Fractional diffusions with time-varying coefficients

TL;DR

This paper explores fractional diffusion equations with time-varying coefficients, providing solutions that extend the law of fractional Brownian motion and analyzing associated probabilistic properties using advanced fractional operators.

Contribution

It introduces new solutions to fractional diffusion equations with time-varying coefficients and links these solutions to distributions of time-changed random variables.

Findings

Extended probability laws for fractional Brownian motion.

Solutions involve Erdélyi–Kober fractional integrals.

Distributions relate to time-changed random variables.

Abstract

This paper is concerned with the fractionalized diffusion equations governing the law of the fractional Brownian motion $B_H(t)$. We obtain solutions of these equations which are probability laws extending that of $B_H(t)$. Our analysis is based on McBride fractional operators generalizing the hyper-Bessel operators $L$ and converting their fractional power $L^{\alpha}$ into Erd\'elyi--Kober fractional integrals. We study also probabilistic properties of the r.v.'s whose distributions satisfy space-time fractional equations involving Caputo and Riesz fractional derivatives. Some results emerging from the analysis of fractional equations with time-varying coefficients have the form of distributions of time-changed r.v.'s.