Symmetric random walks on the real line

The talk will be devoted to a recent joint work of Bertrand Deroin, Andres Navas, Kamlesh Parwani, and myself (arXiv: 1103.1650). We study random walks on the real line, generated by a finite number of orientation-preserving homeomorphisms of $R$, without any smoothness assumption, but with the symmetry one: $f$ and $f^{-1}$ are equiprobable to be applied.
Excluding some degenerate situations (like the presence of a common fixed point or semi-conjugacy to a group of translations), one can prove, that there never exists a finite stationary measure. On the other hand, one can always find an infinite (a sigma-finite) one. Moreover, it turns out that the walk is always recurrent: a random trajectory almost surely oscillates between plus and minus infinity (thus visiting any sufficiently big interval infinitely many times).
Finally, in the case of a minimal dynamics, after a change of variables that maps the stationary measure to the Lebesgue one, we observe a very interesting effect. Each of the maps become (uniformly on $R$) Lipschitz, and, moreover, with a bounded (uniformly on $R$) displacement $|g(x)-x|$. Finally, we have a Derriennic property: the expectation $\sum p_j g_j(x)$ is equal to $x$ for all $x\in R$ (in fact, this is even stronger than the classic Derriennic propery, where this equality is required only for large $x$).