Statistical properties of stochastic functionals under general resetting

R. Flaquer-Galm\'es; V. M\'endez

REVIEW 2 major objections 3 minor 39 references

Statistical properties of stochastic functionals under general resetting

T0 review · 2 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read For any random walker whose position is reset at random times, a single renewal equation gives the full statistics of every time-integrated observable, and the resetting tail exponent alone decides whether those statistics are…

desk verdict Genuinely new half-occupation-time results for power-law resetting, with convincing simulations, but the Laplace-inversion step behind the phase diagram deserves closer scrutiny. read the letter →

arxiv 2507.05955 v1 pith:QVDZ24HE submitted 2025-07-08 cond-mat.stat-mech

V. Méndez , R. Flaquer-Galmés This is my paper · ORCID

classification cond-mat.stat-mech MSC 60K4082C31 PACS 05.40.-a

keywords stochasticresettingfunctionalshalf-occupationtimeergodicitybreakingpower-lawBrownianmotionsubdiffusionrenewalequation

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 derives one renewal identity that gives the generating function of any stochastic functional of a random walk whose position is reset to the origin at random times drawn from an arbitrary distribution. It then shows that if the resetting distribution has finite moments, every such functional becomes ergodic in the long-time limit: its probability density collapses to a delta function at its mean. If the resetting distribution instead decays as a power law $t^{-1-\alpha}$ with $0<\alpha<1$, the paper obtains the limiting distribution of the half-occupation time $T_+$ (the time spent on the positive side) and shows it takes one of three shapes — ∪, W, or ∩ — depending on $\alpha$ and, for subdiffusive walkers, on the walker exponent $\gamma$. The practical claim is that only the tail exponent of the resetting distribution matters, not its detailed form; Monte Carlo checks with power-law, Mittag-Leffler, and Lévy resetting densities support this. If the paper is right, the ergodic or non-ergodic character of any such observable can be read off from the resetting tail alone.

What carries the argument

The central object is the renewal equation (11) for the double Laplace transform of the functional's characteristic function under resetting, $\tilde Q_r(p,s)=\mathcal L[\varphi^*(t)Q_0(p,t)]/(1-\mathcal L[\varphi(t)Q_0(p,t)])$, where $\varphi$ is the resetting-time density, $\varphi^*$ is the survival probability without reset, and $Q_0$ is the characteristic function without reset. It reduces the problem to two input ingredients: the resetting density and the reset-free functional statistics. The long-time analysis is carried by two tools: the small-$s$ Tauberian expansion of the resetting Laplace transform ($1-b_\alpha s^{\alpha}$ for $0<\alpha<1$, $1-\langle t\rangle_R s$ for $1<\alpha<2$) and, for the limiting density, the bulk inversion formula $P(z)_r = -\frac{1}{\pi z}\lim_{\epsilon\to 0}\operatorname{Im} g_\alpha(-1/z+i\epsilon)$, with $g_\alpha$ the scaling function built from the ratio of integrals over the reset-free density $f$. The shape classification follows from the boundary exponents $P(z)_r\sim z^{\alpha+\gamma/2-1}$, where $\gamma=1$ for Brownian motion.

What would settle it

Measure the half-occupation-time density for Brownian motion with power-law resetting at $\alpha=0.4$ and times long enough that $P(z)$ has converged, then compare the measured $P(z)$ near $z=10^{-6}$ with the predicted $P(z)\sim z^{-0.1}$; a clear disagreement in that boundary exponent would falsify the uniformity assumption behind Eq. (53). Alternatively, for a subdiffusive walker with $\gamma=0.7$, test whether the W-to-∩ transition occurs at $\alpha^*=0.65$; finding it elsewhere would falsify the phase-diagram claim.

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Extended reading notes

Core claim

The central result is that the double Laplace transform of the characteristic function of any functional under general resetting satisfies $\tilde Q_r(p,s)=\mathcal L[\varphi^*(t)Q_0(p,t)]/(1-\mathcal L[\varphi(t)Q_0(p,t)])$, where $\varphi$ is the resetting-time density, $\varphi^*$ is the no-reset survival probability, and $Q_0$ is the characteristic function without resetting. From this renewal equation, finite-moment resetting implies the long-time distribution is $\delta(Z-\langle Z\rangle_r)$ and the ergodicity-breaking parameter vanishes. For power-law resetting with $0<\alpha<1$, the half-occupation-time density is given by Eq. (53) in terms of integrals $C_\beta(z)$ over the reset-free limiting density $f(u)=P(T_+/t)$; for an isotropic random walk this formula depends on the resetting distribution only through $\alpha$. Specializing $f$ to the Lévy arcsine law (Brownian motion) yields boundary exponents $P(z)_r\sim z^{\alpha-1/2}$, so the limiting density is ∪ for $\alpha<\alpha_c\simeq0.269$, W for $\alpha_c<\alpha<1/2$, and ∩ for $\alpha>1/2$; specializing $f$ to the Lamperti distribution (subdiffusion with exponent $\gamma$) gives boundary exponents $z^{\alpha+\gamma/2-1}$ and a W-to-∩ transition at $\alpha^*=1-\gamma/2$.

