We treat branching random walks in random environment using the framework of Linear Stochastic Evolution. In spatial dimensions three or larger, we establish diusive behaviour in the entire growth phase. This can be seen through a Central Limit Theorem with respect to the population density as well as through an invariance principle for a path measure we introduce.
"Branching Random Walks in Random Environment are Diffusive in the Regular Growth Phase." Electron. J. Probab. 16 1318 - 1340, 2011. https://doi.org/10.1214/EJP.v16-922