Positive harmonic functions of transformed random walks

TL;DR

This paper investigates how the space of positive harmonic functions for a random walk on a discrete group is affected by changing the probability measure through bounded randomized stopping times, showing invariance of this space.

Contribution

It demonstrates that the space of positive harmonic functions remains unchanged under measure transformations induced by bounded randomized stopping times.

Findings

The space of positive harmonic functions is invariant under bounded randomized stopping times.

Changing the measure via such stopping times does not alter the harmonic function space.

The results apply to random walks on discrete groups.

Abstract

In this paper, we will study the behavior of the space of positive harmonic functions associated with the random walk on a discrete group under the change of probability measure by a randomized stopping time. We show that this space remains unchanged after applying a bounded randomized stopping time.