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Occupation time statistics for non-Markovian random walks

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Pith's one-line read A single memory kernel determines occupation-time statistics for arbitrary non-Markovian random walks.

desk verdict A useful memory-kernel toolkit for occupation times in the continuum CTRW limit, with real but localized holes around the right-edge asymptotics and the half-line boundary expansion. read the letter →

arxiv 2412.05247 v1 pith:ZH6ZHG5D submitted 2024-12-06 cond-mat.stat-mech

Vicenç Méndez , Rosa Flaquer-Galmés , Arnab Pal This is my paper · ORCID
classification cond-mat.stat-mech MSC 60K4082C4160J60 PACS 05.40.Fb
keywords occupationtimecontinuous-timerandomwalkFeynman-Kacequationnon-Markovianwalksstochasticresettingarcsinelawmemorykernelanomalousdiffusion
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

This paper aims to give a unified description of occupation time—the total time a random walker spends inside a given region up to time $t$—for continuous-time random walks with arbitrary waiting-time distributions between jumps. It derives a backward Feynman-Kac equation for the characteristic function of any time-integrated functional and solves it for two observables: the occupation time in a symmetric interval and the half-occupation time on the positive line. The resulting formulas, Eqs. (23) and (57), are written entirely in terms of a memory kernel $K(s)=s\varphi(s)/(1-\varphi(s))$, so Markovian, power-law, and Mittag-Leffler waiting times all fall out of one expression. From these formulas the paper obtains moments, probability-density tails, limiting distributions such as the arcsine and Lamperti laws, ergodicity-breaking parameters, and the modifications produced by stochastic resetting.

What carries the argument

The load-bearing object is the memory kernel $K(s)=s\varphi(s)/(1-\varphi(s))$, the Laplace-domain encoding of the waiting-time distribution. The derivation starts from the Montroll-Weiss propagator, closes the jump distribution at second order in Fourier space ($\Psi(k)\simeq 1-\sigma^2 k^2/2$), and converts the exact renewal equation into the backward Feynman-Kac equation (19): $sQ-1=(\sigma^2/2)K(s+pU(x_0))\,\partial^2 Q/\partial x_0^2-pU(x_0)Q$. Solving that ordinary differential equation in the spatial variable with the indicator $U$ of the interval or the half-line produces the characteristic functions (23) and (57), and the same kernel appears in the resetting renewal relation (68).

What would settle it

Simulate a CTRW with a jump-length distribution of infinite second moment, such as Cauchy jumps, and compare the interval-occupation-time PDF with Eq. (23); the predictions should fail near the interval edges because the $\Psi(k)\simeq 1-\sigma^2 k^2/2$ closure no longer holds. A sharper check is to take a finite-variance jump distribution with strongly non-Gaussian shape and verify that Eq. (23) still matches the simulation once $a\gg\sigma$, which would confirm the second-moment closure is the operative condition.

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Extended reading notes

Core claim

The paper's central discovery is that for a continuous-time random walk whose jumps have finite second moment $\sigma^2$, the generating function $Q(p,s)$ of the occupation time is determined by the memory kernel alone: for the interval $[-a,a]$, $Q(p,s)$ solves the boundary-value problem built on Eq. (19) and yields Eq. (23), and for the half-line it yields Eq. (57). The paper then claims that all long-time statistical properties follow from these closed forms. In particular, the half-occupation time has mean $t/2$ for every isotropic walk, its limiting distribution is the arcsine law (the classic U-shaped distribution for the fraction of time spent on one side) whenever waiting times have finite first moment, and the Lamperti distribution when the power-law exponent satisfies $0<\alpha<1$. Interval-occupation moments show distinct regimes in $\alpha$, and under Poissonian resetting the same kernels predict that the occupation-time fraction becomes Gaussian at intermediate times and collapses to a delta function at its mean at long times, restoring ergodicity at the level of the occupation time even though the reset-free walk is non-ergodic.

