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Partial versus total resetting for L\'evy flights in d dimensions: similarities and discrepancies

T0 review · 2 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A spline-mixture formula for the propagator shows that partial stochastic resetting of Lévy flights matches total resetting in stationary tails, fractional moments, and the Brownian dynamical phase transition, while differing sharply in…

desk verdict Solid d-dimensional extension of partial resetting for Lévy flights, but the core proofs live in a companion paper; worth refereeing if the companion is also checked. read the letter →

arxiv 2501.01139 v2 pith:T3C6WGGK submitted 2025-01-02 cond-mat.stat-mech

Costantino Di Bello , Aleksei Chechkin , Tomasz Grzywny , Karol Szczypkowski , Bartosz Trojan , Zbigniew Palmowski This is my paper · ORCID
classification cond-mat.stat-mech MSC 60G5160G5260J6560F10 PACS 05.40.-a02.50.-r05.10.Gg
keywords partialstochasticresettingLévyflightsα-stableprocessessplinerepresentationstationarydistributiondynamicalphasetransitionBrownianmotionboundedness
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 studies d-dimensional symmetric α-stable Lévy processes (Lévy flights) subjected to partial stochastic resetting, where at exponential times the position is multiplied by a factor c ∈ (0,1) rather than being sent to a fixed point. The authors establish a new spline representation for the propagator, expressing it as an integral of the free propagator against an explicit time-dependent measure, and use it to derive the stationary distribution, its fractional moments, large-|x| tails, and the large-deviation function in the Brownian case α = 2. The payoff is a clean comparison with total resetting: the two mechanisms share stationary tails, the limit γ → 0 of fractional moments, and a dynamical phase transition for Brownian motion, but they differ near the resetting position—total resetting produces an unbounded propagator for d ≥ α, while partial resetting stays bounded. A careful reader should note that the proof of the spline representation itself is deferred to a companion paper, so the results here are conditional on that proof.

What carries the argument

The load-bearing object is the spline representation of the propagator (eqs. (20)–(21)): p_r(x,t) = ∫ p_0(x,u) μ_t(du), with μ_t(du) = $e^{{-rt}}$δ(t)du + $e^{{-rt}}$∑_{j≥1} r^j t^j P_j(u/t) du/t. The splines P_n are defined recursively by P_1(u) = 1_{[m,1]}(u)/(1-m) and P_{n+1}(u) = max(u-$m^{{n+1}}$,0)^n ∫$_u^{1}$ P_n(v)/(v-$m^{{n+1}}$)^{n+1} dv, and each is supported on [m^n,1] with P_n(u) = 0 near u = 0. That vanishing at u = 0 is what makes the limit |x| → 0 finite in eq. (43), and the representation is what turns the renewal equation (17) into a tractable mixture of free propagators.

What would settle it

Run an exact, discretization-free Monte Carlo simulation of a d = 3 Brownian motion with partial resetting at parameters, say, D = 0.5, r = 0.5, c = 0.5, t = 5, and measure the empirical density at positions |x| < ε for small ε: the paper's prediction is that this density stays finite, while the same simulation for total resetting gives a diverging peak. Alternatively, compare the numerically obtained large-deviation function at α = 2, d = 2 with eq. (38); a jump in the derivative at y_* = 2√(Dr) that does not match would falsify the spline-based expansion (36).

Watch

Extended reading notes

Core claim

The central discovery is that for any c > 0 the propagator of an isotropic α-stable process with partial resetting can be written as p_r(x,t) = ∫_0^∞ p_0(x,u) μ_t(du), where μ_t is an explicit measure built from recursively defined splines P_n supported on [m^n,1] (m = c^α). From this representation the paper obtains closed-form expressions: the stationary density (22), the fractional moments (29) with q-Gamma functions, the power-law tail (27) with prefactor 1/(r(1-m)), the Brownian tail (28), and the Brownian large-deviation function (38) that is identical to the total-resetting one, including the dynamical phase transition at y_* = 2√(Dr). The same representation yields the paper's sharpest contrast: at |x| → 0 the propagator for total resetting diverges when d ≥ α, whereas for partial resetting it stays finite because every spline vanishes in a neighbourhood of u = 0.