Load-bearing premise

The U/W/∩ classification of the limiting density assumes that the bulk long-time limit can be taken uniformly for all $z\in[0,1]$, including the boundaries $z\to0$ and $z\to1$; the paper states this uniformity rather than proving it, and separately only conjectures the finite-first-moment relaxation for other functionals.

Editorial extensions

If this is right

If the resetting-time distribution has finite moments, the long-time distribution of any time-integrated observable is a delta function at its mean, so trajectory-to-trajectory fluctuations of time averages vanish.
For power-law resetting with $0<\alpha<1$, the limiting half-occupation-time distribution is universal in the tail exponent: the same Eq. (53) fits simulations for power-law, Mittag-Leffler, and Lévy resetting densities.
The ergodicity-breaking parameter of $T_+$ is $(1-\alpha)(2-\gamma)/2$ in the non-ergodic phase and zero for $\alpha>1$, so $\alpha=1$ is a sharp boundary between random and deterministic occupation times.
For subdiffusive walkers the W-to-∩ transition lies at $\alpha^*=1-\gamma/2$, tying the shape of the occupation-time distribution to the mean of the first-passage time to the origin.
The same machinery applies to any functional whose no-reset limiting density has the scaling form $t^{-1}f(Z/t)$, including observables such as the time-averaged position or the area under the trajectory.

Reading between the lines

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

A natural extension the paper leaves implicit is that the tail-only universality in Eq. (53) should hold for any renewal process with power-law inter-event times, not just positional resetting; a numerical sweep of other functionals of the same walk under Mittag-Leffler resets would test this without new analytic work.
The sharp ergodic transition at $\alpha=1$ could be used in reverse: estimating whether trajectory-to-trajectory variance of $T_+/t$ vanishes gives a direct empirical test that the resetting mechanism has a finite mean reset time.
The boundary exponents are the most exposed part of the derivation because the paper's uniformity assumption is strained as $z\to0$ and $z\to1$; a dedicated simulation at very small $z$ would either confirm the predicted $z^{\alpha-1/2}$ scaling or locate exactly where the inversion limit fails.
If the paper's conjecture on relaxing the finite-moment condition to a finite first moment for functionals with $\mu<1$ is correct, the ergodic phase becomes much larger than the theorem proves, and the same delta-function limit should be observed for resetting densities with infinite second moments but finite means.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here's the short version. The paper's genuinely new content is the half-occupation-time analysis for power-law resetting: the ergodicity-breaking parameter, the limiting distribution Eq. (53), and the U/W/inverted-U shape transitions. The renewal equation (11) itself is a clean generalization of the Poissonian result to arbitrary resetting PDFs, and the ergodic delta theorem for finite-moment resetting is proved under a stated condition. The Monte Carlo evidence across power-law, Mittag-Leffler, and Levy resetting is convincing, and the claim that the long-time distribution depends only on the tail exponent alpha is well supported.

What the paper does well: the derivations are transparent and the moment table, apart from one missing row, reproduces the known special cases. The EB parameters (40) and (42) are exact and match simulations. The phase diagram for the subdiffusive case, with the transition at alpha* = 1 - gamma/2, ties nicely to the known first-passage transition in Ref. [16]. Credit is given to the earlier Poissonian and interval-occupation work.

Soft spots. The stress-test concern about the uniformity of the Laplace inversion near z=0,1 is real but less damaging than it looks: the boundary exponents (57) and (59) are derived from the explicit formula (53) using the small-z behavior of C_beta(z), not by interchanging limits in the inversion itself. Still, the derivation of (53) from (49) via the Godreche-Luck formula is a standard but unproved branch-cut calculation for non-integer alpha; a rigorous referee would want that step justified. The conclusion's claim that the delta result should extend to functionals with mu<1 under only a finite first moment is explicitly labeled as unproved expectation - fine, but it should be flagged as a conjecture, not a result. Table I's right panel is missing the row for mu

Who this is for: statistical physicists working on stochastic resetting and functionals. It deserves serious peer review; the weaknesses are addressable and the central claims are well supported. I'd send it to a referee, asking them to scrutinize the Laplace inversion and the boundary asymptotics.

