Simple procedure for correcting equations of evolution: Application to Markov processes

A general procedure is proposed for correcting evolution equations, arising in different branches of science. Its application to Markov processes shows that the coefficients of the third- and higher-order derivatives in the Kramers-Moyal expansion are, in general, not small; nevertheless, the macroscopic-time evolution of the process is completely described by a differential equation of second order. For Brownian motion, this equation is Galilean invariant, while the Fokker-Planck equation is not. Finally, a correction is derived for the master equation.