Kolmogorov's refined similarity hypotheses for turbulence and general stochastic processes

Abstract

Kolmogorov's refined similarity hypotheses are shown to hold true for a variety of stochastic processes besides high-Reynolds-number turbulent flows, for which they were originally proposed. In particular, just as hypothesized for turbulence, there exists a variable $V$ whose probability density function attains a universal form. Analytical expressions 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 properties of $V$ for the case of antipersistent fractional Brownian motion with the Hurst exponent of $\frac{1}{3}$ are similar in many details to those of high-Reynolds-number turbulence in atmospheric boundary layers a few meters above the ground. The one conspicuous difference between turbulence and the antipersistent fractional Brownian motion is that the latter does not posses the required skewness. Broad implications of these results are discussed.