KOLMOGOROV REFINED SIMILARITY HYPOTHESES FOR TURBULENCE AND GENERAL STOCHASTIC-PROCESSES

Kolmogorov's refined similarity hypotheses are shown to hold true for a variety of stochastic processes besides high-Reynolds-number turbule nt flows, for which they were originally proposed. In particular, just as hypothesized for turbulence, there exists a variable V whose proba bility density function attains a universal form. Analytical expressio ns for the probability density function of V are obtained for Brownian motion as well as for the general case of fractional Brownian motion- the latter under some mild assumptions justified a posteriori. The pro perties of V for the case of antipersistent fractional Brownian motion with the Hurst exponent of 1/3 are similar in many details to those o f high-Reynolds-number turbulence in atmospheric boundary layers a few meters above the ground. The one conspicuous difference between turbu lence and the antipersistent fractional Brownian motion is that the la tter does not possess the required skewness. Broad implications of the se results are discussed.