Scaling limits in dependent random environments: relating a random walk, a branching process and a spatial branching process

Abstract: We extend existing connections between random walks, branching processes, and spatial branching processes, and their respective scaling limits, to include processes in dependent random environments. More specifically, we prove new scaling limits of a random walk in a dependent random environment, an associated branching process in a dependent random environment, and a spatial branching process in a dependent random environment. We show that the scaling limits are related in ways reminiscent of existing results in fixed environments. A Ray-Knight Theorem relates the scaling limits of the random walk in the random environment and the branching process in the random environment. The Brownian snake relates the scaling limit of the spatial branching process in the random environment to the scaling limit of the random walk in the random environment.