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

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