The Diffusion Approximation for Markov Processes N. G. van Kampen∗

Ich schreibe nicht, euch zu gefallen, Ihr sollt was lernen! – Goethe Markov processes in physics, chemistry and biology are often regarded as generalized diffusion and ac-cordingly described by a Kolmogorov equation or a Langevin equation. This procedure is exact if the Lindeberg condition is obeyed. It is here argued, how-ever, that it is at best only approximately obeyed and that therefore the mathematical theory derived from it should be replaced with an approximation proce-dure. Such a procedure, in the form of a power series expansion in a parameter Ω, is described. The con-clusion is that the diffusion approximation is inconsis-tent, except for a special subclass of Markov processes. This distinction has nothing to do with the distinction between discrete and continuous variables. Markov Processes and the Diffusion Ap-proximation Let Y (t) be a continuous time Markov process whose range consists of the real numbers. The range may be a discrete countable set, or it may be continuous, covering a finite or infinite interval on the real line. A combination of both may also occur, but will not be considered here. For t> t0, let P (y, t|y0, t0) be the transition prob-ability, i.e., the probability (or probability density) of Y (t) conditional on Y (t0) = y0. As a conse-quence of the Markov property it obeys the Chapman-Kolmogorov or master equation ∂P (y, t|y0, t0