Stochastic solutions to abstract telegraph-type equations involving fractional dynamics
Alessandro De Gregorio, Roberto Garra
TL;DR
The paper addresses abstract time-fractional telegraph-type equations and Euler–Poisson–Darboux equations in a Hilbert-space setting, modeling memory effects via Caputo derivatives and non-local operators. It leverages spectral theory for positive self-adjoint $A$ to derive analytical solution formulas involving Mittag–Leffler functions and to construct stochastic representations through inverse stable subordinators and related processes, including Brownian and relativistic subordinators. Key contributions include explicit convolution representations $u_\alpha(t,\cdot)=(f*\Phi_\alpha(t))$, stochastic solutions $u_\alpha(t)=\mathbb{E}[v(\mathcal{L}^\alpha(t))]$ for $\alpha\in(0,\tfrac{1}{2}]$, and specialized results for the space–time fractional telegraph equation, Bessel–Riesz operators, and relativistic diffusion, together with Erdélyi–Kober integral representations for the EPD equation. These results provide a probabilistic interpretation of fractional telegraph-type dynamics and extend the classical connections between EPD, D'Alembert solutions, and fractional integrals, with potential applications to anomalous diffusion modeling.
Abstract
This paper investigates abstract integro-differential hyperbolic equations, focusing on the probabilistic representation of their solutions. Our analysis is based on fractional derivatives and non-local operators, which are powerful tools for modeling the anomalous behavior and non-Markovian dynamics observed in various phenomena. We first analyze a time-fractional version of the abstract telegraph equation (involving the Caputo derivative), restricting our analysis to positive self-adjoint operators to leverage spectral theory, which includes key operators in applications, such as the fractional Laplace operator. We derive analytical representations for the solution and provide a stochastic solution to the telegraph-diffusion equation for a specific range of the fractional parameter $α$, thereby generalizing existing results. We discuss particular cases involving the fractional Laplace and Bessel-Riesz operators. Furthermore, we consider the abstract Euler-Poisson-Darboux (EPD) equation, characterized by a singular time coefficient. We demonstrate that the stochastic solution to this EPD equation can be represented in terms of the solution of the abstract wave equation. Crucially, we prove that the solution to the EPD equation admits a representation by means of the Erdelyi-Kober fractional integral. Finally, this work provides a comprehensive analysis of both time-fractional and singular-coefficient abstract telegraph-type equations, offering new analytical and stochastic representation formulas.