Time-Averaged Statistics-Based Methods for Anomalous Diffusive Exponent Estimation of Fractional Brownian Motion

Abstract

The anomalous diffusive processes are widely discussed in the research papers. In contrast to the diffusive processes with the linear second moment, they are characterized by the nonlinear variance. The anomalous diffusive processes exhibit many interesting properties which are not adequate to the diffusive systems and thus they have found various applications including, among others, biology, physics and environmental engineering. Very useful in the testing of the anomalous diffusive behavior are the time-averaged statistics which are based on the sample trajectory of the given process. Similar as the empirical second moment, they exhibit different behavior for anomalous diffusive and diffusive processes. Thus, they can be very effective tools for the estimation and statistical testing of the anomalous diffusive behavior. One can find different theoretical anomalous diffusive processes. One of the classical examples is the fractional Brownian motion. In this chapter, we demonstrate how the selected time-averaged statistics behave for the fractional Brownian motion and show how they can be applied in order to estimate the Hurst exponent (responsible for the anomalous diffusive behavior). By using Monte Carlo simulations, we compare the effectiveness of the presented estimation methods for the considered stochastic process. The described methodology can be applied to any other anomalous diffusive processes.