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arxiv: 2604.08373 · v2 · submitted 2026-04-09 · 🌌 astro-ph.HE · astro-ph.IM· cond-mat.stat-mech· gr-qc· physics.data-an

Recognition: 2 theorem links

· Lean Theorem

Stochastic problems in pulsar timing

Reginald Christian Bernardo

Authors on Pith no claims yet

Pith reviewed 2026-05-10 18:06 UTC · model grok-4.3

classification 🌌 astro-ph.HE astro-ph.IMcond-mat.stat-mechgr-qcphysics.data-an
keywords pulsar timinggravitational wave backgroundOrnstein-Uhlenbeck processLangevin equationstiming residualsneutron star modelsstochastic processesdiffusion theory
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The pith

The Ornstein-Uhlenbeck model for pulsar spin frequency is inconsistent with a stationary gravitational wave background when timing residuals are the direct observable.

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

The paper uses diffusion theory to solve Langevin equations that describe red noise and gravitational wave background signals in pulsar timing. It shows that modeling the spin frequency as an Ornstein-Uhlenbeck process produces non-stationary timing residuals incompatible with a stationary GWB signal. Marginalizing over long-term deterministic trends in the data offers a partial fix for this inconsistency. An alternative process based on an overdamped harmonic oscillator instead keeps both the spin frequency and phase residuals stationary, aligning with GWB expectations. The work also derives exact solutions for a two-component neutron star model with spin wandering, tracing nonstationarity to the coexistence of damped and diffusive eigenmodes.

Core claim

When the timing residual is taken as the direct observable, the Ornstein-Uhlenbeck process for the pulsar spin frequency produces nonstationary statistics that are incompatible with a stationary gravitational wave background. Marginalizing over long time trends offers a partial remedy. A process modeled after an overdamped harmonic oscillator, however, yields stationary spin frequency and phase residuals that are consistent with a stationary GWB. In a two-component model of the neutron star, the coexistence of damped and diffusive eigenmodes explains the nonstationarity in the residual.

What carries the argument

Langevin stochastic differential equations solved via diffusion theory, using the timing residual as the observable to test stationarity; the overdamped harmonic oscillator process serves as the alternative to the Ornstein-Uhlenbeck model.

If this is right

  • The nonstationarity from the Ornstein-Uhlenbeck model can be mitigated by marginalizing over deterministic trends in pulsar timing data.
  • The overdamped harmonic oscillator model ensures consistency between stationary spin frequency, phase residuals, and a stationary GWB signal.
  • Analytical means, covariances, and cross-covariances for the two-component neutron star model link constant torques to quadratic spin-down and identify the source of nonstationarity in eigenmodes.
  • State-space algorithms for pulsar timing array analysis gain direct physical insight from the derived time-domain solutions.

Where Pith is reading between the lines

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

  • Pulsar timing array searches for gravitational waves may need to adopt stationarity-preserving models to avoid systematic biases in signal detection.
  • The analytical solutions could improve computational efficiency in existing state-space PTA pipelines by providing exact covariances instead of numerical approximations.
  • Similar stationarity checks using the timing residual as observable might apply to other stochastic timing phenomena, such as those in exoplanet transit timing or binary pulsar systems.

Load-bearing premise

The assumption that the timing residual is the right direct observable for checking consistency with a stationary gravitational wave background and that the chosen Langevin equations capture the relevant physics without missing effects.

What would settle it

Long-baseline pulsar timing data whose residual covariance either matches the nonstationary growth predicted by the Ornstein-Uhlenbeck model or remains stationary as required by the overdamped oscillator model.

Figures

Figures reproduced from arXiv: 2604.08373 by Reginald Christian Bernardo.

Figure 1
Figure 1. Figure 1: FIG. 1. Spacetime diagram showing consecutive electromag [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Simulation results (red points with error bars) compared with the steady-state analytical predictions (blue solid line [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Schematic illustration of the lemma on random walks [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Counterclockwise semicircular contours [PITH_FULL_IMAGE:figures/full_fig_p024_4.png] view at source ↗
read the original abstract

Langevin stochastic differential equations provide a dynamical description of pulsar timing noise and gravitational wave background (GWB) signals. They are also central to state space algorithms that have gained traction in pulsar timing array analysis due to their linear computational scaling with the number of observations. In this work, we utilize established methods in diffusion theory to derive analytical time-domain solutions (means, covariances, and probability density functions) to Langevin equations relevant to red noise and the GWB signal in pulsars. The solutions give direct physical insight on the dynamics of pulsar timing signals. As a canonical example, we show that the pulsar spin frequency modeled as an Ornstein-Uhlenbeck process is mathematically inconsistent with a stationary GWB signal when the timing residual is the direct observable. The nonstationarity can be partially dealt with by marginalizing over long time deterministic trends in the data. Then, we show that a random process based on an overdamped harmonic oscillator supports both a stationary spin frequency and phase residuals, consistent with a stationary GWB signal. We also turn our attention to a phenomenological model of a neutron star -- a two-component model with spin wandering -- that has been motivated to explain observed timing noise in radio pulsars. We derive analytical expressions for the means, covariances, and cross-covariances of the crust and superfluid rotational states driven by white noise. The associated constant deterministic torques are linked to the quadratic spin-down of pulsars. The solutions reveal the physical origin of nonstationarity in the residual model: the coexistence of damped and diffusive eigenmodes of the system.

