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Martin boundary theory of some quantum random walks. (English) Zbl 1051.31005

The Martin boundary is an important subject in analytic or probabilistic potential theory since it makes possible an integral representation for positive harmonic functions. The present paper defines in a reasonably general setting the notion of quantum Martin boundary for quantum random walks. The corresponding integral representation is obtained. Moreover it is proved that, for a special quantum random walk, the Martin boundary coincides with the minimal Martin boundary and is homeomorphic with an Euclidean sphere. In this respect, it extends several results of P. Biane [see, for example, Probab. Theory Relat. Fields 89, No. 1, 117–129 (1991; Zbl 0746.46058)].

MSC:

31C35 Martin boundary theory
46L53 Noncommutative probability and statistics
81S25 Quantum stochastic calculus

Citations:

Zbl 0746.46058
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