On derivations and solutions of master equations and asymptotic representations

Abstract

A linear semi-group for both: Markov processes and non-Markov processes as they occur in the description of macroscopic systems is introduced. The elegance of the semigroup approach is demonstrated by the derivation of the master equation for a Markov process which undergoes continuous and discontinuous jumps. By use of nonlinear transformations of stochastic processes a class of processes is found for which the whole stochastic kinetics reduces mainly to the kinetics of a general Gauss-Markov process. Further the convergence of sequences of Markov processes to a limiting Markov process is studied. In this context, a semi-group formulation for the validity of various expansion methods of master equations developed recently is given and the convergence of functionals of the original process to a limiting transformed process is investigated. Some results are illustrated for the behaviour of the stochastics in a bistable tunnel diode. A model for macroscopic irreversibility is introduced using a sequence of non-Markov processes which converges to a Fokker-Planck process. Finally a few accomplishments on some recent related works are given.