Almost sure convergence for stochastically biased random walks on trees

作者：Faraud Gabriel^*; Hu Yueyun; Shi Zhan

来源：Probability Theory and Related Fields, 2012, 154(3-4): 621-660.

DOI：10.1007/s00440-011-0379-y

• Biased random walk on a Galton-Watson tree
• Branching random walk
• Slow movement
• Random walk in a random environment

We are interested in the biased random walk on a supercritical Galton-Watson tree in the sense of Lyons (Ann. Probab. 18:931-958, 1990) and Lyons, Pemantle and Peres (Probab. Theory Relat. Fields 106:249-264, 1996), and study a phenomenon of slow movement. In order to observe such a slow movement, the bias needs to be random; the resulting random walk is then a tree-valued random walk in random environment. We investigate the recurrent case, and prove, under suitable general integrability assumptions, that upon the system%26apos;s non-extinction, the maximal displacement of the walk in the first n steps, divided by (log n)(3), converges almost surely to a known positive constant.

• 出版日期2012-12