Intrinsic branching structure within random walk on $\mathbb{Z}$

TL;DR

This paper uncovers a branching process structure within a non-homogeneous random walk on integers, enabling explicit formulas for invariant densities and velocities in random environments with bounded jumps.

Contribution

It introduces a novel branching structure for non-homogeneous random walks with bounded jumps, linking ladder times to multitype branching processes and deriving explicit invariant measures.

Findings

Ladder time expressed via multitype branching process

Law of large numbers established for the walk in random environment

Explicit invariant density and velocity formulas derived

Abstract

In this paper, we reveal the branching structure for a non-homogeneous 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 non-homogeneous multitype branching process. As an application of the branching structure, we prove a law of large numbers of random walk in random environment with bounded jumps and specify the explicit invariant density for the Markov chain of ``the environment viewed from the particle" .The invariant density and the limit velocity could be expressed explicitly in terms of the environment.