Load-bearing premise

The formulas assume jumps have finite second moment and that the interval width is large compared with a typical jump, so the expansion of the jump distribution to second order in Fourier space is legitimate; if that expansion fails, the characteristic functions no longer describe the exact continuous-time random walk.

Editorial extensions

If this is right

  • For any waiting-time distribution with finite first moment, the half-occupation-time PDF approaches the arcsine law; only when the first moment diverges ($0<\alpha<1$) does the Lamperti distribution replace it.
  • The mean interval-occupation time in the $1<\alpha<2$ power-law regime matches normal diffusion, but the second moment separates off until $\alpha>3/2$, so the universality class depends on which moment one measures.
  • Reset-free occupation times are non-ergodic for every waiting-time distribution considered, with explicitly computed ergodicity-breaking parameters.
  • With Poissonian resetting, occupation-time fluctuations become Gaussian with variance linear in time, and the long-time PDF is a delta function centered on the mean, making the time-averaged occupation ergodic.
  • Because all results are expressed through $K(s)$, inserting any other waiting-time PDF into Eqs. (23) and (57) immediately yields that model's occupation-time statistics.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same backward equation should apply to other additive functionals such as local time, area under the path, or first-passage functionals by changing $U$, suggesting the paper's machinery is a template for a wider class of non-Markovian functional statistics.
  • The paper notes that near $\alpha=1$ its moment results diverge and leaves that case aside; a separate boundary-layer analysis could reveal logarithmic corrections in the occupation-time moments.
  • The authors conjecture that any time-integrated functional under Poissonian resetting converges to a delta at its mean; testing this with nonlinear or unbounded functionals would show whether the mechanism is the renewal structure or a special property of bounded occupation indicators.
  • Since the diffusion approximation requires $a\gg\sigma$, the formulas are inherently long-wavelength results; lattice or strongly intermittent walks may need finite-$\sigma$ corrections.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper develops a backward Feynman-Kac formalism for occupation-time statistics of continuous-time random walks (CTRWs) with arbitrary waiting-time distributions, using the generalized master equation (GME) with a memory kernel K(s). The authors derive characteristic functions for the occupation time in a finite interval (Eq. 23) and for the half-line occupation time (Eq. 57), then compute moments, limiting PDFs, and ergodicity-breaking parameters for normal, power-law, and Mittag-Leffler waiting times. They also extend the results to Poissonian stochastic resetting, obtaining long-time Gaussian and delta-peak limits for the occupation-time PDFs. All analytical predictions are compared with numerical simulations for several parameter regimes.

Significance. If correct, the paper provides a unified generating-function framework for occupation-time statistics of non-Markovian random walks, connecting the memory kernel to the Feynman-Kac equation and recovering known results (arcsine law, Lamperti distribution) as special cases. The derivation is self-contained and the paper ships explicit, machine-checkable analytic formulas for moments in Laplace space, together with extensive numerical verification for multiple waiting-time distributions. The novel resetting results and the universality claims for finite-mean waiting times are of interest to the statistical-physics community. The main limitation is that the derivation is performed in a small-jump (diffusion) approximation, which is not clearly stated as a validity condition.