Referee Report

2 major / 3 minor

Summary. The paper derives a renewal equation for the characteristic function of a stochastic functional of a random walk under general resetting, and uses it to obtain long-time results: a delta-function limit when resetting has finite moments, moment scaling for power-law resetting, and for the half-occupation time a universal limiting PDF (Eq. 53) with a phase diagram of shapes (U, W, inverted U). The theoretical results are compared with Monte Carlo simulations for Brownian and subdiffusive walkers under power-law, Mittag-Leffler, and Lévy resetting.

Significance. If the main results hold, the paper establishes a remarkably universal statement: for resetting with finite moments, all positive functionals become ergodic in the long-time limit; and for heavy-tailed resetting with exponent α<1, the limiting distribution of the half-occupation time depends only on the tail exponent α and the walker exponent γ, not on the detailed form of the resetting distribution. This universality is supported by simulations with three different resetting distributions. The renewal equation (Eq. 11) and the moment formulas (Eqs. 15–16) are clean and correctly reduce to the Poissonian limit (Eq. 12). The paper is also honest about its unproved expectations in Section VI.

major comments (2)

[Section V.B (Eqs. 46–53)] The derivation of the limiting PDF (Eq. 53) from Eq. (49) via the Godrèche–Luck inversion (Eq. 51) requires that the small-argument Tauberian expansion (Eq. 46) holds uniformly for arguments s+pu with u in [0,1], and that the limit ε→0 in Eq. (51) can be interchanged with the u-integration defining C_β and D_β in Eq. (52), including near z=0 and z=1. This uniformity is asserted but not proved. The boundary exponents (57) and (59), and hence the U/W/∩ classification and the transition lines α_c and α*, are derived precisely from this boundary behavior. Please provide a proof of the needed uniformity or state explicitly the conditions under which Eq. (53) is proven, and discuss how the phase diagram could be affected if the interchange fails.
[Section IV.B, Table I] The right half of Table I, intended for functionals with 1<μ≤2, lists only the cases 0<α≤1 and 1<α≤μ. For μ<α≤2, the asymptotic analysis of Eqs. (15)–(16) yields a different behavior: L[φ⟨Z⟩0] is then O(1) while L[φ*⟨Z⟩0] contributes subdominantly, giving ⟨Z⟩_r ∼ t and ⟨Z^2⟩_r ∼ t^2 rather than the power laws shown in the table. This range is missing from the table, so the claim that the table gives the temporal scaling for any stochastic functional with 1<μ≤2 is not supported as stated. Please add the missing row or restrict the claim accordingly.

minor comments (3)

[Eq. (7)] There is an index typo: the last term in the sum should be τ_{N+1}, and the notation for the subscript r on Z(t|x0)_r is introduced before its definition; please clarify.
[Section V.B.2, text after Eq. (59)] The sentence 'The transition between W and ∩ shapes is attained at α = αc' appears to be a typo; from the preceding classification, the ∪/W transition is at α_c and the W/∩ transition is at α* = 1 − γ/2.
[Section V.A, paragraph after Eq. (40)] The statement that the condition can be relaxed to require only a finite first moment is proved for T+ in Section V.B (Eq. 48) but is worded in V.A as if it were a general result; consider adding a cross-reference to Section V.B to avoid ambiguity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the renewal-equation derivation is self-contained, the inputs are known external distributions, and the limiting PDFs are not fitted.

full rationale

The central claim, Eq. (11), follows from first principles by summing the renewal decomposition over reset events; it contains no fitted parameter and no term that is defined in terms of the target result. The ergodic delta limit (31) is obtained by expanding Eq. (11) for finite-moment resetting PDFs, and the nonergodic result (53) is obtained by substituting the Tauberian expansion (46) and the known unreset limiting densities (arcsine or Lamperti) into Eq. (49), followed by the Godrèche–Luck inversion (51). Since f(u), alpha, mu and gamma are all model inputs rather than fitted quantities, and the Monte Carlo simulations are independent of the analytic construction, no prediction reduces by construction to an input. The self-citations (Refs. [7], [23], [27]) are used for comparison or as supporting interpretation, not as the load-bearing derivation: the transition alpha* = 1 - gamma/2 follows from the boundary behavior (59) of the paper's own expression for P(z), and Refs. [16]/[7] merely provide physical interpretation of that transition. The paper's main weaknesses are proof gaps, not circularity: the bulk Laplace inversion is asserted to hold uniformly up to z -> 0,1 in Section V.B, and Section VI explicitly states 'we expect, though have not proved' that the delta result extends to functionals with mu < 1. These are correctness/rigor concerns, not instances of a claim being equivalent to its own input. Accordingly, no circular step is identified.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters or invented entities. The theory's inputs are known model parameters (alpha, mu, gamma) and known no-resetting distributions (arcsine and Lamperti). The only load-bearing assumption not fully justified in the text is the uniformity of the bulk Laplace-inversion limit used for Eq. (53), which is marked as ad_hoc_to_paper. The renewal property and the moment-growth assumption are standard domain assumptions; Tauberian expansions are standard mathematics.