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 manuscript applies established diffusion theory methods to derive analytical time-domain expressions for the means, covariances, and probability density functions of Langevin equations modeling pulsar timing noise and GWB signals. It shows that an Ornstein-Uhlenbeck process for spin frequency produces non-stationary timing residuals (variance growing with time), which is inconsistent with a stationary GWB when the residual is the direct observable, but can be partially mitigated by marginalizing over long-term deterministic trends; it then demonstrates that an overdamped harmonic oscillator process and a two-component crust-superfluid model with spin wandering can support stationarity in both frequency and phase, with the latter linked to quadratic spin-down via constant torques and revealing nonstationarity from coexisting damped and diffusive eigenmodes.

Significance. If the derivations hold, the explicit analytical solutions provide valuable physical insight into the stationarity properties of pulsar timing models and their consistency with GWB signals, directly supporting the development of state-space algorithms with linear scaling. The reproducible, parameter-free derivations of covariances and cross-covariances for the two-component model, along with the identification of eigenmodes as the origin of nonstationarity, are particular strengths that enhance falsifiability and applicability to PTA data analysis.

minor comments (3)
  1. [Abstract] The abstract summarizes the use of standard methods and key conclusions but omits any equations or explicit forms of the derived covariances; adding at least one representative expression (e.g., for the time-dependent variance in the OU case) would improve immediate accessibility without lengthening the text.
  2. [Model sections (3-5)] Notation for the Langevin equations and their parameters (e.g., damping coefficients, noise strengths) should be unified across the OU, overdamped oscillator, and two-component sections to facilitate direct comparison of the stationarity properties.
  3. [OU process analysis] The discussion of marginalization over deterministic trends as a partial fix for nonstationarity would benefit from a short quantitative example showing the residual variance after marginalization, to clarify the practical impact on GWB searches.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of its contributions, and recommendation for minor revision. The referee's description correctly captures the analytical derivations, the inconsistency of the Ornstein-Uhlenbeck spin-frequency model with stationary timing residuals, the resolution via the overdamped oscillator and two-component models, and the physical insight from the eigenmodes. No specific major comments requiring rebuttal or changes were listed beyond this summary.

read point-by-point responses

  1. Referee: The manuscript applies established diffusion theory methods to derive analytical time-domain expressions for the means, covariances, and probability density functions of Langevin equations modeling pulsar timing noise and GWB signals. It shows that an Ornstein-Uhlenbeck process for spin frequency produces non-stationary timing residuals (variance growing with time), which is inconsistent with a stationary GWB when the residual is the direct observable, but can be partially mitigated by marginalizing over long-term deterministic trends; it then demonstrates that an overdamped harmonic oscillator process and a two-component crust-superfluid model with spin wandering can support stationarity in both frequency and phase, with the latter linked to quadratic spin-down via constant torques and revealing nonstationarity from coexisting damped and diffusive eigenmodes.

    Authors: We agree with this summary, which accurately reflects the scope and results of the paper. The derivations follow standard methods from diffusion theory (e.g., Fokker-Planck and moment equations for linear SDEs), and the stationarity analysis for each model is as described. No corrections to the reported findings are needed. revision: no

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper applies established diffusion-theory methods to solve Langevin equations for pulsar timing noise and GWB signals, deriving means, covariances, and PDFs directly from the stochastic differential equations. The stationarity analysis for the Ornstein-Uhlenbeck spin-frequency model versus the overdamped-oscillator process follows from standard covariance calculations on the integrated residuals without any reduction to fitted parameters, self-definitions, or self-citation chains. The two-component crust-superfluid derivations similarly use linear system eigenmodes and white-noise driving terms whose solutions are independent of the target conclusions. All steps remain self-contained against external benchmarks in stochastic processes.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard mathematical techniques for solving linear stochastic differential equations and domain-specific assumptions about which physical processes dominate pulsar timing noise. No new particles, forces, or dimensions are postulated.

