Recurrence of random walks on completely simple semigroups (Q1084742)

This paper studies a problem of recurrence of random walks on completely simple semigroups. These semigroups have a special structure of the form \(X\times G\times Y\), where the middle factor is a group. See for their description and some results in this context the book by the reviewer and \textit{N. A. Tserpes} [Measures on topological semigroups: Convolution products and random walks. (1976; Zbl 0342.43001)]. The author studies the following problem: Given a recurrent random walk on such a semigroup \(X\times G\times Y\), under what conditions G is also recurrent, and conversely? Similar questions were first studied by \textit{J. Larisse} [Ann. Inst. Henri Poincaré, Sect. B 8, 107-125 (1972; Zbl 0241.60053); ibid. 127-173 (1972; Zbl 0241.60054) and ibid. 229-240 (1972; Zbl 0248.60065)].