On an interpolation model for the transition operator for Markov and non-Markov processes

Abstract

A phenomenological interpolation model for the transition operator of a stationary Markov process is shown to be equivalent to the simplest difference approximation in the master equation for the conditional density. Comparison with the formal solution of the Fokker-Planck equation yields a criterion for the choice of the correlation time in the approximate solution. The interpolation model is shown to be form-invariant under an iteration-cum-rescaling scheme. Next, we go beyond Markov processes to find the effective time-development operator (the counterpart of the conditional density) in the following very general situation: the stochastic interruption of the systematic evolution of a variable by an arbitrary stationary sequence of randomizing pulses. Continuous-time random walk theory with a distinct first-waiting-time distribution is used, along with the interpolation model for the transition operator, to obtain the solution. Convenient closed-form expressions for the 'averaged' time-development operator and the autocorrelation function are presented in various special cases. These include (i) no systematic evolution, but correlated pulses; (ii) systematic evolution interrupted by an uncorrelated (Poisson) sequence of pulses.