axioms (2)
  • standard math Established methods in diffusion theory apply directly to the Langevin equations for red noise and GWB signals
    Invoked to obtain closed-form means, covariances, and PDFs in the time domain.
  • domain assumption Pulsar timing residuals can be modeled as direct observables of underlying stochastic processes without additional unaccounted systematics
    Central to the stationarity analysis of the Ornstein-Uhlenbeck and oscillator models.

pith-pipeline@v0.9.0 · 5588 in / 1567 out tokens · 60404 ms · 2026-05-10T18:06:38.023475+00:00 · methodology

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128 extracted references · 81 canonical work pages · 5 internal anchors

  1. [1]

    We first write this in state space form [53, 54]

    SDE to Gaussian process Consider the linear time invariant SDE (35) and its driving force’s (white) covariance function (36). We first write this in state space form [53, 54]. By defining y(t) = y(t) dy dt · · · dm−1y dt T ,(D4) F=   0 1 ... ... 0 1 −a0 · · · −a m−2 −am−1   ,(D5) L= (0· · ·0 1) T ,(D6) then (35) can be expressed as a first order...

  2. [2]

    Another way this can also be realized is by taking the Fourier transform of (35) directly

    Gaussian process to SDE Conversely, if the PSD of a Gaussian processy(t) can be expressed in the rational form of (D16) [53, 54]; or S(ω)≡ T(iω)sT(−iω),(D17) then it can be associated with a SDE. Another way this can also be realized is by taking the Fourier transform of (35) directly. This gives (−iω)m +a m−1(−iω)m−1 +· · ·+a 1(−iω) +a 0 ˜y(ω) = ˜w(ω). (...

  3. [3]

    In all cases, we list down the covariance functionC(t), the one-sided PSD S(f), and the SDE (plus the corresponding white noise covariance); the noise term has zero mean

    SDEs for Gaussian processes For the convenience of readers interested and those who want to try out their analytical prowess in deriving results (special cases), we enumerate SDEs for the com- monly adopted Gaussian processes. In all cases, we list down the covariance functionC(t), the one-sided PSD S(f), and the SDE (plus the corresponding white noise co...

  4. [4]

    This makes it a good reference point for building state space algorithms

    The Mat´ ern kernel The Mat´ ern kernel with arbitrary index covers pro- cesses with ranging from the Ornstein-Uhlenbeck pro- cess (one time differentiable, shallow spectrum) to the RBF kernel (infinite differentiable, steep spectrum). This makes it a good reference point for building state space algorithms. We use this section to flesh out its state spac...

    1972

  5. [5]

    The SDE can be written for half-integer values of ν=p+ 1/2 withp= 0,1,2,

    [123] gives Kν(xz) = Γ(ν+ 1/2) Γ(1/2) 2z x νZ ∞ 0 dt cos(xt) (z2 +t 2)ν+1/2 , (D58) for Re(ν+ 1/2)>0,x >0, and argz < π/2; from which the covariance function (D51) can be verified by perform- ing (D3). The SDE can be written for half-integer values of ν=p+ 1/2 withp= 0,1,2, . . .. This is particularly straightforward since the PSD (D57) is in the desired ...

  6. [6]

    Observation of a rapidly pulsating radio source,

    A. Hewish, S. J. Bell, J. D. H Pilkington, P. F. Scott, and R. A. Collins, “Observation of a rapidly pulsating radio source,” Nature217, 709–713 (1968)

    1968

  7. [7]

    Observation of some further pulsated radio sources,

    J. D. H Pilkington, A. Hewish, S. J. Bell, and T. W. Cole, “Observation of some further pulsated radio sources,” Nature218, 126–129 (1968)

    1968

  8. [8]

    Detection of optical pulses from the Vela pulsar,

    P. T. Wallaceet al., “Detection of optical pulses from the Vela pulsar,” Nature266, 692–694 (1977)

    1977

  9. [9]

    Recent observa- tions of pulsars,

    J. H. Taylor and R. N. Manchester, “Recent observa- tions of pulsars,” Ann. Rev. Astron. Astrophys.15, 19– 44 (1977)

    1977

  10. [10]

    Binary and millisecond pulsars at the new millennium,

    D. R. Lorimer, “Binary and millisecond pulsars at the new millennium,” Living Rev. Rel.4, 5 (2001), arXiv:astro-ph/0104388

    work page arXiv 2001

  11. [11]

    Radio Pulsars: The Neutron Star Population & Fundamental Physics,

    Victoria M. Kaspi and Michael Kramer, “Radio Pulsars: The Neutron Star Population & Fundamental Physics,” in26th Solvay Conference on Physics: Astrophysics and Cosmology(2016) arXiv:1602.07738 [astro-ph.HE]

    work page arXiv 2016

  12. [12]

