Generalized Wright Analysis: Stochastic and Applications
W. Bock, L. Cristofaro, J. L. da Silva
TL;DR
The paper introduces generalized Fox-$H$ processes within the generalized Wright analysis framework, defining a non-Gaussian, non-Markovian class of stochastic processes via the measure $\mu_\Psi$ on a nuclear triple $(\mathcal{N}', \mathcal{C}_\sigma(\mathcal{N}'))$. It derives finite-dimensional distributions, covariance structures, and representations as time-changed fractional Brownian motion with an independent positive random variable $Y_{A,B}$, and it proves Hölder continuity and non-semimartingale behavior for $\mathbbm{H} \neq \tfrac{1}{2}$. The work further establishes the existence of a generalized Fox-$H$ noise $N_t^{\mathbbm{H},A,B}$ in the distribution space, and demonstrates applications to local times and anomalous diffusion, revealing sub-, normal-, and super-diffusive regimes depending on $\mathbbm{H}$. Overall, the framework extends classical Gaussian analyses to a broad non-Gaussian setting and opens avenues for SPDEs, Green-function analysis, and Malliavin-calculus approaches in anomalous-diffusion models.
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{ö}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 $(\mathcal{S})^{-1}_{μ_Ψ}$. We conclude by establishing the existence of local times and discussing their anomalous diffusion properties.