From Sturm-Liouville problems to fractional and anomalous diffusions

TL;DR

This paper explores explicit solutions to fractional and anomalous diffusion equations involving fractional derivatives, connecting them to stable processes, Sturm-Liouville problems, and a new perspective on Bochner's subordination rule.

Contribution

It introduces a novel approach to representing solutions of fractional diffusions related to fractional Sturm-Liouville problems and extends Bochner's subordination rule in this context.

Findings

Explicit laws of stable processes are fundamental for solutions.

Established connections between subordination and space-fractional operators.

Provided new representations for fractional diffusion solutions.

Abstract

Some fractional and anomalous diffusions are driven by equations involving fractional derivatives in both time and space. Such diffusions are processes with randomly varying times. In representing the solutions to those diffusions, the explicit laws of certain stable processes turn out to be fundamental. This paper directs one's efforts towards the explicit representation of solutions to fractional and anomalous diffusions related to Sturm-Liouville problems of fractional order associated to fractional power function spaces. Furthermore, we study a new version of the Bochner's subordination rule and we establish some connections between subordination and space-fractional operator