Based on equations, symmetry and boundary conditions of quantum statistical correlation functions of renormalized spin wave excitations in a HEISENBERG ferromagnet including dipolar coupling, the parallel pumping effect under special considering the nonlinear three magnon interaction is discussed. An asymptotic approximation procedure enables us to interpret the fundamental equations in terms of the LANGEVIN theory of BROWNian motion and to find solutions by seperation ansatzes. For all cases of practical interest, the differences between the classical and the quantum statistical threshold condition can be neglected. The validity of SCHLÖMANN'S subthreshold χ″ formula is proved. Above the threshold, the steady state χ″ is larger than proposed by semiclassical theories because the strongly excited spin waves are influenced by a certain nonlinear fluctuating field; for YIG, a sufficient agreement between experimental and theoretical χ″ values and a qualitative explanation of the parameter dependence are found. Some non-equilibrium correlation functions are given explicitly.