Random mass splitting and a quenched invariance principle

We will investigate a random mass splitting model and the closely related random walk in a random environment (RWRE). The heat kernel for the RWRE at time t is the mass splitting distribution at t. We prove a quenched invariance principle (QIP) for the RWRE which gives us a quenched central limit theorem for the mass splitting model. Our RWRE has an environment which is changing with time. We follow the outline for proving a QIP for a random walk in a space-time random environment laid out by Rassoul-Agha and Seppäläinen (2005) which in turn was based on the work of Kipnis and Varadhan (1986) and others.