Radon stationary measures for a random walk on $\mathbb{T}^d \times \mathbb{R}$

TL;DR

This paper classifies Radon stationary measures for a specific random walk on a product space involving a torus and real line, revealing their rigidity and structure under certain conditions.

Contribution

It provides a classification of Radon stationary measures for a random walk on d6^d imes d6, demonstrating their rigidity and homogeneity under irreducibility and recurrence assumptions.

Findings

Radon stationary measures are classified under the given dynamics.

Stationary measures exhibit rigidity and homogeneity.

Results depend on irreducibility and recurrence conditions.

Abstract

We classify Radon stationary measures for a random walk on $\mathbb{T}^d \times \mathbb{R}$. This walk is realised by a random action of $SL_{d}(\mathbb{Z})$ on the $\mathbb{T}^d$ component, coupled with a translation on the $\mathbb{R}$ component. We show, under assumptions of irreducibility and recurrence, the rigidity and homogeneity of Radon ergodic stationary measures.