For a simply connected plane Greenian domain $D$ we give a natural description of the means of exit of a Brownian path from $D$. We give a related discussion of the connection between classical and probabilistic Fatou theorems.
M. Cranston. "On the Means of Approach to the Boundary of Brownian Motion." Ann. Probab. 15 (3) 1009 - 1013, July, 1987. https://doi.org/10.1214/aop/1176992077