pith. sign in

arxiv: 2404.09800 · v2 · submitted 2024-04-15 · 🧮 math.PR

Fractional derivatives of local times for some Gaussian processes

Minhao Hong , Qian Yu This is my paper

Pith reviewed 2026-05-24 02:32 UTC · model grok-4.3

classification 🧮 math.PR
keywords local timefractional derivativesMarchaud derivativeGaussian processesHölder continuitystrong local nondeterminismcontour integration
0
0 comments X

The pith

A condition tied to strong local nondeterminism determines when Marchaud fractional derivatives of local times exist in L^p for Gaussian processes.

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

The paper gives a condition under which the Marchaud fractional derivatives of local times for d-dimensional centered Gaussian processes exist in L^p for p at least 1. These derivatives are Hölder continuous in both time and space variables and continuous with respect to the fractional order. Under further assumptions the condition is also necessary, proved via contour integration.

Core claim

For d-dimensional centered Gaussian processes satisfying a strong local nondeterminism property, a certain condition ensures the existence of Marchaud fractional derivatives of the local time in L^p spaces for p ≥ 1. These derivatives are Hölder continuous with respect to time and space variables and continuous with respect to the order of derivatives. Under additional assumptions, this condition is also necessary for existence of the derivatives, established using contour integration.

What carries the argument

Marchaud fractional derivative of the local time, whose existence is governed by a condition linked to the strong local nondeterminism property of the Gaussian process

Load-bearing premise

The d-dimensional centered Gaussian processes satisfy a strong local nondeterminism property.

What would settle it

An explicit Gaussian process satisfying the strong local nondeterminism property for which the Marchaud fractional derivatives of local time fail to exist in some L^p, or exist without the condition under the additional assumptions used for necessity.

Figures

Figures reproduced from arXiv: 2404.09800 by Minhao Hong, Qian Yu.

Figure 4.1
Figure 4.1. Figure 4.1: The shadow stands for the area {(s, t) : 0 < s < T , s < t < e se(s)} Then using Lemma 2.3, because Hd > 1, we get 2+2H P P ℓ∈I αℓ ℓ∈I αℓ+|α|+d − 2H < 0 and E |L (α) +,ε(T, 0)| 2 ≥ cε 2+2H P P ℓ∈I αℓ ℓ∈I αℓ+|α|+d−2H for any ε ∈ (0, 1 2 ), which completes statements (2) and (3) in Theorem 1.4. Part III: The case that αℓ is integer and xℓ 6= 0 for some ℓ ∈ {1, 2, · · · , d} In this part, we assume X ∈ Ged,… view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: The contour of (z) −α−1 0 (1 − e −z ) Now assume 0 < r < R, by Cauchy-Goursat Theorem for analytic function we have Z γ1+γr+γ2+γR (z) −α−1 0 (1 − e −z )dz = 0, where the four curves γ1, γr, γ2, γR(see [PITH_FULL_IMAGE:figures/full_fig_p025_5_1.png] view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: The contour of (z + ι x a ) me − 1 2 az2 when x > 0 (left) and x < 0 (right) Because F(z) is analytic on C, we have Z γ1 F(z)dz + Z γ2 F(z)dz = Z γ4 F(z)dz + Z γ3 F(z)dz, where Z γ1 F(z)dz = ιsgn (x) Z |x| a 0 [PITH_FULL_IMAGE:figures/full_fig_p027_5_2.png] view at source ↗
read the original abstract

In this article, we consider fractional derivatives of local time for $d-$dimensional centered Gaussian processes satisfying certain strong local nondeterminism property. We first give a condition for existence of fractional derivatives of the local time defined by Marchaud derivatives in $L^p(p\ge1)$ and show that these derivatives are H\"older continuous with respect to both time and space variables and are also continuous with respect to the order of derivatives. Moreover, under some additional assumptions, we show that this condition is also necessary for existence of derivatives of the local time with the help of contour integration.

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 / 0 minor

Summary. The paper studies Marchaud fractional derivatives of local times for d-dimensional centered Gaussian processes satisfying a strong local nondeterminism property. It establishes a sufficient condition for existence of these derivatives in L^p (p≥1), proves they are Hölder continuous in both time and space variables and continuous in the derivative order, and shows necessity of the condition (under additional assumptions) via contour integration.

Significance. If the technical arguments hold, the results extend the regularity theory of local times for Gaussian processes to fractional orders, with the necessity direction providing a sharp characterization. The approach relies on the strong local nondeterminism assumption and standard contour-integration techniques; this could be useful for further work on sample-path properties of Gaussian processes.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work, recognition of its significance in extending regularity theory for local times of Gaussian processes to fractional orders, and recommendation to accept the manuscript. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper conditions its existence results for Marchaud fractional derivatives of local time on an external strong local nondeterminism property of the Gaussian processes, proves Hölder continuity and continuity in order from that plus a stated condition, and establishes necessity via contour integration (a standard external technique) under further assumptions. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citation chains appear; the derivation chain is self-contained against external benchmarks and does not reduce any claimed result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the domain assumption that the processes are centered Gaussian and obey strong local nondeterminism; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption The processes are d-dimensional centered Gaussian processes satisfying a strong local nondeterminism property.
    This property is required for the existence condition on the fractional derivatives.

