A Fundamental Limitation of Markov Models

Abstract

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