Bounds on the Speed and on Regeneration Times for Certain Processes on Regular Trees

Abstract

We develop a technique that provides a lower bound on the speed of transient random walk in a random environment on regular trees. A refinement of this technique yields upper bounds on the first regeneration level and regeneration time. In particular, a lower and upper bound on the covariance in the annealed invariance principle follows. We emphasize the fact that our methods are general and also apply in the case of once-reinforced random walk. Durrett, Kesten and Limic (2002) prove an upper bound of the form $b/(b+δ)$ for the speed on the $b$-ary tree, where $δ$ is the reinforcement parameter. For $δ>1$ we provide a lower bound of the form $γ^2 b/(b+δ)$, where $γ$ is the survival probability of an associated branching process.