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

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.
Comment: 25 pages