Let S be a group which is a semi-direct product of R^n and R^d, R^d acting of R^n in a appropriate way (that will be specified
during the talk). Given a left-invariant second order elliptic operator L on S (under suitable assumptions) there is smooth, positive
integrable function on R^n called a Poisson kernel that reproduces bounded L harmonic functions on S. It turns out that P dx is the
stationary measure for a random recursion X_{n+1}=f_{n+1}(X_n) where f_1, f_2,... are independent equally distributed random transfor
-mations of R^n, and X_n are defined recursively, X_0 being a constant vector. The law of f_1 is closely related to the heat kernel for
L. P dx is the law of the limit of X_n. The situation can be generalized to transformations f having the law with no relation to a diffe
-rential operator giving rise to a family of affine stochastic recursions who's stationary measures are of interest. Among them are the
classical ones considered by Kesten, Vervaat, Grincevicius, Goldie and more recent ones considered by Alsmeyer - Mentemeier, Guivarc'h
- Le Page, Guivarc'h - Buraczewski and myself. Asymptotics at infinity of stationary measures will be described.