Law of large numbers for random walk with unbounded jumps and BDP with bounded jumps in random environment

TL;DR

This paper establishes a law of large numbers for random walks with unbounded jumps and for birth-death processes with bounded jumps in random environments, under specific ergodic and elliptic conditions, providing explicit velocities.

Contribution

It extends the law of large numbers to processes with unbounded jumps and explicitly characterizes velocities in bounded jump birth-death processes in random environments.

Findings

Law of large numbers proven for unbounded jump random walk.

Explicit velocity formula derived for birth-death processes.

Positive velocity established under finite mean hitting time condition.

Abstract

We study random walk with unbounded jumps in random environment. The environment is stationary and ergodic, uniformly elliptic and decays polynomially with speed $Dj^{-(3+\varepsilon_0)}$ for some small $\varepsilon_0>0$ and proper $D>0.$ We prove a law of large number with positive velocity under the condition that the annealed mean of the hitting time of the positive half lattice is finite. Secondly, we consider birth and death process with bounded jumps in stationary and ergodic environment. Under the uniformly elliptic condition, we prove a law of large number and give the explicit formula of its velocity.