Some applications of the Mellin transform to branching processes

TL;DR

This paper explores the use of Mellin transforms in analyzing branching processes, providing new formulas for lifetime distributions and type distributions, and linking Mellin and Fourier transforms of stable distributions.

Contribution

It introduces a Mellin transform approach to branching processes and derives new formulas for distributions in Bellman-Harris and Luria-Delbrück processes.

Findings

Derived the lifetime distribution of particles in Bellman-Harris processes.

Proved a formula for the Laplace transform of types in Luria-Delbrück processes.

Showed how to obtain Mellin transforms of stable distributions from Fourier transforms.

Abstract

We introduce a Mellin transform of functions which live on all of $\bR$ and discuss its applications to the limiting theory of Bellman-Harris processes, and specifically Luria-Delbr\"uck processes. More precisely, we calculate the life-time distribution of particles in a Bellman-Harris process from their first-generation offspring and limiting distributions, and prove a formula for the Laplace transform of the distribution of types in a Luria-Delbr\"uck process in the Mittag-Leffler limit. As a by-product, we show how to easily derive the (classical) Mellin transforms of certain stable probability distributions from their Fourier transform.