A fundamental limitation of Markov models

A basic question in turbulence theory is whether Markov models produce stat istics that differ systematically from dynamical systems. The conventional wisdom is that Markov models are problematic at short time intervals, but p recisely what these problems are and when these problems manifest themselve s do not seem to be generally recognized. A barrier to understanding this i ssue is the lack of a closure theory for the statistics of nonlinear dynami cal systems. Without such theory, one has difficulty stating precisely how dynamical systems differ from Markov models. It turns out, nevertheless, th at certain fundamental differences between Markov models and dynamical syst ems can be understood from their differential properties. It is shown than any stationary. ergodic system governed by a finite number of ordinary diff erential equations will produce time-lagged covariances with negative curva ture over short lags and produce power spectra that decay faster than any p ower of frequency. In contrast, Markov models (which necessarily include wh ite noise terms) produce covariances with positive curvature over short lag s, and produce power spectra that decay only with some integer power of fre quency. Problems that arise from these differences in the context of statis tical prediction and turbulence modeling are discussed.