Generalized Wright Analysis: Stochastic and Applications

TL;DR

This paper introduces the generalized Fox-H process, a new class of non-Gaussian, non-Markovian stochastic processes extending fractional Brownian motion, with analysis of their properties and applications in anomalous diffusion.

Contribution

It defines and studies the generalized Fox-H process, extending existing stochastic models and analyzing their probabilistic and path properties, including local times and anomalous diffusion.

Findings

The generalized Fox-H process has stationary increments.

Paths of the process are Hölder continuous.

The process exhibits anomalous diffusion properties.

Abstract

In this paper, we investigate the stochastic counterpart of the generalized Wright analysis introduced in Beghin et al.~ in Integral Equations and Operator Theory, {\bf 97}, 2025. We define a new class of non-Gaussian and non-Markovian processes, called the generalized Fox-$H$ process, which extends well-known processes such as fractional Brownian motion and generalized grey Brownian motion. We study their joint probability density and covariance, showing the stationarity of their increments. In addition, this process has H{\"o}lder continuous paths and is represented as a time-change of fractional Brownian motion. We characterize the generalized Fox-$H$ noise as an element in the distribution space $(\mathcal{S})^{-1}_{\mu_\Psi}$. We conclude by establishing the existence of local times and discussing their anomalous diffusion properties.