Uniqueness of the mixing measure for a random walk in a random environment on the positive integers

Maren Eckhoff,Silke Rolles

doi:10.1214/ecp.v14-1441

Abstract

Consider a random walk in an irreducible random environment on the positive integers. We prove that the annealed law of the random walk determines uniquely the law of the random environment. An application to linearly edge-reinforced random walk is given.

Highlights

Introduction and resultsRandom walk in a random environment
We prove that the annealed law of the random walk determines uniquely the law of the random environment
We say that (X t )t∈ 0 is a random walk in a random environment on 0 if there exists a probability measure on [0, 1] such that

Summary

Introduction

Introduction and resultsRandom walk in a random environment. Let G = ( 0, E) be the graph with vertex set 0 and set of undirected edges E = {{n, n + 1}, n ∈ 0}. We consider random walk in a random environment on G defined as follows: Let Ω0 ⊆ We say that (X t )t∈ 0 is a random walk in a random environment on 0 if there exists a probability measure on [0, 1] such that

Results

Conclusion