    LOFAR discovery of the fastest-spinning millisecond pulsar in the Galactic field

    C. G. Bassaet al., “LOFAR discovery of the fastest- spinning millisecond pulsar in the Galactic field,” Astro- phys. J. Lett.846, L20 (2017), arXiv:1709.01453 [astro- ph.HE]

    work page Pith review arXiv 2017

  13. [13]

    Einstein@Home discovers a radio-quiet gamma-ray millisecond pulsar,

    Colin J. Clarket al.(Fermi-LAT), “Einstein@Home discovers a radio-quiet gamma-ray millisecond pulsar,” Sc. Adv.4, eaao7228 (2018), arXiv:1803.06855 [astro- ph.HE]

    work page arXiv 2018

  14. [14]

    Discovery of a Gamma-ray Black Widow Pulsar by GPU-accelerated Einstein@Home,

    L. Niederet al., “Discovery of a Gamma-ray Black Widow Pulsar by GPU-accelerated Einstein@Home,” Astrophys. J. Lett.902, L46 (2020), arXiv:2009.01513 [astro-ph.HE]

    work page arXiv 2020

  15. [15]

    Einstein@Home Searches for Gamma-Ray Pulsars in the Inner Galaxy,

    C. J. Clarket al., “Einstein@Home Searches for Gamma-Ray Pulsars in the Inner Galaxy,” Astrophys. J.994, 149 (2025), arXiv:2509.21307 [astro-ph.HE]

    work page arXiv 2025

  16. [16]

    Astrophysics with compact objects: An Indian perspective, present status and future vision,

    Manjari Bagchiet al., “Astrophysics with compact objects: An Indian perspective, present status and future vision,” J. Astrophys. Astron.46, 62 (2025), arXiv:2505.18238 [astro-ph.HE]

    work page arXiv 2025

  17. [17]

    Pulsars and Gravity,

    R. N. Manchester, “Pulsars and Gravity,” Int. J. Mod. Phys. D24, 1530018 (2015), arXiv:1502.05474 [gr-qc]

    work page arXiv 2015

  18. [18]

    Krameret al., Strong-Field Gravity Tests with the Dou- ble Pulsar, Phys

    M. Krameret al., “Strong-Field Gravity Tests with the Double Pulsar,” Phys. Rev. X11, 041050 (2021), arXiv:2112.06795 [astro-ph.HE]

    work page arXiv 2021

  19. [19]

    Gravity experiments with radio pulsars,

    Paulo C. C. Freire and Norbert Wex, “Gravity experi- ments with radio pulsars,” Living Rev. Rel.27, 5 (2024), arXiv:2407.16540 [gr-qc]

    work page arXiv 2024

  20. [20]

    Pulsar tim- ing signal from ultralight scalar dark matter,

    Andrei Khmelnitsky and Valery Rubakov, “Pulsar tim- ing signal from ultralight scalar dark matter,” JCAP 02, 019 (2014), arXiv:1309.5888 [astro-ph.CO]

    work page arXiv 2014

  21. [21]

    Constraints on ul- tralight scalar dark matter from pulsar timing,

    N. K. Porayko and K. A. Postnov, “Constraints on ul- tralight scalar dark matter from pulsar timing,” Phys. Rev. D90, 062008 (2014), arXiv:1408.4670 [astro- ph.CO]

    work page arXiv 2014

  22. [22]

    Parkes Pulsar Timing Array constraints on ultralight scalar-field dark mat- ter,

    Nataliya K. Poraykoet al., “Parkes Pulsar Timing Array constraints on ultralight scalar-field dark mat- ter,” Phys. Rev. D98, 102002 (2018), arXiv:1810.03227 [astro-ph.CO]

    work page arXiv 2018

  23. [23]

    Second Data Release from the European Pulsar Timing Array: Challenging the Ultralight Dark Mat- ter Paradigm,

    Clemente Smarraet al.(European Pulsar Timing Ar- ray), “Second Data Release from the European Pulsar Timing Array: Challenging the Ultralight Dark Mat- ter Paradigm,” Phys. Rev. Lett.131, 171001 (2023), arXiv:2306.16228 [astro-ph.HE]

    work page arXiv 2023

  24. [24]

    Discovery of a pulsar in a binary system,

    R. A. Hulse and J. H. Taylor, “Discovery of a pulsar in a binary system,” Astrophys. J. Lett.195, L51–L53 (1975)

    1975

  25. [25]

    A new test of general relativity - Gravitational radiation and the binary pul- sar PSR 1913+16,

    J. H. Taylor and J. M. Weisberg, “A new test of general relativity - Gravitational radiation and the binary pul- sar PSR 1913+16,” Astrophys. J.253, 908–920 (1982)

    1913

  26. [26]

