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arxiv: 2511.17284 · v2 · submitted 2025-11-21 · 🧮 math.PR

Regularity of multiplicative processes on infinite-dimensional Lie groups

Anita Behme , Markus Riedle , Shend Thaqi This is my paper

Pith reviewed 2026-05-17 20:25 UTC · model grok-4.3

classification 🧮 math.PR
keywords multiplicative stochastic processesinfinite-dimensional Lie groupscàdlàg modificationsregularity propertiesexponential mapBanach spacesstochastic analysisBanach-Lie groups
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The pith

Multiplicative stochastic processes on infinite-dimensional Lie groups admit càdlàg modifications under suitable conditions by transferring results from Banach spaces.

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

The paper tries to prove that multiplicative stochastic processes on infinite-dimensional Lie groups can be made to have càdlàg paths and controlled local behavior when certain conditions hold. This would matter because many models in physics, control, and probability live on curved infinite-dimensional structures rather than flat vector spaces, so regularity tools need to travel there too. The authors achieve the transfer by using the fact that the exponential and logarithm maps make the group locally look like its tangent Banach space, letting known analytic estimates carry over. A reader would care if this means simulation and analysis of jumps become feasible without building a new theory from scratch for each group.

Core claim

Under suitable conditions, multiplicative stochastic processes on infinite-dimensional Lie groups admit càdlàg modifications and satisfy local behavior bounds. This follows from transferring analytic estimates and structural results from Banach spaces via the local equivalence of Banach-Lie groups and Banach spaces given by the exponential and logarithm maps.

What carries the argument

the local equivalence between Banach-Lie groups and Banach spaces via the exponential and logarithm maps, which permits direct transfer of analytic estimates and structural results

If this is right

  • The processes possess right-continuous paths with left limits almost surely.
  • Local behavior bounds become available by direct appeal to the corresponding Banach-space results.
  • Existence of regular modifications can be verified by checking conditions in the tangent space.
  • Jump processes on curved infinite-dimensional geometries inherit regularity from their linear approximations.

Where Pith is reading between the lines

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

  • Numerical simulation of these processes could proceed by working in local charts that reduce to Banach-space calculations.
  • The approach may extend to time-inhomogeneous or more general driving noises on the same groups.
  • Global regularity questions arise once local bounds are glued across overlapping charts.

Load-bearing premise

The exponential and logarithm maps create a local equivalence that allows analytic estimates to transfer from Banach spaces to the Lie group without major loss or distortion.

What would settle it

A concrete multiplicative stochastic process on an infinite-dimensional Lie group such as the diffeomorphism group that meets the transferred conditions from its Banach-space approximation yet fails to possess a càdlàg modification.

read the original abstract

This article studies regularity properties of multiplicative stochastic processes on infinite-dimensional Lie groups. We investigate conditions under which these processes admit c\`adl\`ag modifications and derive bounds on their local behavior. Our approach builds on the local equivalence of Banach-Lie groups and Banach spaces via the exponential and logarithm, allowing us to transfer analytic estimates and structural results.

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

1 major / 2 minor

Summary. The manuscript studies regularity properties of multiplicative stochastic processes on infinite-dimensional Lie groups. It claims that under suitable conditions these processes admit càdlàg modifications and satisfy local behavior bounds, obtained by transferring analytic estimates and structural results from Banach-space theory via the local equivalence of Banach-Lie groups and their Lie algebras given by the exponential and logarithm maps.

Significance. If the transfer of càdlàg modifications and local bounds can be made rigorous with appropriate localization, the result would usefully extend existing Banach-space regularity theory to the setting of infinite-dimensional Lie groups. The approach of exploiting the local chart equivalence is a natural one and, if successful, could support further work on stochastic flows and multiplicative processes in this context.

major comments (1)
  1. [main transfer argument (abstract and §3–4)] The central transfer argument (outlined in the abstract and presumably developed in the main results section) constructs a càdlàg version in the Lie algebra and pulls it back via the logarithm. However, the exponential map is a diffeomorphism only on a neighborhood U of the identity. For a multiplicative process with independent increments, jumps may exit U in finite time. No stopping-time localization that preserves the multiplicative structure is described to control the exit probability or to glue the local càdlàg versions; without such controls the global càdlàg property on the group does not follow automatically from the Banach-space result.
minor comments (2)
  1. [preliminaries] Notation for the domain of the logarithm should be introduced explicitly (e.g., a fixed ball in the Banach norm) and used consistently when stating the local equivalence.
  2. [introduction] The abstract refers to “local behavior bounds”; the precise form of these bounds (e.g., moment estimates or modulus of continuity) should be stated in the introduction or theorem statements.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need for explicit localization in the transfer argument. We address this point in detail below and will strengthen the exposition in the revised version.

read point-by-point responses

  1. Referee: The central transfer argument (outlined in the abstract and presumably developed in the main results section) constructs a càdlàg version in the Lie algebra and pulls it back via the logarithm. However, the exponential map is a diffeomorphism only on a neighborhood U of the identity. For a multiplicative process with independent increments, jumps may exit U in finite time. No stopping-time localization that preserves the multiplicative structure is described to control the exit probability or to glue the local càdlàg versions; without such controls the global càdlàg property on the group does not follow automatically from the Banach-space result.

    Authors: We agree that the exponential map is a local diffeomorphism and that jumps of a multiplicative process may exit any fixed neighborhood U of the identity. The manuscript performs the transfer of càdlàg modifications and local bounds first in the Lie algebra (via the Banach-space theory) and then pulls the paths back locally via the logarithm. To obtain the global statement on the group we rely on the independent-increments property to restart the process after each exit from U. We acknowledge, however, that the construction of the successive stopping times, the control of exit probabilities, and the verification that the glued process remains multiplicative were only sketched rather than fully detailed. In the revision we will add a dedicated subsection that (i) defines the sequence of exit times from a countable exhaustion of neighborhoods, (ii) shows that the independent-increments property is preserved under these stopping times, and (iii) verifies that the resulting global process is càdlàg on the group and satisfies the stated local bounds. This will make the localization argument fully rigorous. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation transfers standard Banach-space results via local exp/log equivalence without self-definition or fitted predictions.

full rationale

The paper derives càdlàg modifications and local bounds for multiplicative processes on infinite-dimensional Lie groups by transferring analytic estimates from Banach spaces using the local diffeomorphism property of the exponential and logarithm maps near the identity. This is a standard structural fact in infinite-dimensional Lie theory and is invoked as an external input rather than being defined in terms of the target regularity properties. No equations reduce the claimed predictions to fitted parameters or prior self-citations by construction; the approach remains self-contained against external benchmarks in Banach-space stochastic analysis. The skeptic concern about chart exit is a potential technical gap but does not indicate circularity in the given derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that local equivalence via exponential and logarithm transfers estimates without distortion; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Local equivalence of Banach-Lie groups and Banach spaces via the exponential and logarithm allows transfer of analytic estimates and structural results
    Invoked in the abstract as the basis for moving results from Banach spaces to the group setting.

pith-pipeline@v0.9.0 · 5338 in / 1204 out tokens · 32080 ms · 2026-05-17T20:25:41.645459+00:00 · methodology

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