FRACTIONAL BROWNIAN MOTIONS DESCRIBED BY SCALED LANGEVIN EQUATION

An ideal system is considered in simulating the dynamics of a random process. The system is composed of a set of clusters, where a cluster is an aggregation of elements activated in a random manner. The time evolution of each duster is described by the Langevin equation, which characterizes a family of the Brownian motion. A scaling rule is introduced to the set of the Langevin equations in order to model the complexity of the random system. The response of this system is considered as an illustration of the fractional Brownian motion. Fractal dimension D defined by the scaling constant is related to the Hurst exponent H empirically introduced to specify the fractional Brownian motion as 1 - D = 2H - 2 (D > 0). Solutions of the scaled Langevin equation constitute an orthonormal system of functions. The theory can be developed to describe a more complicated system in which local clusters are characterized by plural scaling rules. A special case is considered where scaling parameters are defined by complex numbers. A complex random system thus derived from the complex Langevin equation illustrates the complex fractional Brownian motion. The complex Brownian motion in this study is characterized by a complex fractal dimension defined by the scaling parameter. The complex system shows the limit-cycle behavior of the Brownian motion and the instability of the Brownian motion.