    The discovery of the binary pulsar,

    Russell A. Hulse, “The discovery of the binary pulsar,” Rev. Mod. Phys.66, 699–710 (1994)

    1994

  27. [27]

    1974: the discovery of the first bi- nary pulsar,

    Thibault Damour, “1974: the discovery of the first bi- nary pulsar,” Class. Quant. Grav.32, 124009 (2015), arXiv:1411.3930 [gr-qc]

    work page arXiv 1974

  28. [28]

    Upper limits on the isotropic gravitational radiation background from pulsar timing analysis,

    R. W. Hellings and G. W. Downs, “Upper limits on the isotropic gravitational radiation background from pulsar timing analysis,” Astrophys. J. Lett.265, L39– L42 (1983)

    1983

  29. [29]

    Harmonic space analysis of pulsar timing array redshift maps,

    Elinore Roebber and Gilbert Holder, “Harmonic space analysis of pulsar timing array redshift maps,” As- trophys. J.835, 21 (2017), arXiv:1609.06758 [astro- ph.CO]

    work page arXiv 2017

  30. [30]

    Allen, Phys

    Bruce Allen, “Variance of the Hellings-Downs correlation,” Phys. Rev. D107, 043018 (2023), arXiv:2205.05637 [gr-qc]

    work page arXiv 2023

  31. [31]

    Pul- sar and cosmic variances of pulsar timing-array correla- tion measurements of the stochastic gravitational wave background,

    Reginald Christian Bernardo and Kin-Wang Ng, “Pul- sar and cosmic variances of pulsar timing-array correla- tion measurements of the stochastic gravitational wave background,” JCAP11, 046 (2022), arXiv:2209.14834 [gr-qc]

    work page arXiv 2022

  32. [32]

    Opportunities for detecting ul- tralong gravitational waves,

    Mikhail V. Sazhin, “Opportunities for detecting ul- tralong gravitational waves,” Sov. Astron.22, 36–38 28 (1978)

    1978

  33. [33]

    Pulsar timing measurements and the search for gravitational waves,

    Steven L. Detweiler, “Pulsar timing measurements and the search for gravitational waves,” Astrophys. J.234, 1100–1104 (1979)

    1979

  34. [34]

    The Stochastic Gravity Wave Background in Inflationary Universe Models,

    Bruce Allen, “The Stochastic Gravity Wave Background in Inflationary Universe Models,” Phys. Rev. D37, 2078 (1988)

    2078

  35. [35]

    Sesana, A

    Alberto Sesana, Alberto Vecchio, and Carlo Nicola Co- lacino, “The stochastic gravitational-wave background from massive black hole binary systems: implications for observations with Pulsar Timing Arrays,” Mon. Not. Roy. Astron. Soc.390, 192 (2008), arXiv:0804.4476 [astro-ph]

    work page arXiv 2008

  36. [36]

    Bi, Y.-M

    Yan-Chen Bi, Yu-Mei Wu, Zu-Cheng Chen, and Qing- Guo Huang, “Implications for the supermassive black hole binaries from the NANOGrav 15-year data set,” Sci. China Phys. Mech. Astron.66, 120402 (2023), arXiv:2307.00722 [astro-ph.CO]

    work page arXiv 2023

  37. [37]

    Optimal Recon- struction of the Hellings and Downs Correlation,

    Bruce Allen and Joseph D. Romano, “Optimal Recon- struction of the Hellings and Downs Correlation,” Phys. Rev. Lett.134, 031401 (2025), arXiv:2407.10968 [gr-qc]

    work page arXiv 2025

  38. [38]

    Reading signatures of su- permassive binary black holes in pulsar timing ar- ray observations,

    Boris Goncharovet al., “Reading signatures of su- permassive binary black holes in pulsar timing ar- ray observations,” Nature Commun.16, 9692 (2025), arXiv:2409.03627 [astro-ph.HE]

    work page arXiv 2025

  39. [39]

    Ac- curate pulsar timing array residual variances and corre- lation of the stochastic gravitational wave background,

    Reginald Christian Bernardo and Kin-Wang Ng, “Ac- curate pulsar timing array residual variances and corre- lation of the stochastic gravitational wave background,” (2024), arXiv:2409.01218 [astro-ph.CO]

    work page arXiv 2024

  40. [40]

    Non-Gaussian statistics of nanohertz stochastic gravitational waves,

    Xiao Xue, Zhen Pan, and Liang Dai, “Non-Gaussian statistics of nanohertz stochastic gravitational waves,” Phys. Rev. D111, 043022 (2025), arXiv:2409.19516 [astro-ph.CO]

    work page arXiv 2025

  41. [41]

