusions and boundary behavior of harmonic functions
Zhen-Qing Chen, Richard Durrett, Gang Ma
Ann. Probab. 25(3): 1103-1134 (July 1997). DOI: 10.1214/aop/1024404507


We study a family of differential operators ${L^{\alpha}, \alpha \geq 0}$ in the unit ball $D$ of $C^n$ with $n \geq 2$ that generalize the classical Laplacian, $\alpha = 0$, and the conformal Laplacian, $\alpha = 1/2$ (that is, the Laplace-Beltrami operator for Bergman metric in $D$). Using the diffusion processes associated with these (degenerate) differential operators, the boundary behavior of $L^{\alpha}$-harmonic functions is studied in a unified way for $0 \leq \alpha \leq 1/2$. More specifically, we show that a bounded $L^{\alpha}$-harmonic function in $D$ has boundary limits in approaching regions at almost every boundary point and the boundary approaching region increases from the Stolz cone to the Korányi admissible region as $\alpha$ runs from 0 to 1/2. A local version for this Fatou-type result is also established.


Download Citation

Zhen-Qing Chen. Richard Durrett. Gang Ma. "Holomorphic diffusions and boundary behavior of harmonic functions." Ann. Probab. 25 (3) 1103 - 1134, July 1997.


Published: July 1997
First available in Project Euclid: 18 June 2002

zbMATH: 0891.60072
MathSciNet: MR1457613
Digital Object Identifier: 10.1214/aop/1024404507

Primary: 31B25 , 60J45
Secondary: 31B10 , 60J60

Keywords: approaching region , boundary limit , conditional process , harmonic measure , Harnack inequality , hitting probability , holomorphic and $L$-harmonic functions , Holomorphic diffusions , martingale

Rights: Copyright © 1997 Institute of Mathematical Statistics


Vol.25 • No. 3 • July 1997
Back to Top