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 α-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.Comment: 24, accepted for publicatio