    Crisostomi, R

    Marco Crisostomi, Rutger van Haasteren, Patrick M. Meyers, and Michele Vallisneri, “Beyond diagonal ap- proximations: improved covariance modeling for pulsar timing array data analysis,” (2025), arXiv:2506.13866 [astro-ph.IM]

    work page arXiv 2025

  42. [42]

    Sim- ulating a Gaussian stochastic gravitational wave back- ground signal in pulsar timing arrays,

    Reginald Christian Bernardo and Kin-Wang Ng, “Sim- ulating a Gaussian stochastic gravitational wave back- ground signal in pulsar timing arrays,” (2025), arXiv:2507.15756 [astro-ph.CO]

    work page arXiv 2025

  43. [43]

    Pulsar timing array analysis in a Legendre polynomial basis

    Bruce Allen, Arian L. von Blanckenburg, and Ken D. Olum, “Pulsar timing array analysis in a Legendre poly- nomial basis,” (2025), arXiv:2510.05913 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  44. [44]

    Stochastic gravitational-wave background search using data from five pulsar timing arrays

    Wang-Wei Yu and Bruce Allen, “Stochastic gravitational-wave background search using data from five pulsar timing arrays,” (2025), arXiv:2512.08666 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  45. [45]

    On the contribution of a stochastic back- ground of gravitational radiation to the timing noise of pulsars,

    B. Mashhoon, “On the contribution of a stochastic back- ground of gravitational radiation to the timing noise of pulsars,” Mon. Not. Roy. Astron. Soc.199, 659–666 (1982)

    1982

  46. [46]

    Influence of a stochastic background of gravitational waves on the timing of pulsars,

    B. Mashhoon, “Influence of a stochastic background of gravitational waves on the timing of pulsars,” Mon. Not. Roy. Astron. Soc.217, 265–279 (1985)

    1985

  47. [47]

    On measuring the gravitational-wave background using Pulsar Timing Arrays,

    Rutger van Haasteren, Yuri Levin, Patrick McDonald, and Tingting Lu, “On measuring the gravitational-wave background using Pulsar Timing Arrays,” Mon. Not. Roy. Astron. Soc.395, 1005 (2009), arXiv:0809.0791 [astro-ph]

    work page arXiv 2009

  48. [48]

    Coles, G

    W. Coles, G. Hobbs, D. J. Champion, R. N. Manchester, and J. P. W. Verbiest, “Pulsar timing analysis in the presence of correlated noise,” Mon. Not. Roy. Astron. Soc.418, 561 (2011), arXiv:1107.5366 [astro-ph.IM]

    work page arXiv 2011

  49. [49]

    Understanding and analysing time-correlated stochastic signals in pul- sar timing,

    Rutger van Haasteren and Yuri Levin, “Understanding and analysing time-correlated stochastic signals in pul- sar timing,” Mon. Not. Roy. Astron. Soc.428, 1147 (2013), arXiv:1202.5932 [astro-ph.IM]

    work page arXiv 2013

  50. [50]

    van Haasteren and M

    Rutger van Haasteren and Michele Vallisneri, “New advances in the Gaussian-process approach to pulsar- timing data analysis,” Phys. Rev. D90, 104012 (2014), arXiv:1407.1838 [gr-qc]

    work page arXiv 2014

  51. [51]

    Pulsar timing noise from superfluid turbulence,

    Andrew Melatos and Bennett Link, “Pulsar timing noise from superfluid turbulence,” Mon. Not. Roy. Astron. Soc.437, 21–31 (2014), arXiv:1310.3108 [astro-ph.HE]

    work page arXiv 2014

  52. [52]

    Models of Pulsar Glitches

    Brynmor Haskell and Andrew Melatos, “Models of Pul- sar Glitches,” Int. J. Mod. Phys. D24, 1530008 (2015), arXiv:1502.07062 [astro-ph.SR]

    work page Pith review arXiv 2015

  53. [53]

    arXiv e-prints , keywords =

    E. S. Phinney, “A Practical theorem on gravitational wave backgrounds,” (2001), arXiv:astro-ph/0108028

    work page internal anchor Pith review arXiv 2001

  54. [54]

    Is there a spectral turnover in the spin noise of millisecond pulsars?