Load-bearing premise

All new formulas rest on the spline representation (20)–(21), which the paper does not prove here but attributes to a companion paper; if that representation fails for a dimension d ≥ 2 or for c > 0, the stationary measure, tails, moments, phase transition, and boundedness conclusions all collapse.

Editorial extensions

If this is right

  • The propagator can be evaluated numerically in any dimension d without resorting to numerically unstable special functions.
  • The stationary fractional moments (29) provide a closed-form formula for all −1 < γ < α, matching total resetting as m → 0.
  • For Brownian motion, the large deviation function and the dynamical phase transition are independent of the dimension d and of the partial resetting parameter c, in agreement with the total-resetting result.
  • The boundedness of the propagator at the resetting position holds for all c > 0 and all d, in contrast to total resetting where it diverges for d ≥ α.
  • The tail asymptotics (27) coincide for partial and total resetting, with a universal 1/(r(1−m)) prefactor multiplying the Lévy measure.

Reading between the lines

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

  • Because the spline representation relies only on the scaling property of the underlying process, it is plausible the same mixture-of-free-propagators form extends to other scale-invariant Markov processes beyond stable laws.
  • The absence of a dynamical phase transition for α < 2 suggests that heavy-tailed jump processes cannot freeze near the resetting point; a similar conclusion should hold for other heavy-tailed resetting mechanisms with finite-time singular resetting.
  • The boundedness contrast at the origin could have operational consequences for search and restart strategies that use partial resetting to avoid an infinite accumulation of probability at the resetting point.
  • A natural testable extension is to asymmetric Lévy flights, where the spline representation may carry an angular dependence; the technique should generalize if the free propagator retains the needed self-similarity.
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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

2 major / 5 minor

Summary. The paper studies d-dimensional isotropic α-stable Lévy flights under partial stochastic resetting, where at rate r the position is multiplied by c ∈ (0,1). The central object is a new spline representation of the propagator, Eq. (20)-(21), from which the authors derive the stationary distribution (22), its tail asymptotics (27)-(28), fractional moments (29), the Brownian large-deviation function with a dynamical phase transition (36)-(38), and boundedness of the propagator near |x|=0 (43). The main qualitative message is that partial and total resetting coincide in stationary tails, in the m→0 limit of the moments, and in the Brownian dynamical phase transition, but differ in the behavior at the resetting position: total resetting diverges for d ≥ α while partial resetting remains bounded. The analytical arguments are claimed to be proven in the companion paper [30], and the numerical checks are Monte Carlo simulations in d=1.

Significance. If the spline representation (20)-(21) is correct, the paper provides a substantial and useful extension of the one-dimensional results of [19] to arbitrary dimension, with concrete formulas for the stationary measure, tails, moments, and the Brownian large-deviation function. The paper is interesting because it identifies a clear physical distinction between partial and total resetting — the boundedness at the resetting point — while showing that several other asymptotic features are shared. Strengths include the absence of fitted free parameters, the use of exact Monte Carlo sampling of resetting times, and the explicit q-series formulas. The main caveat is that essentially all of the analytical novelty rests on proofs deferred to a companion preprint, so the present manuscript is not self-contained at the level of its central claims.

