  We provide a complex-analytic approach to the classification of positive
stationary measures on $S^1$ with respect to complex finite Borel measures on
$PSU(1,1)$ satisfying the finite first moment condition by studying their
Poisson-like and Cauchy-like transforms from the perspective of generalized
analytic continuation. This approach allows us to establish a
functional-analytic criterion for singularity of a hitting measure for random
walks on lattices in $PSU(1,1)$. In addition, this approach suggests
alternative proofs to the several well-known related results by Furstenberg,
Kaimanovich, Guivarc'h-le Jan and Bourgain, among several others.
