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arxiv: 2405.10193 · v4 · submitted 2024-05-16 · 🧮 math.PR

The Lamperti transformation in the infinite-dimensional setting, self-similar populations, and coalescents

Arno Siri-J\'egousse , Alejandro Hern\'andez Wences This is my paper

Pith reviewed 2026-05-24 01:30 UTC · model grok-4.3

classification 🧮 math.PR
keywords Lamperti transformationself-similar populationsmeasure-valued processesLambda Fleming-Viot processesLambda coalescentsduality relationspopulation frequency processes
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The pith

Extending the Lamperti transformation to measure-valued processes describes the frequency processes of self-similar populations using general Lambda Fleming-Viot processes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shifts analysis of population models away from the branching property toward a newly introduced self-similarity property for measure-valued processes. It extends the classical Lamperti transformation to this infinite-dimensional setting to express the frequency process of a broader class of self-similar populations in terms of Lambda Fleming-Viot processes. This framework supports models in which total population size varies stochastically and individual reproduction depends on current size. The work also establishes a new duality between the measure-valued processes and Lambda-coalescents that generalizes the known duality for Lambda Fleming-Viot processes.

Core claim

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 into the infinite dimensional setting, we were able to embed and extend known results in population genetics within the self-similarity framework: we describe the frequency process of a larger class of measure-valued SS populations in terms of general Lambda Fleming-Viot processes. We also uncover a new duality relation between measure-valued processes and Lambda-coa

What carries the argument

The Lamperti transformation extended to infinite-dimensional measure-valued self-similar processes, which maps their frequency evolution to general Lambda Fleming-Viot processes.

If this is right

  • Frequency processes of self-similar populations with stochastically varying total size are given by Lambda Fleming-Viot processes.
  • Reproduction dynamics modulated by total population size fall inside the same framework.
  • A duality relation holds between the resulting measure-valued processes and Lambda-coalescents.
  • The self-similar perspective applies to populations whose individuals do not reproduce independently.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same transformation might supply explicit transition rates or simulation schemes for self-similar models once the corresponding Lambda Fleming-Viot process is identified.
  • The new duality could be used to translate questions about genealogies in size-dependent populations into questions about the underlying coalescent.

Load-bearing premise

The self-similarity property can be rigorously defined for measure-valued processes in a manner that permits a meaningful extension of the Lamperti transformation while preserving the link to Lambda Fleming-Viot processes.

What would settle it

Constructing an explicit measure-valued self-similar population whose frequency process after the extended Lamperti map fails to satisfy the generator or martingale problem of any Lambda Fleming-Viot process would falsify the central claim.

read the original 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 into the infinite dimensional setting, we were able to embed and extend known results in population genetics within the self-similarity framework: we describe the frequency process of a larger class of measure-valued SS populations in terms of general Lambda Fleming-Viot processes. Our results demonstrate the potential power of the self-similar perspective for the study of populations whose total size varies stochastically over time, and in which the reproduction dynamics of the individuals are not independent from one another but are modulated by the total size of the population, allowing for more complex and realistic models. We also uncover a new duality relation between measure-valued processes and Lambda-coalescents which extends the well-known duality relation between Lambda Fleming-Viot processes and Lambda coalescents.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces a self-similarity property for measure-valued processes (for the first time in this setting) and extends the classical Lamperti transformation to the infinite-dimensional case. It claims that this framework embeds and extends population-genetics results by expressing the frequency process of a larger class of measure-valued self-similar populations in terms of general Lambda Fleming-Viot processes, and that it yields a new duality between such measure-valued processes and Lambda-coalescents.

Significance. If the central constructions are rigorous, the work supplies a new analytic perspective on population models whose total mass evolves stochastically and whose reproduction is size-modulated, thereby accommodating more complex dependence structures than branching-based models. The claimed duality extension would also broaden the scope of known Lambda-coalescent dualities beyond the standard Fleming-Viot case.

major comments (2)
  1. [section introducing the self-similarity framework] The section introducing the self-similarity framework: the definition of self-similarity for measure-valued processes must be shown to render the infinite-dimensional Lamperti map well-defined and invertible on the state space while ensuring that the time-changed and scaled process has generator exactly matching that of a general Lambda Fleming-Viot process, without extra regularity assumptions on the driving measure. If the definition only holds when total mass is a deterministic function of time, the embedding of a “larger class” fails.
  2. [section on the duality relation] The section on the duality relation: the claimed new duality between measure-valued self-similar processes and Lambda-coalescents must be derived explicitly from the Lamperti-transformed generator; the abstract supplies no steps, so the derivation must be checked for measurability issues in the infinite-dimensional setting and for whether it reduces to the known FV-coalescent duality when the self-similarity parameter is constant.
minor comments (1)
  1. Notation for the infinite-dimensional state space and the precise form of the Lamperti time change should be introduced with explicit measurability statements.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive major comments. We address each point below and will incorporate the requested clarifications and proofs into a revised manuscript.

read point-by-point responses

  1. Referee: [section introducing the self-similarity framework] The section introducing the self-similarity framework: the definition of self-similarity for measure-valued processes must be shown to render the infinite-dimensional Lamperti map well-defined and invertible on the state space while ensuring that the time-changed and scaled process has generator exactly matching that of a general Lambda Fleming-Viot process, without extra regularity assumptions on the driving measure. If the definition only holds when total mass is a deterministic function of time, the embedding of a “larger class” fails.

    Authors: We will add a new subsection containing a self-contained proof that the definition of self-similarity renders the infinite-dimensional Lamperti map well-defined and invertible on the relevant state space. The argument will verify that the generator of the time-changed and scaled process coincides exactly with that of a general Lambda Fleming-Viot process, without imposing further regularity conditions on the driving measure. The definition is formulated to permit stochastic evolution of total mass; the revision will include an explicit example in which total mass is a non-deterministic process, thereby confirming that the claimed embedding of a larger class holds. revision: yes

  2. Referee: [section on the duality relation] The section on the duality relation: the claimed new duality between measure-valued self-similar processes and Lambda-coalescents must be derived explicitly from the Lamperti-transformed generator; the abstract supplies no steps, so the derivation must be checked for measurability issues in the infinite-dimensional setting and for whether it reduces to the known FV-coalescent duality when the self-similarity parameter is constant.

    Authors: The revised manuscript will expand the duality section with an explicit derivation starting from the generator obtained after the Lamperti transformation. The steps will include verification that all maps are measurable in the infinite-dimensional setting. We will also prove that the duality reduces to the classical Fleming-Viot–Lambda-coalescent duality when the self-similarity parameter is constant, providing a direct consistency check. revision: yes

Circularity Check

0 steps flagged

No circularity: new definitions and extensions are self-contained.

full rationale

The paper introduces the self-similarity property for measure-valued processes for the first time and extends the Lamperti transformation to obtain descriptions of frequency processes via Lambda Fleming-Viot processes plus a new duality with Lambda-coalescents. The abstract and provided text contain no equations, fitted parameters, or self-citations that reduce any claimed prediction or result to its own inputs by construction. The derivation chain rests on explicit constructions of the time-change, scaling, and generator matching, which are presented as novel rather than tautological. This qualifies as a standard self-contained mathematical development with no load-bearing circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review prevents identification of specific free parameters, axioms, or invented entities. No numerical fitting or new postulated objects are mentioned.

pith-pipeline@v0.9.0 · 5717 in / 1109 out tokens · 24431 ms · 2026-05-24T01:30:39.952624+00:00 · methodology

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Reference graph

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