Return Probabilities of Random Walks

TL;DR

This paper investigates whether the set of return probabilities uniquely identifies a random walk on integers, establishing conditions under which this is true and exploring implications in representation theory.

Contribution

It proves that, except for trivial cases, return probabilities uniquely determine the random walk on , and connects this to 's representation theory.

Findings

Return probabilities generally determine the random walk uniquely.

Trivial cases where different walks share the same return probabilities are characterized.

Application to representation theory of .

Abstract

Associated to a random walk on $\mathbb{Z}$ and a positive integer $n$, there is a return probability of the random walk returning to the origin after $n$ steps. An interesting question is when the set of return probabilities uniquely determines the random walk. There are trivial situations where two different random walks have the same return probabilities. However, among random walks in which these situations do not occur, our main result states that the return probabilities do determine the random walk. As a corollary, we will obtain a result dealing with the representation theory of $\mathrm{U}(1)$.