A class of processes defined in the white noise space through generalized fractional operators

Abstract

The generalization of fractional Brownian motion in the white and grey noise spaces has been recently carried over, following the Mandelbrot–Van Ness representation, through Riemann–Liouville type fractional operators. Our aim is to extend this construction by means of more general fractional derivatives and integrals, which we define as Fourier-multiplier operators and then specialize by means of Bernstein functions. More precisely, we introduce a general class of kernel-driven processes which encompasses, as special cases, a number of models in the literature, including fractional Brownian motion, tempered fractional Brownian motion, Ornstein–Uhlenbeck process. The greater generality of our model, with respect to the previous ones, allows a higher flexibility and a wider applicability. We derive here some properties of this class of processes (such as continuity, occupation density, variance asymptotics and persistence) according to the conditions satisfied by the Fourier symbol of the operator or the Bernstein function chosen. On the other hand, these processes are proved to display short- or long-range dependence, if obtained by means of a derivative or an integral type operator, respectively, regardless of the kernel used in their definition. Finally, this kind of construction allows us to define the corresponding noise and to solve a Langevin-type integral equation.