Mellin transform and subordination laws in fractional diffusion processes

TL;DR

This paper explores the application of the Mellin transform to fractional diffusion processes, revealing integral formulas that describe their distributions through subordination laws, thus advancing understanding of self-similar stochastic processes.

Contribution

It introduces new integral formulas involving Mellin transforms for fractional diffusion processes, linking them to subordination laws and expanding their analytical framework.

Findings

Derived integral formulas for fractional diffusion distributions

Connected Mellin transform techniques to subordination laws

Enhanced analytical tools for self-similar stochastic processes

Abstract

The Mellin transform is usually applied in probability theory to the product of independent random variables. In recent times the machinery of the Mellin transform has been adopted to describe the L\'evy stable distributions, and more generally the probability distributions governed by generalized diffusion equations of fractional order in space and/or in time. In these cases the related stochastic processes are self-similar and are simply referred to as fractional diffusion processes. We provide some integral formulas involving the distributions of these processes that can be interpreted in terms of subordination laws.