Survival of a Branching Random Walk in Random Environment

We consider a particular Branching Random Walk in Random Environment (BRWRE) on $N_0$ started with one particle at the origin. Particles reproduce according to an offspring distribution (which depends on the location) and move either one step to the right (with a probability in $(0,1]$ which may also depend on the location) or stay in the same place. We give criteria for local and global survival and show that global survival is equivalent to exponential growth of the moments.