major comments (3)
  1. [Section II, Appendix A, Eqs. (6) and (A4)] The central derivation is not valid for an arbitrary CTRW, but only in the continuum limit where the jump-length distribution is expanded to second order, Ψ(k) ≈ 1 − σ²k²/2, and Q(p,s|x0±σ) is expanded to O(σ²). This requires σ to be much smaller than the interval half-width a and, for the half-occupation problem, requires the starting point x0 = 0 not to lie exactly on the discontinuity of U(x0). The paper does not state this validity condition, and its own simulations use σ/a ≈ 0.18 (a = 0.055, σ = 0.01), which is not asymptotically small. Please state explicitly that Eqs. (19), (23), and (57) hold for σ/a → 0 (and for x0 not at a boundary), and discuss the expected error at finite σ/a, ideally with a numerical test for larger σ.
  2. [Section IV.B, Eqs. (48) and (52)] The claim that Eq. (52) reduces to Eq. (48) when Ta ∼ t is incorrect. Setting Ta = t − ε with ε ≪ t in Eq. (52) gives Q ∼ Aα/(ε^{1+α/2} t^α), whereas Eq. (48) gives Q ∼ Aα/(ε^{1−α/2} t^α). The discrepancy is a factor (t−Ta)^{−α/2} (or equivalently the exponents of ε differ by α). Thus the 'central bulk' formula does not match the right-edge formula in the overlapping regime, and the statement 'if Ta ∼ t ... Eq. (52) reduces to Eq. (48)' is false. This must be corrected: either Eq. (52) is wrong, or the reduction claim is wrong, and the numerical agreement shown in Figure 5 should be re-examined in the right-edge region.
  3. [Section IV.B, Eq. (43) and Appendix D] The left-edge PDF (43) is asserted to hold for Ta ≪ (a/√Kα)^{2/α}, which is a very narrow region near y = 0. However, the derivation of Eq. (43) uses the intermediate formula Q(p,s) ≃ 1/p + 2 e^{−a p^{α/2}/√Kα}/(s^{1−α/2} p^{α/2}) and then Laplace inverts term by term. The validity of this double-Laplace inversion approximation, and its consistency with the normalization of Q(Ta,t), are not discussed. Please provide a more precise statement of the error terms or a numerical check of the normalisation of Eq. (43).
minor comments (4)
  1. [Section VI.A, Eq. (74)] Equation (74) contains a typographical error: the coefficient multiplying ⟨Z(s+r)⟩² should be 2r(s+r)/s, not 2r/[s(s+r)]. The subsequent long-time result in Eq. (75) and the variance formula (76) are consistent with the corrected coefficient, so this appears to be a simple misprint, but it should be fixed.
  2. [Section II, Eq. (6)] The jump-length distribution is expanded to second order in Fourier space, but the notation σ is introduced as a 'characteristic dispersal distance' without discussing the case of asymmetric jump distributions. For an asymmetric distribution with nonzero first moment, the expansion should include a drift term; please clarify that the analysis assumes symmetric jumps (or state the generalization).
  3. [Section V, Eq. (59)] The Lamperti distribution in Eq. (59) is written with a denominator y^α + (1−y)^α + 2y^{α/2}(1−y)^{α/2} cos(απ/2), while the standard Lamperti form often appears with a factor sin(απ/2)/π times y^{α/2−1}(1−y)^{α/2−1} divided by the same denominator. Please verify the normalization and the convention for α; the current expression appears to be missing a factor of 2 in the denominator when compared to Ref. [14].
  4. [Appendix D, Eq. (D3)] The series expansion in Eq. (D3) has a summand (−a T_a^{−α/2} a√Kα)^n, which appears to contain a repeated factor a; please check whether this should be (−a T_a^{−α/2}/√Kα)^n or a different expression.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Feynman-Kac equations are derived from the CTRW renewal equation, and all subsequent results follow by algebra; the only self-citation is non-load-bearing.

full rationale

The derivation chain is self-contained. Equation (19) is obtained in Appendix A from the CTRW renewal equation (A1) by expanding Q(p,s|x0 ± σ) to second order (A4); this is a small-jump continuum-limit approximation, not an input disguised as a prediction. The characteristic functions (23) and (57) are then solved from (19) with stated boundary conditions, and all moments and PDFs follow by differentiation and Laplace inversion, with no fitted parameters. Numerical simulations use the same model parameters (sigma, a, t0, tau*) and serve as genuine checks rather than fits. The only self-citation with author overlap, Ref. [28], is cited as an earlier derivation of Eqs. (32) and (33) and for ergodicity-breaking results; the present paper rederives those quantities from its own Eq. (19), so the citation is not load-bearing. No uniqueness theorem or ansatz is imported from the authors' prior work; the arcsine and Lamperti limiting distributions are standard external results rederived in Appendix H. The notable limitation, namely that the sigma^2 expansion in Appendix A requires jump lengths much smaller than other length scales and its validity at x0 = 0 for the half-line is not quantified, is an approximation-validity concern rather than a circular one. The algebraic misprint in Eq. (74) is also not a circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new entities and fits no free parameters to data. The central formulas rest on the CTRW renewal assumption, the diffusion-limit expansion of the jump distribution, the Poissonian resetting renewal relation, and standard boundary conditions for the second-order Fokker-Planck-type equations.

