The central topic of the talk is the class of Markov processes associated with the random walk on the group of the affine transformations of the real line. The corresponding generators L have a form of functional-differential operators with linearly transformed argument. These or similar operators were studied by Poincare, Birkhoff, Kato and other classics from the pure analytical point of view. We will present the complete analysis of the bounded L-harmonic functions (i.e. Martin boundary of our Markov processes).