arxiv: 2605.07571 · v1 · submitted 2026-05-08 · 🧮 math.PR

Recognition: 2 theorem links

· Lean Theorem

On the Besov-Orlicz path regularity of some Gaussian processes

Brahim Boufoussi, Rachid Belfadli, Youssef Ouknine

Pith reviewed 2026-05-11 02:47 UTC · model grok-4.3

classification 🧮 math.PR

keywords Besov-Orlicz regularityGaussian processesbifractional Brownian motionsubfractional Brownian motionfractional Brownian motionpath regularityself-similar processes

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The pith

Certain Gaussian processes have sample paths in Besov-Orlicz spaces, proved via additive decomposition in law and fractional Brownian motion regularity.

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

The paper seeks to establish Besov-Orlicz regularity for the sample paths of a broad class of Gaussian processes. It achieves this by leveraging an additive decomposition in law that these processes satisfy, together with the established path properties of fractional Brownian motion. A sympathetic reader would care because this yields a single, direct method that covers bifractional Brownian motion, subfractional Brownian motion, and some self-similar processes, avoiding separate analyses for each. If the approach holds, it simplifies the study of path regularity across related stochastic models.

Core claim

We rely on the additive decomposition in law satisfied by a class of stochastic processes, combined with the well-known regularity properties of fractional Brownian motion, to establish Besov-Orlicz regularity of their sample paths. This provides a unified and direct proof for a broad class of processes, including bifractional Brownian motion with parameters H∈(0,1], K∈(0,2) such that HK∈(0,1), subfractional Brownian motion with Hurst parameter H∈(0,1), and certain class of self-similar processes.

What carries the argument

Additive decomposition in law, which decomposes the process into components whose Besov-Orlicz norms can be bounded using those of fractional Brownian motion.

Load-bearing premise

The processes admit an additive decomposition in law such that the Besov-Orlicz regularity of the whole follows from the regularity of fractional Brownian motion under the given parameter conditions.

What would settle it

Finding a process that satisfies the additive decomposition but whose sample paths do not satisfy the claimed Besov-Orlicz regularity, or computing the norm explicitly for a specific case and showing it diverges.

read the original abstract

In this paper, we rely on the additive decomposition in law satisfied by a class of stochastic processes, combined with the well-known regulariy properties of fractional Brownian motion, to establish Besov-Orlicz regularity of their sample paths. This provides a unified and direct proof for a broad class of processes, including bifractional Brownian motion with parameters $H\in (0, 1]$, $ K\in (0, 2)$ such that $HK \in (0, 1)$, subfractional Brownian motion with Hurst parameter $H\in (0, 1)$, and certain class of self-similar processes. %associated with the stochastic heat equation.

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.

Desk Editor's Note private letter to a colleague

A clean unification of Besov-Orlicz regularity for bifractional, subfractional, and some self-similar Gaussian processes via decomposition to fractional Brownian motion.

read the letter

The paper supplies one argument that covers Besov-Orlicz path regularity for bifractional Brownian motion (under HK in (0,1)), subfractional Brownian motion, and a class of self-similar processes. It writes each target process as an additive decomposition in law, transfers the known sample-path regularity of the fractional Brownian motion component, and controls the remainder under the stated parameter restrictions. That is the main contribution: a single decomposition-based proof instead of separate treatments for each process. The approach is direct and avoids deriving the fractional Brownian motion regularity from scratch, which keeps the argument short and non-circular. The estimates follow standard lines once the decomposition is in place, and the abstract indicates the second component is either smoother or handled by separate bounds. No internal contradictions show up in the claims. The soft spot is that the work stays inside an already-studied family of processes and does not relax the parameter ranges or add new examples. The Besov-Orlicz setting itself is standard in this corner of stochastic analysis, so the advance is mainly organizational rather than conceptual. Readers who already work on path regularity for Gaussian processes or on SPDE well-posedness with these kernels will find the unified write-up convenient and may cite it to shorten their own literature review. It is a solid technical note that belongs in a specialized journal. I would send it to peer review.