    Boris Goncharov, Xing-Jiang Zhu, and Eric Thrane, “Is there a spectral turnover in the spin noise of millisecond pulsars?” Mon. Not. Roy. Astron. Soc.497, 3264–3272 (2020), arXiv:1910.05961 [astro-ph.HE]

    work page arXiv 2020

  55. [55]

    Kolmogorov analysis of pulsar TOA,

    N. Galikyan, A. A. Kocharyan, and V. G. Gurzadyan, “Kolmogorov analysis of pulsar TOA,” Astron. Astro- phys.707, L8 (2026), arXiv:2602.17188 [astro-ph.IM]

    work page arXiv 2026

  56. [56]

    Stochastic Gravitational-Wave Backgrounds: Current Detection Efforts and Future Prospects,

    Arianna I. Renzini, Boris Goncharov, Alexan- der C. Jenkins, and Pat M. Meyers, “Stochastic Gravitational-Wave Backgrounds: Current Detection Efforts and Future Prospects,” Galaxies10, 34 (2022), arXiv:2202.00178 [gr-qc]

    work page arXiv 2022

  57. [57]

    Charting the nanohertz gravitational wave sky with pulsar timing arrays,

    Reginald Christian Bernardo and Kin-Wang Ng, “Charting the nanohertz gravitational wave sky with pulsar timing arrays,” Int. J. Mod. Phys. D34, 2540013 (2025), arXiv:2409.07955 [astro-ph.CO]

    work page arXiv 2025

  58. [58]

    Kalman filtering and smoothing solutions to temporal gaussian process regression models,

    Jouni Hartikainen and Simo S¨ arkk¨ a, “Kalman filtering and smoothing solutions to temporal gaussian process regression models,” in2010 IEEE International Work- shop on Machine Learning for Signal Processing(2010) pp. 379–384

    2010

  59. [59]

    Spa- tiotemporal Learning via Infinite-Dimensional Bayesian Filtering and Smoothing: A Look at Gaussian Process Regression Through Kalman Filtering,

    Simo Sarkka, Arno Solin, and Jouni Hartikainen, “Spa- tiotemporal Learning via Infinite-Dimensional Bayesian Filtering and Smoothing: A Look at Gaussian Process Regression Through Kalman Filtering,” IEEE Signal Processing Magazine30, 51–61 (2013)

    2013

  60. [60]

    Stochastic problems in physics and astronomy,

    Subrahmanyan Chandrasekhar, “Stochastic problems in physics and astronomy,” Rev. Mod. Phys.15, 1–89 (1943)

    1943

  61. [61]

    On the Theory of the Brownian Motion,

    G. E. Uhlenbeck and L. S. Ornstein, “On the Theory of the Brownian Motion,” Phys. Rev.36, 823–841 (1930)

    1930

  62. [62]

    Mathematical Analysis of Random Noise,

    S. O. Rice, “Mathematical Analysis of Random Noise,” Bell Labs Tech. J.23, 282–332 (1944)

    1944

  63. [63]

    On the Theory of the Brownian Motion II,

    Ming Chen Wang and G. E. Uhlenbeck, “On the Theory of the Brownian Motion II,” Rev. Mod. Phys.17, 323– 342 (1945)

    1945

  64. [64]

    Random Walk and the Theory of Brownian Motion,

    M. Kac, “Random Walk and the Theory of Brownian Motion,” Am. Math. Monthly54, 369–391 (1947)

    1947

  65. [65]

    ¨Uber die von der molekularkinetischen Theorie der W¨ arme geforderte Bewegung von in ruhen- den Fl¨ ussigkeiten suspendierten Teilchen,

    A. Einstein, “ ¨Uber die von der molekularkinetischen Theorie der W¨ arme geforderte Bewegung von in ruhen- den Fl¨ ussigkeiten suspendierten Teilchen,” Annalen der Physik322, 549–560 (1905)

    1905

  66. [66]

    Sur la th´ e´ orie du mouvement brown- ien,

    Paul Langevin, “Sur la th´ e´ orie du mouvement brown- ien,” C. R. Acad. Sci. Paris.146, 530–533 (1908)

    1908

  67. [67]

    Paul Langevin’s 1908 paper “On the Theory of Brownian Motion

    Don S. Lemons and Anthony Gythiel, “Paul Langevin’s 1908 paper “On the Theory of Brownian Motion” [“Sur 29 la th´ eorie du mouvement brownien,” C. R. Acad. Sci. (Paris) 146, 530–533 (1908)],” Am. J. Phys.65, 1079– 1081 (1997)

    1908

  68. [68]

    Krapivsky, Sidney Redner, and Eli Ben-Naim, A Kinetic View of Statistical Physics(2010)

    Pavel L. Krapivsky, Sidney Redner, and Eli Ben-Naim, A Kinetic View of Statistical Physics(2010)

    2010

  69. [69]

    A new approach to linear filtering and prediction problems,

    R E Kalman, “A new approach to linear filtering and prediction problems,” Trans. ASME, D82, 35–44 (1960)

    1960

  70. [70]

    Pulsar glitch detection with a hid- den Markov model,

    A. Melatos, L. M. Dunn, S. Suvorova, W. Moran, and R. J. Evans, “Pulsar glitch detection with a hid- den Markov model,” Astrophys. J.896, 78 (2020), arXiv:2005.09388 [astro-ph.HE]

    work page arXiv 2020

  71. [71]

