[Paper Review] Williams' decomposition of the Lévy continuous random tree and simultaneous extinction probability for populations with neutral mutations

Romain Abraham, Jean‐François Delmas|ArXiv.org|Apr 11, 2007

Stochastic processes and statistical mechanics12 references4 citations

TL;DR

This paper establishes a Williams' decomposition of the Lévy continuum random tree to analyze the genealogy of a population with neutral mutations, where the total population and the ancestral Eve-population go extinct simultaneously. It derives a closed-form formula for the simultaneous extinction probability using the exploration process and excursion measures of continuous-state branching processes with general branching mechanisms.

ABSTRACT

We consider an initial Eve-population and a population of neutral mutants, such that the total population dies out in finite time. We describe the evolution of the Eve-population and the total population with continuous state branching processes, and the neutral mutation procedure can be seen as an immigration process with intensity proportional to the size of the population. First we establish a Williams' decomposition of the genealogy of the total population given by a continuous random tree, according to the ancestral lineage of the last individual alive. This allows us give a closed formula for the probability of simultaneous extinction of the Eve-population and the total population.

Motivation & Objective

• To characterize the joint genealogy of an ancestral Eve-population and its neutral mutant descendants using a continuum random tree (CRT) framework.
• To extend known results on simultaneous extinction probability beyond the quadratic branching mechanism case to general subcritical or critical continuous-state branching processes (CSBP) with infinite variation.
• To establish a Williams’ decomposition of the CRT based on the ancestral lineage of the last surviving individual in the total population.
• To derive a closed-form expression for the probability that the Eve-population and the total population go extinct simultaneously, under general conditions on the branching mechanism.
• To leverage excursion measures and exploration processes to analyze extinction times and pathwise dynamics in CSBP with immigration from the Eve-population.

Proposed method

• Uses the height process $ H $ and exploration process $ \rho_t $ to code the genealogy of a continuous-state branching process (CSBP), extending the framework of Le Gall and Le Jan.
• Applies the canonical measure $ \mathbb{N} $ and excursion measure to describe the law of the total population process $ Y $, with $ \tau_Y $ denoting its extinction time.
• Introduces a decomposition of the CRT along the ancestral lineage of the last individual alive, using the time $ T_0 $ when the last individual is born.
• Employs the Laplace transform and the solution $ u(\lambda,t) $ of the integral equation $ \int_{u(\lambda,t)}^{\lambda} \frac{dv}{\psi(v)} = t $ to characterize the distribution of $ Y $.
• Derives the extinction time distribution via $ \mathbb{P}_x(\tau_Y < t) = \exp(-x \mathbb{N}[\tau_Y \geq t]) $, with $ c(t) = \mathbb{N}[\tau_Y \geq t] $ solving $ \int_{c(t)}^{\infty} \frac{dv}{\psi(v)} = t $.
• Uses the total local time $ L^a $ at level $ a $ and the function $ w_m(t) $, $ w_m^*(t) $ to model the evolution of the population size along the ancestral lineage, leading to the final formula via integration over the excursion path.

Experimental results

Research questions

• RQ1 What is the probability that the ancestral Eve-population and the total mutant population go extinct simultaneously in a population with neutral mutations?
• RQ2 How can the genealogy of a population with neutral mutations be decomposed using the last individual’s ancestral lineage?
• RQ3 Can the Williams’ decomposition of the Lévy continuum random tree be applied to general CSBP with infinite variation and a.s. extinction?
• RQ4 What is the role of the exploration process and excursion measure in characterizing extinction times in CSBP with immigration?
• RQ5 How does the branching mechanism $ \psi $ influence the simultaneous extinction probability in a multi-type CSBP with irreversible mutations?

Key findings

• The paper establishes a closed-form expression for the simultaneous extinction probability of the Eve-population and the total population, valid for general subcritical or critical CSBP with infinite variation and a.s. extinction.
• The key result is derived via a Williams’ decomposition of the Lévy continuum random tree, where the ancestral lineage of the last surviving individual serves as the decomposition axis.
• The simultaneous extinction probability is shown to be equal to $ w(0) $, where $ w(0) $ is the limit of the function $ w_m(0) $ as $ m \to \infty $, derived from the solution of a system of ODEs involving the branching mechanisms.
• The derivation relies on the identification of the density of the excursion measure and the use of local time at level $ a $, leading to the final formula through integration over the excursion path.
• The result generalizes previous formulas known only for the quadratic branching mechanism (e.g., $ \psi(u) = \alpha_0 u + \beta u^2 $) to the full class of CSBP with $ \int^{\infty} \frac{dv}{\psi(v)} < \infty $.
• The method successfully handles the non-Markovian nature of the height process $ H $ by using the exploration process $ \rho_t $ and the associated measure-valued Markov process.