Holomorphic diffusions and boundary behavior of harmonic functions

We study a family of differential operators {L^α, 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 L^α-harmonic functions is studied in a unified way for 0 ≤ α ≤ 1/2. More specifically, we show that a bounded L^α-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.