Intrinsic Branching Structure within Random Walk on Z

Abstract

We reveal the branching structure for a nonhomogeneous random walk with bounded jumps. The ladder time $T_1,$ the first hitting time of $[1,\infty)$ by the walk starting from $0,$ could be expressed in terms of a nonhomogeneous multitype branching process. As an application of the branching structure, we prove a law of large numbers of random walk with bounded jumps in random environment and specify the explicit invariant density for the Markov chain of “the environment viewed from the particles.” The invariant density and the limit velocity could be expressed explicitly in terms of the environment.