assumptions (5)

domain assumption Resetting times are iid with PDF phi(tau) and independent of the walker's trajectory; after each reset the walk restarts from x0.
Used throughout Section III to write the renewal equation (Eqs. 7-11). This is the standard resetting model.
domain assumption The functional Z(t) is additive over reset intervals and its no-resetting moments grow as powers of time: <Z^n>0 ~ t^(n mu).
Needed for the moment scalings in Section IV (Eqs. 27-30 and Table I).
standard math Tauberian expansions of the Laplace transforms: phi_tilde(s) approx 1 - b_alpha s^alpha for 0<alpha<1 and approx 1 - <t> s for 1<alpha<2.
Used in Eqs. (46)-(47) to derive the bulk limiting PDF; standard Tauberian theory for regularly varying distributions.
domain assumption The unreset half-occupation-time PDF has the scaling form P(T+,t)_0 approx (1/t) f(T+/t) with f symmetric (arcsine for Brownian, Lamperti for subdiffusive).
Needed to reduce the renewal equation to Eqs. (44) and (49); known results cited as Refs. 28 and 29.
ad hoc to paper The double Laplace inversion can be computed in the bulk scaling limit (s->0, p->0, p/s fixed), and the resulting formula holds uniformly over z in [0,1], including the boundary singularities.
Asserted in Section V.B around Eqs. (49)-(53); the uniformity near z=0,1 is not proven. This is the main fragile premise for the shape classification.

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Pith. "Pith review of Statistical properties of stochastic functionals under general resetting." pith.science (2026). https://pith.science/paper/QVDZ24HE

@misc{pith2026250705955,
  author       = {Pith},
  title        = {Pith review of: Statistical properties of stochastic functionals under general resetting},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QVDZ24HE}},
  note         = {Machine review of arXiv:2507.05955}
}

read the original abstract

We derive the characteristic function of stochastic functionals of a random walk whose position is reset to the origin at random times drawn from a general probability distribution. We analyze the long-time behavior and obtain the temporal scaling of the first two moments of any stochastic functional of the random walk when the resetting time distribution exhibits a power-law tail. When the resetting times PDF has finite moments, the probability density of any functional converges to a delta function centered at its mean, indicating an ergodic phase. We explicitly examine the case of the half-occupation time and derive the ergodicity breaking parameter, the first two moments, and the limiting distribution when the resetting time distribution follows a power-law tail, for both Brownian and subdiffusive random walks. We characterize the three different shapes of the limiting distribution as a function of the exponent of the resetting distribution. Our theoretical findings are supported by Monte Carlo simulations, which show excellent agreement with the analytical results.

Figures

Figures reproduced from arXiv: 2507.05955 by the authors.

**Figure 2.** FIG. 2. Exponent [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗

**Figure 3.** FIG. 3. a) Mean ( [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4. Limiting PDF of the half occupation time of a Brownian Motion with resetting for a [PITH_FULL_IMAGE:figures/full_fig_p023_4.png]

**Figure 5.** Figure 5: FIG. 5. a) Phase diagram of the limiting PDF of [PITH_FULL_IMAGE:figures/full_fig_p025_5.png]

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [16]

Mas´ o-Puigdellosas, D

A. Mas´ o-Puigdellosas, D. Campos, and V. m. c. M´ endez, Transport properties and first-arrival statistics of random motion with stochastic reset times, Phys. Rev. E 99, 012141 (2019)

work page 2019

[2] [1]

In the limit u → 0, f (u) ∼ u−1/2 and from (56) we see that Cβ(z) ∼ zβ+1/2 as z → 0

Brownian walker If, for example, the underlying random walk is a Brownian motion then the limiting PDF of T +/t without resetting obeys the L´ evy arcsine law [28] f (u) = 1 π p u(1 − u) (55) if 0 < u <1 and from (52) with (55) we obtain Cβ(z) = Γ (1 +β)√πΓ 3 2 + β z 1 2 +β 2F1 1 2 , 1 2 ; 3 2 + β; z , (56) where 2F1(·) is the Gauss’s hypergeometric funct...

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Subdiffusive walker When the underlying random walker performs a subdiffusive motion, then the limiting PDF in the absence of resetting is given by the Lamperti distribution [29] f (u) = sin(γπ/2) π [u(1 − u)] γ 2 −1 uγ + (1 − u)γ + 2 [u(1 − u)] γ 2 cos(γπ/2) . (58) Recall that the exponent γ is related to the index of fractional derivative, or alternativ...

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B. ˙Zbik and B. Dybiec, L´ evy flights and l´ evy walks under stochastic resetting, Phys. Rev. E 109, 044147 (2024)

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