BIASED RANDOM-WALKS ON GALTON-WATSON TREES

We consider random walks with a bias toward the root on the family tre e T of a supercritical Galton-Watson branching process and show that t he speed is positive whenever the walk is transient. The corresponding harmonic measures are carried by subsets of the boundary of dimension smaller than that of the whole boundary. When the bias is directed aw ay from the root and the extinction probability is positive, the speed may be zero even though the walk is transient; the critical bias for positive speed is determined.