pith-pipeline@v0.9.0 · 5609 in / 1217 out tokens · 42151 ms · 2026-05-24T02:32:56.096669+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages · 1 internal anchor

  1. [1]

    Houdr´ e and J

    C. Houdr´ e and J. Villa: An example of infinite dimensional quasi-helix . Contemporary Mathematics 336, 195-202, 2003

    work page 2003

  2. [2]

    Bojdeckia, L

    T. Bojdeckia, L. Gorostizab and A. Talarczyk: Sub-fractional Brownian motion and its relation to occupation times. Statistics & Probability Letters 69, 405–419, 2004

    work page 2004

  3. [3]

    J. Song, F. Xu and Q. Yu: Limit theorems for functionals of two ind ependent Gaussian processes. Stochastic Processes and their Applications 129(11), 4791–4836, 2019

    work page 2019

  4. [4]

    Samko, A

    S. Samko, A. Kilbas and O. Marichev: Fractional integrals and derivatives: theory and applications, Gordon and Breach, Switzerland, 1993

    work page 1993

  5. [5]

    Yamada: On the fractional derivative of Brownian local times

    T. Yamada: On the fractional derivative of Brownian local times. Jounal of Mathemat- ics of Kyoto University 25(1), 49–58, 1985. 29

    work page 1985

  6. [6]

    S. M. Berman: Local times and sample function properties of sta tionary Gaussian processes, The Annals of Mathematical Statistics 41(4), 1260-1272, 1970

    work page 1970

  7. [7]

    Yan: The fractional derivative for fractional Brownian local time with Hurst index large than 1/2

    L. Yan: The fractional derivative for fractional Brownian local time with Hurst index large than 1/2. Mathematische Zeitschrift 283, 437–468, 2016

    work page 2016

  8. [8]

    Yamada: On some limit theorems for occupation times of one-dim ensional Brow- nian motion and its continuous additive functionals locally of zero ener gy

    T. Yamada: On some limit theorems for occupation times of one-dim ensional Brow- nian motion and its continuous additive functionals locally of zero ener gy. Jounal of Mathematics of Kyoto University 26(2), 302–322, 1986

    work page 1986

  9. [9]

    Fitzsimmons and R

    P. Fitzsimmons and R. Getoor: Limit theorems and variation prope rties for fractional derivatives of the local time of a stable process. Annales de l’Institut Henri Poincare (B) Probability and Statistics 28, 311–333, 1992

    work page 1992

  10. [10]

    M. A. Ouahra and M. Ouali: Occupation time problems for fraction al Brownian mo- tion and some other self-similar processes. Random Operators and Stochastic Equations 17(1), 69–89, 2009

    work page 2009

  11. [11]

    Sghir: Limit theorems and strong approximation for occupat ion times problem of sub-fractional Brownian motion in a class of Besov spaces

    A. Sghir: Limit theorems and strong approximation for occupat ion times problem of sub-fractional Brownian motion in a class of Besov spaces. Random Operators and Stochastic Equations 21, 111–124, 2013

    work page 2013

  12. [12]

    Nualart and S

    D. Nualart and S. Ortiz-Latorre: Intersection local time for t wo independent fractional Brownian motions, Journal of Theoretical Probability 20(4), 759–767, 2007

    work page 2007

  13. [13]

    Wu and Y

    D. Wu and Y. Xiao: Regularity of intersection local times of fract ional Brownian mo- tions. Journal of Theoretical Probability 23(4), 972–1001, 2010

    work page 2010

  14. [14]

    Derivative for the intersection local time of fractional Brownian Motions

    L. Yan: Derivative for the intersection local time of fractional Brownian motions. arXiv:1403.4102., 2014

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  15. [15]

    J. Guo, Y. Hu and Y. Xiao: Higher-Order Derivative of Intersec tion Local Time for Two Independent Fractional Brownian Motions. Journal of Theoretical Probability 32, 1190–1201, 2019

    work page 2019

  16. [16]

    Hong and F

    M. Hong and F. Xu: Derivatives of local times for some Gaussian fi elds. Journal of Mathematical Analysis and Applications 484, 123716, 2020

    work page 2020

  17. [17]

    Hong and F

    M. Hong and F. Xu: Derivatives of local times for some Gaussian fi elds II. Statistics & Probability Letters 172, 109063, 2021

    work page 2021

  18. [18]

    M. Hong, H. Liu and F. Xu: Limit theorems for additive functionals of some self-similar Gaussian processes, arXiv:2305.13146, 2023

    work page arXiv 2023

  19. [19]

    Cuzick and J.P

    J. Cuzick and J.P. Du Preez: Joint continuity of Gaussian local tim es. Annals of Prob- ability 10, 810–817, 1982

    work page 1982

  20. [20]

    Kalbasi and T

    K. Kalbasi and T. Mountford: On the probability distribution of t he local times of diagonally operator-self-similar Gaussian fields with stationary incre ments. Bernoulli 26(2), 1504–1534, 2020. 30 Minhao Hong School of Science, Shanghai Maritime University, Shanghai 20130 6, China hongmhmath@163.com Qian Yu Department of Mathematics, Nanjing University o...

    work page 2020