Probability density of the complex-valued fractional Brownian motion of order n via the maximum entropy principle in R+1/n

In previous work, it has been proposed a model of complex-valued fractional Brownian motion with independent increments which is defined by means of a random walk on the complex roots of the unity. Here one shows that, exactly like the Gaussian probability function can be obtained as a result of the maximum entropy principle, the probability density of the complex-valued fractional noise of order n maximizes the informational entropy of a complex-valued random variable defined in an isotropic complex space of order n. In addition one defines and derives the main properties of the so-called coloured fractional noise of order n and its probability density also is obtained by using the maximum entropy principle. Lastly, as an illustrative example of application, one analyzes how the effects of fractal noises disturbances, in the form of fractal noises, on the stability of dynamical systems, vary depending upon of their order. (C) 2003 Elsevier Ltd. All rights reserved.