major comments (2)
  1. [§V, Eq. (22)] The central spline representation of the propagator and the large-time asymptotic expansion (36) are not proved in this paper; the text explicitly says 'We do not report here the proof' and 'This result is proved in 30'. Since Eqs. (22), (27)-(28), (29), (36)-(38), and (43) all depend on this representation and on the support property of the splines P_n, the load-bearing part of the paper is an external preprint by four of the six authors. I am not claiming the companion proof is wrong, but the present manuscript should either include a statement of the relevant theorems with a sufficient sketch (especially the support property of P_n on [m^n,1] and the Stein-method estimate behind (36)) or otherwise make clear to the reader which results are assumptions imported from [30]. As written, a reader cannot verify the main claims from the manuscript alone.
  2. [§V, Eq. (22)] The stationary distribution formula (22) as printed is not correct for general d, r, and D. The exponent e^{-m^{-k} u r^{d/α}} has dimensions T^{1-d/α}, which is not dimensionless except when d=α, and the argument r^{1/α}x in p0(r^{1/α}x,u) does not produce the required scaling in D. For example, in d=1, α=2, the Fourier transform of the right-hand side of (22) is proportional to 1/(m^{-k} r^{1/2} + D r k^2), which is not proportional to the factor r/(r + D m^k k^2) required by Eq. (13); the discrepancy disappears only for special choices such as D=1 and r=1. The correct representation should be p_s(x) = r/(m;m)_∞ ∑_{k=0}^∞ (-1)^k m^{k(k-1)/2}/(m;m)_k ∫_0^∞ e^{-r m^{-k} u} p_0(x,u) du, which does reduce to Eq. (24) for d=1 Brownian motion. Please correct Eq. (22) or clarify the convention under which the printed formula is intended.
minor comments (5)
  1. [§IV, Eq. (21)] In the definition of μ_t(du), the term δ(t)du should presumably be δ(u-t)du (or δ(t-u)du); as written, the delta function does not have the integration variable as its argument.
  2. [Table I] In the row for the Fokker-Planck equation, the term (r/c) p_r(r/c, t) should be (r/c) p_r(x/c, t).
  3. [Fig. 4 caption] The caption states 'y = x/rt', but Eq. (38) uses y = |x|/t; please correct the caption or, if |x|/(rt) is intended, rescale the large-deviation function accordingly.
  4. [§V.B, Eq. (29)] The condition '-1 < γ < α' appears to be the one-dimensional condition; in d dimensions the natural integrability condition near the origin is -d < γ < α. Please state the d-dimensional condition or explicitly restrict Eq. (30)-(33) to d=1.
  5. [Figs. 2, 4, 5] The Monte Carlo histograms and time-collapse plots are presented without error bars or sampling uncertainty estimates. Adding error bars (or reporting the standard error) would strengthen the numerical evidence, particularly for the phase-transition collapse.

Circularity Check

3 steps flagged · score 4.0 of 10

Central spline representation and all derived asymptotics are deferred to a companion paper co-authored by the present authors; no fitted-input circularity, but the main derivation chain is self-citation load-bearing.

  1. self citation load bearing [Section IV, eqs. (18)-(21)]
    "As shown in the companion paper 30 (see section 3), a possible way to find the solution to the renewal equation is to define recursively the sequence ( Pn : n ∈ N) of splines via ... In 30 it was proved that the solutions (17) reads pr(x, t) = Z ∞ 0 p0(x, u) µt(du), x ∈ Rd, ... We do not report here the proof, but just specify that it makes use of the renewal equation (17) and of the properties of α-stable distributions (see 37 chapter 3)."

    The spline representation (20)-(21) is the paper's central new tool: every subsequent main result — stationary distribution (22), tail asymptotics (26)-(28), fractional moments (29), Brownian large-deviation function (36), and boundedness at x=0 (43) — is derived from it. Yet the proof of (20)-(21) is not contained in this paper; it is asserted to be proved in companion arXiv:2412.15626, whose authors include four of the present authors. Within this manuscript, the main derivation chain therefore reduces to an unverified self-citation, with independent grounding only from a d=1 Monte Carlo check at one parameter set.

  2. self citation load bearing [Section VI, eq. (36) and eq. (38)]
    "In our companion paper 30 we were able to prove that for α = 2 we have pr(x, t) = ... (36) ... This suggests that the LDF in eq. (35) can be written as ... (38), which is precisely the same LDF obtained in 40 for the total resetting. ... The result is proved in 30 by making use of (20) and of the Stein method41."

    The claimed Brownian dynamical phase transition, including the explicit large-deviation function (38), is one of the paper's advertised results. The proof is explicitly deferred to the same companion paper [30] by four of the present authors, and the in-paper justification is only that the formula 'suggests' the LDF plus numerical plots. Thus the central claim of a dimension-independent, c-independent phase transition is not independently derived in this paper; it is imported from a self-citation whose proof is not reproduced here.

