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The Lamperti transformation in the infinite-dimensional setting and the genealogies of self-similar Markov processes

Arno Siri-Jégousse, Alejandro Hernández Wences

TL;DR

This work develops a comprehensive framework linking measure-valued population models with self-similar Markov processes by extending the Lamperti transformation to infinite dimensions. It constructs a four-parameter class of measure-valued self-similar populations whose total mass evolves as a positive self-similar Markov process and whose frequency dynamics are described by $$(\Lambda+\frac{\sigma^2}{2}\delta_0)$$-Fleming-Viot processes, while genealogies are captured by dual $\Lambda$-coalescents. A central contribution is the Banach-space Lamperti transform, establishing a bijection between $\alpha$-SS processes, Markov additive processes, and SMH processes, via time changes $c_\alpha$ and $\gamma_\alpha$. The paper also provides both a Poissonian construction for finite-activity jumps and a weak-limit martingale-problem approach to realize the continuum limit, thereby unifying size-dependent reproduction with classical FV/coalescent duality and expanding the modeling toolkit beyond the Beta-subfamily to general $\Lambda$-dynamics.

Abstract

We propose a change in focus from the prevalent paradigm based on the branching property as a tool to analyze the structure of population models, to one based on the self-similarity property, which we also introduce for the first time in the setting of measure-valued processes. By extending the well-known Lamperti transformation for self-similar Markov processes to the Banach-valued case we are able to generalize celebrated results in population genetics that describe the frequency-process of measure-valued stable branching processes in terms of the subfamily of Beta-Fleming-Viot processes. In our work we describe the frequency process of populations whose total size evolves as any positive self-similar Markov process in terms of general $Λ$-Fleming-Viot processes. Our results demonstrate the potential power of the self-similar perspective for the study of population models in which the reproduction dynamics of the individuals depend on the total population size, allowing for more complex and realistic models.

The Lamperti transformation in the infinite-dimensional setting and the genealogies of self-similar Markov processes

TL;DR

This work develops a comprehensive framework linking measure-valued population models with self-similar Markov processes by extending the Lamperti transformation to infinite dimensions. It constructs a four-parameter class of measure-valued self-similar populations whose total mass evolves as a positive self-similar Markov process and whose frequency dynamics are described by -Fleming-Viot processes, while genealogies are captured by dual -coalescents. A central contribution is the Banach-space Lamperti transform, establishing a bijection between -SS processes, Markov additive processes, and SMH processes, via time changes and . The paper also provides both a Poissonian construction for finite-activity jumps and a weak-limit martingale-problem approach to realize the continuum limit, thereby unifying size-dependent reproduction with classical FV/coalescent duality and expanding the modeling toolkit beyond the Beta-subfamily to general -dynamics.

Abstract

We propose a change in focus from the prevalent paradigm based on the branching property as a tool to analyze the structure of population models, to one based on the self-similarity property, which we also introduce for the first time in the setting of measure-valued processes. By extending the well-known Lamperti transformation for self-similar Markov processes to the Banach-valued case we are able to generalize celebrated results in population genetics that describe the frequency-process of measure-valued stable branching processes in terms of the subfamily of Beta-Fleming-Viot processes. In our work we describe the frequency process of populations whose total size evolves as any positive self-similar Markov process in terms of general -Fleming-Viot processes. Our results demonstrate the potential power of the self-similar perspective for the study of population models in which the reproduction dynamics of the individuals depend on the total population size, allowing for more complex and realistic models.
Paper Structure (10 sections, 14 theorems, 123 equations)

This paper contains 10 sections, 14 theorems, 123 equations.

Table of Contents

  1. Introduction and Main Results
  2. The Lamperti Transformation in Normed Spaces
  3. Preliminary objects of study
  4. $\Lambda$-coalescents
  5. The $\Lambda$-Fleming-Viot processes
  6. The Dawson-Watanabe process and its Lamperti transformation
  7. The genealogy of self-similar populations
  8. Main results and comments
  9. Poissonian construction for finite $\zeta^{-2}\Lambda(d\zeta)$
  10. A Weak Limit and Martingale Problem

Key Result

Theorem 1.1

Fix $\alpha\in\mathbb{R}$. There exists a standard measure-valued $\alpha$-SS Markov process $$μ_t$_{t\geq 0}$ with generator of the form where, for $\sigma\geq 0, \kappa\in\mathbb{R}$, and $\Lambda\in\mathtt{M}((0,1))$, the above operators are defined by The total size of the population $$$\left\lVert \mu_t \right\rVert$$_{t\geq 0}$ is a positive $\alpha$-SS Markov process with non-negative jum

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1: SMH $\iff$ MAP
  • proof
  • Theorem 2.2: Self-Similar Lamperti Time Change
  • proof
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 20 more