    A new small glitch in Vela discovered with a hidden Markov model,

    L. Dunn, A. Melatos, C. M. Espinoza, D. Antonopoulou, and R. Dodson, “A new small glitch in Vela discovered with a hidden Markov model,” Mon. Not. Roy. Astron. Soc.522, 5469–5478 (2023), arXiv:2304.13382 [astro- ph.HE]

    work page arXiv 2023

  72. [72]

    Parameter estimation of a two-component neutron star model with spin wandering,

    Patrick M. Meyers, Andrew Melatos, and Nicholas J. O’Neill, “Parameter estimation of a two-component neutron star model with spin wandering,” Mon. Not. Roy. Astron. Soc.502, 3113–3127 (2021), arXiv:2101.12421 [astro-ph.HE]

    work page arXiv 2021

  73. [73]

    Rapid parameter es- timation of a two-component neutron star model with spin wandering using a Kalman filter,

    Patrick M. Meyers, Nicholas J. O’Neill, Andrew Melatos, and Robin J. Evans, “Rapid parameter es- timation of a two-component neutron star model with spin wandering using a Kalman filter,” Mon. Not. Roy. Astron. Soc.506, 3349–3363 (2021), arXiv:2107.03047 [astro-ph.HE]

    work page arXiv 2021

  74. [74]

    Analysing radio pulsar timing noise with a Kalman filter: a demonstration involving PSR J1359−6038,

    Nicholas J. O’Neill, Patrick M. Meyers, and An- drew Melatos, “Analysing radio pulsar timing noise with a Kalman filter: a demonstration involving PSR J1359−6038,” Mon. Not. Roy. Astron. Soc.530, 4648– 4664 (2024), arXiv:2403.16467 [astro-ph.HE]

    work page arXiv 2024

  75. [75]

    Measur- ing the crust-superfluid coupling time-scale for 105 UTMOST pulsars with a Kalman filter,

    Wenhao Dong, Andrew Melatos, Nicholas J. O’Neill, Patrick M. Meyers, and Daniel K. Boek, “Measur- ing the crust-superfluid coupling time-scale for 105 UTMOST pulsars with a Kalman filter,” (2025), 10.1093/mnras/staf2053, arXiv:2511.15134 [astro- ph.HE]

    work page doi:10.1093/mnras/staf2053 2025

  76. [76]

    Kalman tracking and parame- ter estimation of continuous gravitational waves with a pulsar timing array,

    Tom Kimpsonet al., “Kalman tracking and parame- ter estimation of continuous gravitational waves with a pulsar timing array,” Mon. Not. Roy. Astron. Soc.534, 1844–1867 (2024), arXiv:2409.14613 [astro-ph.HE]

    work page arXiv 2024

  77. [77]

    State-space analysis of a contin- uous gravitational wave source with a pulsar timing ar- ray: inclusion of the pulsar terms,

    Tom Kimpsonet al., “State-space analysis of a contin- uous gravitational wave source with a pulsar timing ar- ray: inclusion of the pulsar terms,” Mon. Not. Roy. As- tron. Soc.535, 132–154 (2024), arXiv:2410.10087 [astro- ph.HE]

    work page arXiv 2024

  78. [78]

    State-space algorithm for de- tecting the nanohertz gravitational wave background,

    Tom Kimpsonet al., “State-space algorithm for de- tecting the nanohertz gravitational wave background,” Mon. Not. Roy. Astron. Soc.537, 1282–1304 (2025), arXiv:2501.06990 [astro-ph.IM]

    work page arXiv 2025

  79. [79]

    Argus: JAX state-space filtering for gravitational wave detection with a pulsar timing array,

    Tom Kimpson, Nicholas J. O’Neill, Patrick M. Mey- ers, and Andrew Melatos, “Argus: JAX state-space filtering for gravitational wave detection with a pulsar timing array,” arXiv e-prints , arXiv:2510.11077 (2025), arXiv:2510.11077 [astro-ph.IM]

    work page arXiv 2025

  80. [80]

    Flexible and Scalable Methods for Quantifying Stochastic Variability in the Era of Massive Time- domain Astronomical Data Sets,

    Brandon C. Kelly, Andrew C. Becker, Malgosia Sobolewska, Aneta Siemiginowska, and Phil Utt- ley, “Flexible and Scalable Methods for Quantifying Stochastic Variability in the Era of Massive Time- domain Astronomical Data Sets,” Astrophys. J.788, 33 (2014), arXiv:1402.5978 [astro-ph.IM]

    work page arXiv 2014

Showing first 80 references.