Fractional Diffusion-Telegraph Equations and their Associated Stochastic Solutions

TL;DR

This paper develops stochastic solutions for generalized fractional PDEs, including fractional diffusion and telegraph equations, using Lévy processes and inverse stable subordinators, and explores related stochastic differential equations.

Contribution

It introduces a novel stochastic solution framework for fractional PDEs involving Prabhakar operators, extending existing models with new probabilistic representations.

Findings

Stochastic solutions expressed via Lévy processes and inverse subordinators.

Derivation of related stochastic differential equations.

Application to fractional diffusion and telegraph equations.

Abstract

We present the stochastic solution to a generalized fractional partial differential equation involving a regularized operator related to the so-called Prabhakar operator and admitting, amongst others, as specific cases the fractional diffusion equation and the fractional telegraph equation. The stochastic solution is expressed as a L\'evy process time-changed with the inverse process to a linear combination of (possibly subordinated) independent stable subordinators of different indices. Furthermore a related SDE is derived and discussed.