Radon stationary measures for a random walk on \(\mathbb{T}^d\times\mathbb{R}\). (Mesures de Radon stationnaires pour une marche aléatoire sur \(\mathbb{T}^d\times\mathbb{R}\).) (French. English summary) Zbl 1519.37048
Summary: 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 \(\mathrm{SL}_d(\mathbb{Z})\) 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.
MSC:
| 37H12 | Random iteration |
| 37B20 | Notions of recurrence and recurrent behavior in topological dynamical systems |
| 37A50 | Dynamical systems and their relations with probability theory and stochastic processes |
| 37C40 | Smooth ergodic theory, invariant measures for smooth dynamical systems |
| 60G50 | Sums of independent random variables; random walks |
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