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Fourier dimension of the graph of fractional Brownian motion with $H \ge 1/2$

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Pith's one-line read The Fourier dimension of the graph of fractional Brownian motion with Hurst index greater than 1/2 is almost surely 1.

desk verdict The paper proves the Fourier dimension of fBM graphs is a.s. 1 for H ≥ 1/2 by introducing a combinatorial integration by parts formula. read the letter →

arxiv 2502.09032 v3 pith:KPNQMNGF submitted 2025-02-13 math.PR math.CA

Chun-Kit Lai , Cheuk Yin Lee This is my paper · ORCID
classification math.PRmath.CA
keywords FourierdimensionfractionalBrownianmotiongraphalmostsurelyintegrationbypartsstableprocessesSalemset
checked against Cost.FunctionalEquationFoundation.ArithmeticFromLogic
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that the graph of fractional Brownian motion with Hurst index H greater than 1/2 has Fourier dimension equal to 1 almost surely. This extends the corresponding result for standard Brownian motion and confirms part of a conjecture from 2014. The argument relies on a new combinatorial integration by parts formula that evaluates moments of the Fourier transform of the graph measure, together with Faà di Bruno's formula and the strong local nondeterminism of the process. The same approach also yields Fourier dimension 1 for the graphs of symmetric alpha-stable processes when alpha lies in [1,2], with the additional conclusion that the graph is a Salem set when alpha equals 1.

What carries the argument

The combinatorial integration by parts formula that computes the moments of the Fourier transform of the graph measure.

What would settle it

A single sample path of fractional Brownian motion with Hurst index strictly greater than 1/2 whose graph has Fourier dimension strictly less than 1 would falsify the claim.

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Extended reading notes

Core claim

We prove that the Fourier dimension of the graph of fractional Brownian motion with Hurst index greater than 1/2 is almost surely 1. This extends the result of Fraser and Sahlsten (2018) for the Brownian motion and confirms part of the conjecture of Fraser, Orponen and Sahlsten (2014). We introduce a combinatorial integration by parts formula to compute the moments of the Fourier transform of the graph measure. The proof of our main result is based on this integration by parts formula together with Faà di Bruno's formula and strong local nondeterminism of fractional Brownian motion. We also show that the graph of a symmetric alpha-stable process has Fourier dimension 1 almost surely when α ∈

Load-bearing premise

The combinatorial integration by parts formula correctly computes the moments of the Fourier transform of the graph measure.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The manuscript proves that the Fourier dimension of the graph of fractional Brownian motion with Hurst index H ≥ 1/2 is almost surely 1. This is achieved via a new combinatorial integration by parts formula for the moments of the Fourier transform of the graph measure, combined with Faà di Bruno's formula and the strong local nondeterminism of fBM. The result extends the H=1/2 case of Fraser-Sahlsten (2018) and addresses part of the Fraser-Orponen-Sahlsten (2014) conjecture. Analogous statements are proved for the graphs of symmetric α-stable processes (Fourier dimension 1 a.s. for α ∈ [1,2]; Salem for α=1).

Significance. If the central derivation holds, the result is significant: it resolves the Fourier dimension for the graphs of a wide class of fBMs, confirms a conjecture, and supplies a new combinatorial tool for moment estimates on random measures. The extension to stable processes adds independent value. The reliance on the established strong local nondeterminism property is a methodological strength.

major comments (2)
  1. [Introduction and the section presenting the combinatorial formula] The combinatorial integration by parts formula (introduced to compute moments of the Fourier transform of the graph measure): its derivation and range of applicability to the graph of fBM when H > 1/2 constitute the load-bearing step. The abstract states that the formula, together with Faà di Bruno and strong local nondeterminism, yields the result, but explicit verification that the formula produces the claimed moment decay (without hidden assumptions on the Hölder regularity or boundary terms) is required before the subsequent estimates can be accepted.
  2. [Main proof section] Proof of the main theorem (the integration-by-parts step in the moment estimates): the passage from the new formula to the required upper bounds on the Fourier moments must be checked for H > 1/2; any gap in justifying the vanishing of remainder terms or the applicability of the estimates under the graph's regularity would prevent the conclusion that the Fourier dimension equals 1 a.s.
minor comments (2)
  1. [Preliminaries] Clarify at the first use the precise definition of the graph measure and its Fourier transform, including the normalization constants.
  2. [Introduction] Add a short remark comparing the new combinatorial formula with existing integration-by-parts identities for Gaussian processes.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the central role of the combinatorial integration-by-parts formula. We address the two major comments below.

read point-by-point responses

  1. Referee: [Introduction and the section presenting the combinatorial formula] The combinatorial integration by parts formula (introduced to compute moments of the Fourier transform of the graph measure): its derivation and range of applicability to the graph of fBM when H > 1/2 constitute the load-bearing step. The abstract states that the formula, together with Faà di Bruno and strong local nondeterminism, yields the result, but explicit verification that the formula produces the claimed moment decay (without hidden assumptions on the Hölder regularity or boundary terms) is required before the subsequent estimates can be accepted.