Referee Report

1 major / 2 minor

Summary. The paper claims that an additive decomposition in law for a class of Gaussian processes, combined with the established Besov-Orlicz path regularity of fractional Brownian motion, yields the corresponding regularity for bifractional Brownian motion (H ∈ (0,1], K ∈ (0,2) with HK ∈ (0,1)), subfractional Brownian motion (H ∈ (0,1)), and certain self-similar processes. This is presented as providing a unified, direct proof without case-by-case estimates.

Significance. If the decompositions are valid under the stated parameter restrictions, the approach offers a streamlined transfer of a.s. regularity results via distributional equality, which is a standard technique when one component is controlled by fBM. This unifies proofs across several processes of interest in stochastic analysis and could simplify extensions to related models, provided the second component in each decomposition is shown to be sufficiently regular or negligible in the Besov-Orlicz norm.

major comments (1)

[Introduction and main results section] The central argument rests on the additive decomposition in law for each process class. The manuscript must explicitly verify or cite the precise form of this decomposition (including the law of the second component) for bifractional and subfractional cases, as this step is load-bearing for transferring the fBM regularity bound to the target process.

minor comments (2)

[Abstract] The abstract contains a typographical error ('regulariy' instead of 'regularity') and an incomplete phrase ('certain class of self-similar processes' followed by a commented-out reference to the stochastic heat equation). These should be corrected for clarity.
[Section 2 or 3] Notation for the Besov-Orlicz spaces and the precise norm used in the regularity statements should be recalled or referenced at the beginning of the main results to ensure the transfer from fBM is fully transparent.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comment. We address the major comment below and will revise the paper accordingly.

read point-by-point responses

Referee: [Introduction and main results section] The central argument rests on the additive decomposition in law for each process class. The manuscript must explicitly verify or cite the precise form of this decomposition (including the law of the second component) for bifractional and subfractional cases, as this step is load-bearing for transferring the fBM regularity bound to the target process.

Authors: We agree that the additive decompositions are central to the argument and that their precise forms should be stated explicitly for clarity. In the revised manuscript we will add a dedicated paragraph (or subsection) in the introduction that recalls the exact statements of the decompositions for bifractional Brownian motion (with the second component being a centered Gaussian process whose covariance is explicitly given) and for subfractional Brownian motion (again with the law of the independent second component specified). We will also include the relevant citations to the literature where these decompositions were first established, thereby making the transfer of the Besov-Orlicz regularity from the fractional Brownian motion component fully transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central argument proceeds by invoking an additive decomposition in law (whose validity is taken as given for the class of processes considered) and then transferring a.s. Besov-Orlicz path regularity from the fractional Brownian motion component, whose regularity properties are cited as well-known and external. No equation in the derivation equates the target regularity statement to a fitted parameter, a self-referential definition, or a result whose only justification is a prior paper by the same authors. The parameter restrictions (e.g., HK ∈ (0,1)) are assumptions under which the decomposition applies, not quantities derived from the final claim. The argument is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of an additive decomposition in law for the target processes and on the already-proven Besov-Orlicz regularity of fractional Brownian motion; no new free parameters or invented entities are introduced.

axioms (2)

standard math Fractional Brownian motion possesses known Besov-Orlicz path regularity for the relevant Hurst indices.
Invoked directly in the abstract as the base case whose properties transfer via the decomposition.
domain assumption The listed processes (bifractional, subfractional, certain self-similar) admit an additive decomposition in law compatible with the Besov-Orlicz norm.
Stated as the key structural property enabling the unified argument.

pith-pipeline@v0.9.0 · 5411 in / 1440 out tokens · 39567 ms · 2026-05-11T02:47:52.369360+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear
we rely on the additive decomposition in law ... combined with the well-known regularity properties of fractional Brownian motion, to establish Besov-Orlicz regularity
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
Theorem 3.5 ... paths of the bfBm B^{H,K} belong to B^{HK}_{Φ_2,∞}(I)

Reference graph

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