Long-memory Gaussian processes governed by generalized Fokker-Planck equations

Abstract

It is well-known that the transition function of the Ornstein-Uhlenbeck process solves the Fokker-Planck equation. This standard setting has been recently generalized in different directions, for example, by considering the so-called $\alpha $-stable driven Ornstein-Uhlenbeck, or by time-changing the original process with an inverse stable subordinator. In both cases, the corresponding partial differential equations involve fractional derivatives (of Riesz and Riemann-Liouville types, respectively) and the solution is not Gaussian. We consider here a new model, which cannot be expressed by a random time-change of the original process: we start by a Fokker-Planck equation (in Fourier space) with the time-derivative replaced by a new fractional differential operator. The resulting process is Gaussian and, in the stationary case, exhibits a long-range dependence. Moreover, we consider further extensions, by means of the so-called convolution-type derivative.