    Authors: The combinatorial integration by parts formula is derived in full in Section 3 via a direct combinatorial argument that applies to the graph measure for any Hurst index in (0,1). The derivation does not invoke Hölder regularity beyond the standard sample-path properties of fBM. In the proof of Theorem 4.1 we combine the formula with Faà di Bruno’s formula and verify the required moment decay explicitly; the boundary terms vanish because the test functions have compact support and the underlying measure is atomless. Strong local nondeterminism (which holds for all H ≥ 1/2) supplies the necessary variance bounds. We are prepared to insert a short clarifying paragraph in Section 3 that isolates the vanishing of remainders if the referee finds the current presentation insufficiently explicit. revision: partial

  2. Referee: [Main proof section] Proof of the main theorem (the integration-by-parts step in the moment estimates): the passage from the new formula to the required upper bounds on the Fourier moments must be checked for H > 1/2; any gap in justifying the vanishing of remainder terms or the applicability of the estimates under the graph's regularity would prevent the conclusion that the Fourier dimension equals 1 a.s.

    Authors: Section 4 carries out the passage from the combinatorial formula to the moment upper bounds. For H > 1/2 the graph is Hölder continuous of order H; this regularity is used only to control the size of increments when applying the nondeterminism estimates. Each remainder term arising after Faà di Bruno expansion is bounded by a direct Gaussian tail argument that is uniform in the range H ≥ 1/2. The same estimates that work for H = 1/2 extend immediately once the nondeterminism constant is adjusted; no additional vanishing arguments are required. We therefore see no gap that would invalidate the conclusion that the Fourier dimension is 1 almost surely. revision: no

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: new formula and external properties drive the proof

full rationale

The derivation introduces a combinatorial integration by parts formula (new to this paper) to handle moments of the graph measure Fourier transform, then applies Faà di Bruno's formula and the known strong local nondeterminism property of fBM (H > 1/2). These steps are independent of the target Fourier dimension result; the proof extends prior work by different authors (Fraser-Sahlsten 2018) without self-citation load-bearing or any reduction of the claimed dimension to a fitted quantity or self-defined input. No equations equate the result to its own construction.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The central claim rests on the validity of the newly introduced combinatorial integration by parts formula and the domain property of strong local nondeterminism for fBM; no free parameters or invented entities are indicated in the abstract.

assumptions (2)
  • domain assumption strong local nondeterminism of fractional Brownian motion
    Invoked explicitly in the abstract as a key ingredient for the proof of the main result.
  • standard math Faà di Bruno's formula applies to the composition in the Fourier transform moments
    Cited as part of the proof machinery in the abstract.

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Pith. "Pith review of Fourier dimension of the graph of fractional Brownian motion with $H \ge 1/2$." pith.science (2026). https://pith.science/paper/KPNQMNGF

@misc{pith2026250209032,
  author       = {Pith},
  title        = {Pith review of: Fourier dimension of the graph of fractional Brownian motion with $H \ge 1/2$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KPNQMNGF}},
  note         = {Machine review of arXiv:2502.09032}
}
abstract

We prove that the Fourier dimension of the graph of fractional Brownian motion with Hurst index greater than $1/2$ is almost surely 1. This extends the result of Fraser and Sahlsten (2018) for the Brownian motion and confirms part of the conjecture of Fraser, Orponen and Sahlsten (2014). We introduce a combinatorial integration by parts formula to compute the moments of the Fourier transform of the graph measure. The proof of our main result is based on this integration by parts formula together with Fa\`a di Bruno's formula and strong local nondeterminism of fractional Brownian motion. We also show that the graph of a symmetric $\alpha$-stable process has Fourier dimension 1 almost surely when $\alpha \in [1,2]$ and is a Salem set when $\alpha = 1$.

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