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arxiv: 2604.24664 · v1 · submitted 2026-04-27 · 🧮 math.PR

Absolute continuity of Rosenblatt measures

Petr \v{C}oupek , Tyrone E. Duncan , Bozenna Pasik-Duncan , Jakub Slav\'ik This is my paper

Pith reviewed 2026-05-08 01:37 UTC · model grok-4.3

classification 🧮 math.PR
keywords rosenblattpathmeasureabsolutecontinuitymeasuresprobabilityaddress
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The pith

For Rosenblatt paths shifted by a deterministic term plus a Wiener integral against a related fractional Brownian motion, an equivalent measure exists making the shifted path Rosenblatt again.

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

Rosenblatt processes are non-Gaussian, self-similar random paths with long memory, used in modeling turbulence and other phenomena. The question is whether shifting such a path by a fixed direction still allows an equivalent probability law under which the new path behaves like an original Rosenblatt process. Earlier work showed this fails for a simple linear drift. The present result identifies a larger class of shifts—built from a deterministic function plus a stochastic integral against fractional Brownian motion with matching Hurst index—for which an equivalent measure does exist. The proof constructs the Radon-Nikodym derivative explicitly for these directions. Several concrete examples illustrate the range of admissible shifts. The work sits in the theory of absolute continuity for measures on function spaces induced by non-Gaussian processes.

Core claim

we show that if the Rosenblatt path is shifted in a direction belonging to a class of nontrivial Gaussian variables (that consists of a deterministic shift and a Wiener integral with respect to a fractional Brownian motion with a related Hurst parameter), such a measure exists.

Load-bearing premise

The translation direction must lie in the specified class of Gaussian random variables built from deterministic functions and Wiener integrals against fractional Brownian motion; the result does not hold for arbitrary shifts, as shown by the linear-drift counterexample in the cited prior work.

read the original abstract

In the article, we address the problem of absolute continuity of translated Rosenblatt measures on the path space. In [\v{C}oupek, P., K\v{r}\'i\v{z}, P., Maslowski, B., Stoch. Proc. Appl. 179 (2025) art. no. 104499], it is shown that there is no probability measure that would be equivalent to the original probability measure and under which a Rosenblatt path with a linear drift would again be a Rosenblatt path. Here, we show that if the Rosenblatt path is shifted in a direction belonging to a class of nontrivial Gaussian variables (that consists of a deterministic shift and a Wiener integral with respect to a fractional Brownian motion with a related Hurst parameter), such a measure exists. We also give several examples to demonstrate the scope of the result.

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

0 major / 3 minor

Summary. The paper proves an existence result for equivalent measures under which a Rosenblatt process remains Rosenblatt after translation by shifts belonging to a specific class of Gaussian elements: deterministic functions plus Wiener integrals against a fractional Brownian motion whose Hurst index is related to that of the Rosenblatt process. This is explicitly contrasted with the non-existence for linear drifts established in the cited prior work of Čoupek, Kříž, and Maslowski (Stoch. Proc. Appl. 2025). Several illustrative examples are included to delineate the scope.

Significance. The result supplies a positive, targeted Girsanov-type theorem for the Rosenblatt process, clarifying the boundary between admissible and inadmissible translations in the non-Gaussian setting. By restricting to this Gaussian class and invoking standard properties of Rosenblatt processes and fBM, it provides a concrete, falsifiable characterization that advances the measure-theoretic analysis of self-similar processes with stationary increments.

minor comments (3)
  1. §2 (or wherever the main theorem is stated): the precise relation between the Hurst parameter H of the Rosenblatt process and the Hurst index of the driving fBM in the shift class should be stated explicitly as an assumption or derived from the covariance structure, rather than left implicit in the examples.
  2. The Radon-Nikodym derivative construction in the proof relies on the Cameron-Martin space of the auxiliary fBM; a short remark on why the resulting density is indeed a martingale (or satisfies the necessary integrability) would strengthen the argument without lengthening the paper.
  3. In the examples section, the deterministic component of the shift is sometimes taken to be zero; adding one non-trivial deterministic example would better demonstrate the full scope of the class.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the clear summary of its contributions, and the recommendation for minor revision. We are pleased that the result is viewed as advancing the measure-theoretic analysis of self-similar processes.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained mathematical proof

full rationale

The paper establishes existence of an equivalent measure for Rosenblatt paths translated by shifts in a specific Gaussian class (deterministic plus Wiener integral w.r.t. related fBM), using standard properties of Rosenblatt processes and fractional Brownian motion. It contrasts this with a linear-drift non-existence result from a prior citation (overlapping but distinct authors and result), but the positive existence claim is proved independently via construction of the Radon-Nikodym derivative or Girsanov-type arguments. No self-definitional reduction, fitted prediction, or load-bearing self-citation chain appears; the central result does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim relies on background properties of Rosenblatt processes and fractional Brownian motion that are taken from the literature rather than re-derived here.

axioms (2)
  • domain assumption Rosenblatt process is a well-defined non-Gaussian self-similar process with stationary increments and long-range dependence for Hurst parameter H in (1/2,1)
    Invoked implicitly when discussing translated measures and equivalence; standard in the field but not proved in this paper.
  • domain assumption Fractional Brownian motion with matching Hurst index generates the admissible Gaussian directions via Wiener integrals
    Used to define the class of shifts for which equivalence holds.

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