Random walks on finite volume homogeneous spaces (Q663302)

The authors study random walks on finite-volume homogeneous spaces which belong to products of a real and a \(p\)-adic Lie group. They show that such random walks are recurrent if their transition probabilities have finite exponential moments and their support generates a subgroup whose Zariski closure is semi-simple. This result extends results of \textit{A. Eskin} and \textit{G. Margulis} [in: V. A. Kaimanovich, (ed.), Random walks and geometry. Collected papers. Berlin: de Gruyter.431--444 (2004; Zbl 1064.60092)] and answers a conjecture of them. The proof is based on an inequality in the exterior algebra of a finite dimensional vector space which leads to some contraction principle for the random walks under consideration. Recurrence is then a consequence of this contraction principle.