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Holomorphic diffusions and boundary behavior of harmonic functions

Publication ,  Journal Article
Chen, ZQ; Durrett, R; Ma, G
Published in: Annals of Probability
January 1, 1997

We study a family of differential operators {Lα, α ≥ 0} in the unit ball D of Cn with n ≥ 2 that generalize the classical Laplacian, α = 0, and the conformal Laplacian, α = 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 Lα-harmonic functions is studied in a unified way for 0 ≤ α ≤ 1/2. More specifically, we show that a bounded Lα-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 α runs from 0 to 1/2. A local version for this Fatou-type result is also established.

Duke Scholars

Published In

Annals of Probability

DOI

ISSN

0091-1798

Publication Date

January 1, 1997

Volume

25

Issue

3

Start / End Page

1103 / 1134

Related Subject Headings

  • Statistics & Probability
  • 0104 Statistics
  • 0101 Pure Mathematics
 

Citation

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Chen, Z. Q., Durrett, R., & Ma, G. (1997). Holomorphic diffusions and boundary behavior of harmonic functions. Annals of Probability, 25(3), 1103–1134. https://doi.org/10.1214/aop/1024404507
Chen, Z. Q., R. Durrett, and G. Ma. “Holomorphic diffusions and boundary behavior of harmonic functions.” Annals of Probability 25, no. 3 (January 1, 1997): 1103–34. https://doi.org/10.1214/aop/1024404507.
Chen ZQ, Durrett R, Ma G. Holomorphic diffusions and boundary behavior of harmonic functions. Annals of Probability. 1997 Jan 1;25(3):1103–34.
Chen, Z. Q., et al. “Holomorphic diffusions and boundary behavior of harmonic functions.” Annals of Probability, vol. 25, no. 3, Jan. 1997, pp. 1103–34. Scopus, doi:10.1214/aop/1024404507.
Chen ZQ, Durrett R, Ma G. Holomorphic diffusions and boundary behavior of harmonic functions. Annals of Probability. 1997 Jan 1;25(3):1103–1134.

Published In

Annals of Probability

DOI

ISSN

0091-1798

Publication Date

January 1, 1997

Volume

25

Issue

3

Start / End Page

1103 / 1134

Related Subject Headings

  • Statistics & Probability
  • 0104 Statistics
  • 0101 Pure Mathematics