On uniqueness of invariant measures for random walks on HOMEO+ (R)

We consider random walks on the group of orientation-preserving homeomorphisms of the real line R. In particular, the fundamental question of uniqueness of an invariant measure of the generated process is raised. This problem was studied by Choquet and Deny [Sur l'equation de convolution mu = mu * sigma. C. R. Acad. Sci. Paris 250 (1960), 799-801] in the context of random walks generated by translations of the line. Nowadays the answer is quite well understood in general settings of strongly contractive systems. Here we focus on a broader class of systems satisfying the conditions of recurrence, contraction and unbounded action. We prove that under these conditions the random process possesses a unique invariant Radon measure on R. Our work can be viewed as following on from Babillot et al [The random difference equation X-n = A(n)X(n-1) + B-n in the critical case. Ann. Probab. 25(1) (1997), 478-493] and Deroin et al [Symmetric random walk on HOMEO+ (R). Ann. Probab. 41(3B) (2013), 2066-2089].