A class of infinite-dimensional Gaussian processes defined through generalized fractional operators

TL;DR

This paper introduces a new class of infinite-dimensional Gaussian processes using generalized fractional derivatives and integrals defined via Bernstein functions, extending previous fractional Brownian motion models.

Contribution

It generalizes fractional Brownian motion in infinite-dimensional spaces by employing Bernstein functions for defining fractional derivatives and integrals, enabling new process constructions.

Findings

Processes exhibit short- or long-range dependence based on derivative or integral construction.

Properties like continuity, local times, and variance asymptotics are derived.

Defines corresponding noise and Ornstein-Uhlenbeck type processes.

Abstract

The generalization of fractional Brownian motion in infinite-dimensional 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 general fractional derivatives and integrals, which we define through Bernstein functions. According to the conditions satisfied by the latter, some properties of these processes (such as continuity, local times, variance asymptotics and persistence) are derived. On the other hand, they are proved to display short- or long-range dependence, if obtained by means of a derivative or an integral, respectively, regardless of the Bernstein function chosen. Moreover, this kind of construction allows us to define the corresponding noise and to derive an Ornstein-Uhlenbeck type process, as solution of an integral equation.