Martin Boundary Theory of some Quantum Random Walks

In this paper we define a general setting for Martin boundary theory associated to quantum random walks, and prove a general representation theorem. We show that in the dual of a simply connected Lie subgroup of U(n), the extremal Martin boundary is homeomorphic to a sphere. Then, we investigate restriction of quantum random walks to Abelian subalgebras of group algebras, and establish a Ney-Spitzer theorem for an elementary random walk on the fusion algebra of SU(n), generalizing a previous result of Biane. We also consider the restriction of a quantum random walk on $SU_q(n)$ introduced by Izumi to two natural Abelian subalgebras, and relate the underlying Markov chains by classical probabilistic processes. This result generalizes a result of Biane.