2016/07/19 by Bruère, Caroline
#Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1607.05694
Take G a locally compact second-countable group, and H a subgroup of G. Choose μ a probability measure on G, such that the group spanned by its support is dense in G, and consider the Markov chain on the homogeneous space X=G/H with transition probability Px=μ*δx for x∈ X. Under some conditions on G, H, μ, we know that this Markov chain is either everywhere recurrent or everywhere transient. A natural question is whether such a dichotomy is universally true. The goal of this paper is to show it is not, even when G is finitely generated, through the explicit construction of a counter example. The methods used include the study of stable laws, Gnedenko-Kolmogorov's local limit theorem for stable laws, and the study of the time of first return to equilibrium.