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

TL;DR

This paper introduces a new Gaussian process governed by a generalized Fokker-Planck equation with fractional derivatives, exhibiting long-range dependence and extending previous models involving fractional and time-changed processes.

Contribution

It proposes a novel Gaussian process model driven by a generalized fractional Fokker-Planck equation, not based on random time-changes, and explores its properties and extensions.

Findings

The process is Gaussian with long-range dependence in the stationary case.

The model extends existing fractional and time-changed processes.

It introduces a new fractional differential operator in the Fokker-Planck framework.

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.