Hierarchically Organized Iterative Solutions of the Evolution Equations in QCD

The task of Monte Carlo simulation of the evolution of the parton distributions in QCD and of constructing new parton shower Monte Carlo algorithms requires new way of organizing solutions of the QCD evolution equations, in which quark-gluon transitions on one hand and quark-quark or gluon-gluon transitions (pure gluonstrahlung) on the other hand, are treated separately and differently. This requires certain reorganization of the iterative solutions of the QCD evolution equations and leads to what we refer to as a hierarchic iterative solutions of the evolution equations. We present three formal derivations of such a solution. Results presented here are already used in the other recent works to formulate new MC algorithms for the parton-shower-like implementations of the QCD evolution equations. They are primarily of the non-Markovian type. However, such a solution can be used for the Markovian-type MCs as well. We also comment briefly on the relation of the presented formalism to similar methods used in other branches of physics.