A new approach to complex-valued fractional Brownian motion via rotating white noise

In this paper, a Brownian motion of order n is defined by a probabilistic approach which is different from Mandelbrotʹs and Saintyʹs models. This process is constructed in the form of the integral of a complex Gaussian white noise which itself is defined as the product of a Gaussian white noise by a complex white process which takes on values on the set of the roots of the unity of order n. An Itô-Taylorʹs lemma of order n is obtained; therefore one derives the dynamical equations of the complex Brownian motion moments whereby one can obtain a generalized Fokker-Planck equation or heat equation of order n. A possible relation with Kramers-Moyal expansion is outlined. The framework is essentially applied mathematics.