Projection theorems for hitting probabilities and a theorem of Littlewood (Q1059323)

\textit{J. E. Littlewood} [Proc. Lond. Math. Soc. 28, 383-394 (1928)] showed that a positive superharmonic function u on the unit disc has radial limits a.e. Using techniques due to Doob this result is extended to all rank one symmetric spaces. In addition simplifications are obtained of \textit{J. L. Doob'}s proof [Ann. Inst. Fourier 15, No.1, 113-135 (1965; Zbl 0154.075), errata ibid. 17, No.1, 469 (1967)] of normal convergence a.e. of a positive superharmonic function on a half space. The symmetric space analogue of this half space result is also obtained. The methods used are shown to fail for the potential theory on \({\mathbb{R}}^ n\) associated with \(\Delta u=au\) \((a>0)\). It is an open question as to whether Littlewood's theorem holds in this context. When the methods work they also establish a lower estimate for the probability of Brownian motion hitting a set in terms of the solid angle spanned by the set when viewed from the starting point of the Brownian paths.