assumptions (4)
  • domain assumption CTRW independence structure: waiting times and jump lengths are i.i.d. and independent of each other for each step.
    Section II introduces the CTRW with jump PDF Ψ(x) and waiting time PDF φ(t); all subsequent equations rely on this renewal structure.
  • domain assumption Diffusion limit: the jump length distribution has finite second moment and is expanded as Ψ(k) ≈ 1 − σ²k²/2, and the backward equation is expanded to second order in σ.
    Eq. (6) in Section II and Appendix A Eq. (A4) use this expansion to obtain the GME and the FK equation; it fails for Lévy-type jumps with infinite second moment and for interval widths comparable to σ.
  • domain assumption Poissonian resetting renewal: reset events occur at rate r independently of the walk and restart the process from x0.
    Section VI, Eq. (68) uses the standard renewal relation Q_r(p,s) = Q(p,s+r)/(1−rQ(p,s+r)).
  • standard math Boundary conditions for the FK equations: Q → 1/s or 1/(s+p) at ±∞ and continuity of Q and its derivative at interval boundaries.
    Appendices B and G impose these conditions to solve the second-order ODE for the characteristic functions.

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Pith. "Pith review of Occupation time statistics for non-Markovian random walks." pith.science (2026). https://pith.science/paper/ZH6ZHG5D

@misc{pith2026241205247,
  author       = {Pith},
  title        = {Pith review of: Occupation time statistics for non-Markovian random walks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZH6ZHG5D}},
  note         = {Machine review of arXiv:2412.05247}
}
read the original abstract

We study the occupation time statistics for non-Markovian random walkers based on the formalism of the generalized master equation for the Continuous-Time Random Walk. We also explore the case when the random walker additionally undergoes a stochastic resetting dynamics. We derive and solve the backward Feynman-Kac equation to find the characteristic function for the occupation time in an interval and for the half occupation time in the semi-infinite domain. We analyze the behaviour of the PDFs, the moments, the limiting distributions and the ergodic properties for both occupation times when the underlying random walk is normal or anomalous. For the half occupation time, we revisit the famous arcsine law and examine its validity pertaining to various regimes of the rest period of the walker. Our results have been verified with numerical simulations exhibiting an excellent agreement.

Figures

Figures reproduced from arXiv: 2412.05247 by the authors.

Figure 1
Figure 1. FIG. 1. Mean occupation time in an interval. The points are [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The mean squared displacement (MSD). The points [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Second moment of the mean occupation time in [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (8 more)
Figure 6
Figure 6. Figure 6: FIG. 6. Scaled PDF of occupation time [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Scaled PDF of occupation time [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Universality of the [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Mean half occupation time. All the data points are [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. The second moment of the half occupation time. All [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. First moment (top panel), second moment (middle [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Scaled PDF [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Scaled PDF [PITH_FULL_IMAGE:figures/full_fig_p015_12.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Statistical properties of stochastic functionals under general resetting

    cond-mat.stat-mech 2025-07 conditional novelty 6.0 of 10

    Under general resetting, stochastic functionals of random walks become deterministic at long times if reset times have a finite first moment, and obey new U-shaped, W-shaped, or inverted-U limiting distributions for p...

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Reviewed August 11, 2026 · model on record in the stance chip above.