  We present versions of several classical results on harmonic functions and
Poisson boundaries in the setting of locally compact quantum groups. In
particular, the Choquet--Deny theorem holds for compact quantum groups; also,
the result of Kaimanovich--Vershik and Rosenblatt, which characterizes group
amenability in terms of harmonic functions, admits a non-commutative analogue
in the separable case. We also explore the relation between classical and
quantum Poisson boundaries by investigating the spectrum of the quantum group.
We apply this machinery to find a concrete realization of the Poisson
boundaries of the compact quantum group $SU_{q}(2)$ arising from measures on
its spectrum.
