Random mass splitting and a quenched invariance principle

TL;DR

This paper proves a quenched invariance principle for a random walk in a time-changing environment, establishing a central limit theorem for a related mass splitting model, advancing understanding of stochastic processes in dynamic random environments.

Contribution

It introduces a quenched invariance principle for a novel random walk in a space-time random environment, linking it to a mass splitting model and extending prior theoretical frameworks.

Findings

Established a quenched invariance principle for the model

Proved a quenched central limit theorem for the mass splitting distribution

Extended existing methods to time-changing environments

Abstract

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 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 quenched invariant process for a random walk in a space-time random environment laid out by Rassoul-Agha and Sepp\"al\"ainen which in turn was based on the work of Kipnis and Varadhan and others.