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We consider a particular Branching Random Walk in Random Environment (BRWRE)on $\sN_0$ started with one particle at the origin. Particles reproduceaccording to an offspring distribution (which depends on the location) and moveeither one step to the right (with a probability in $(0,1]$ which also dependson the location) or stay in the same place. We give criteria for local andglobal survival and show that global survival is equivalent to exponentialgrowth of the moments. Further, on the event of survival the number ofparticles grows almost surely exponentially fast with the same growth rate asthe moments.

Survival and Growth of a Branching Random Walk in Random Environment