Local limit theorem and equivalence of dynamic and static points of view for certain ballistic random walks in i.i.d. environments

TL;DR

This paper establishes the equivalence between static and dynamic perspectives for certain ballistic random walks in i.i.d. environments in dimensions four and higher, and proves a local limit theorem relating quenched and annealed measures.

Contribution

It demonstrates the equivalence of static and dynamic viewpoints for ballistic random walks in high dimensions and derives a local limit theorem connecting quenched and annealed measures.

Findings

Equivalence between static and dynamic points of view in dimension d≥4.

A local limit theorem relating quenched and annealed measures.

Results applicable to ballistic random walks in i.i.d. environments.

Abstract

In this work, we discuss certain ballistic random walks in random environments on $\mathbb{Z}^d$, and prove the equivalence between the static and dynamic points of view in dimension $d\geq4$. Using this equivalence, we also prove a version of a local limit theorem which relates the local behavior of the quenched and annealed measures of the random walk by a prefactor.