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 noncommutative 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 SUq(2) arising from measures on its spectrum.
- Choquet-Deny theorem
- Poisson boundary
- harmonic functions
- locally compact quantum groups
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