Statistical-dynamical theory of nonlinear stochastic processes

A new statistical-dynamical theory of nonlinear stochastic processes in nonequilibrium open systems is presented. It is shown by means of the time-convolutionless projector method that multiplicative type stochastic equations of motion for relevant variables Ai(t) can always be transformed exactly into additive type (Langevin type) stochastic equations of motion for Ai(t) and the corresponding master equation for the probability distribution. The Langevin type equation consists of a drift term and a fluctuating force. The fluctuating force is shown to give a new kind of additive stochastic process which has a quite different feature from the ordinary additive processes. The importance of Langevin equations of this type in the multiplicative stochastic process is pointed out from the statistical-dynamical viewpoint. A new cumulant expansion of the master equation in powers of stochastic forces is also found.