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

We classify Radon stationary measures for a random walk on $$\mathbb{T}$^d \times $\mathbb{R}$$\mathrm{.}Thiswalkisrealisedbyarandomactionof$SL_{d}($\mathbb{Z}$)$onthe$$\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.