1 more flagged steps
  1. self citation load bearing [Section VII, eq. (43)]
    "on the other hand, for PSR after substituting the same scaling (41) into eq. (20) we get ... = p0(0, 1) Z ∞ 0 u−d/αµt(u)du (43) where the integral in the last step does not diverge since all splines are identically equal to 0 close to u = 0 (see eqs. (19))."

    The boundedness discrepancy between partial and total resetting is the paper's final distinctive conclusion. The non-divergence in (43) rests entirely on the splines' vanishing near u=0, a support property stated for the Pn in (18)-(19) but proved only in companion [30]. Without an in-paper proof of that property, the claim that PSR stays bounded while total resetting diverges for d ≥ α is not derived here; it is another load-bearing consequence of the self-cited companion work.

full rationale

The paper contains no fitted free parameters, no definitional identification of outputs with inputs, and no renaming of a known result as a new one; the d=1 Monte Carlo checks in Figs. 2, 3, and 6 give some independent empirical grounding. However, the paper explicitly describes itself as a results-focused companion: 'A complete proof of the results that we present here is available in the companion paper30.' The spline representation (20)-(21), the stationary formula (22), the tail asymptotics (26)-(28), the fractional moments (29), the Brownian expansion (36) with its large-deviation function, and the boundedness conclusion (43) are all stated as proved in arXiv:2412.15626, a companion authored by four of the six present authors. That is a substantial load-bearing self-citation chain: the paper's own novel derivation is not self-contained in the submitted text. The cited companion may well be correct and independently verified, and no circularity of the 'defined into existence' sort is present; but under the reviewing rules, an unproved central premise justified only by overlapping-author citation raises the circularity score to 4 rather than 0.

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

The central mathematical machinery is the spline-based measure representation of the propagator. All main results depend on it, and it is proved in a companion paper by the same research group. The model assumptions are standard for partial resetting. There are no fitted free parameters; r, c, D, and alpha are fixed model inputs. The only new object is the spline sequence (18)-(19), which is introduced for this problem and carries the burden of the derivation.

assumptions (4)
  • domain assumption Isotropic alpha-stable Levy process in R^d with characteristic function exp(-D|k|^alpha t), alpha in (0,2].
    This is the process family under study, introduced in eq. (2); Brownian motion is the alpha=2 limit.
  • domain assumption Partial resetting events form a Poisson process of rate r and multiply the position by a fixed factor c in [0,1), with m=c^alpha < 1.
    This defines the model in eq. (1) and the fractional Fokker-Planck equation (6); independence of resetting times from the Levy process is assumed.
  • ad hoc to paper The renewal equation (17) has a unique solution expressible as the spline measure (20)-(21), with the proof in companion [30].
    This spline representation is the central mathematical input for all new results and is not derived in the present manuscript.
  • ad hoc to paper For alpha=2 the asymptotic expansion (36) and the large deviation function (38) are valid via Stein's method, as proved in companion [30].
    The dynamical phase transition conclusion depends on this deferred proof and is not derived in this paper.

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Cite this review

Pith. "Pith review of Partial versus total resetting for L\'evy flights in d dimensions: similarities and discrepancies." pith.science (2026). https://pith.science/paper/T3C6WGGK

@misc{pith2026250101139,
  author       = {Pith},
  title        = {Pith review of: Partial versus total resetting for L\'evy flights in d dimensions: similarities and discrepancies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T3C6WGGK}},
  note         = {Machine review of arXiv:2501.01139}
}
abstract

While stochastic resetting (or total resetting) is less young and more established concept in stochastic processes, partial stochastic resetting (PSR) is a relatively new field. PSR means that, at random moments in time, a stochastic process gets multiplied by a factor between 0 and 1, thus approaching but not reaching the resetting position. In this paper, we present new results on PSR highlighting the main similarities and discrepancies with total resetting. Specifically, we consider both symmetric $\alpha$-stable L\'evy processes (L\'evy flights) and Brownian motion with PSR in arbitrary d dimensions. We derive explicit expressions for the propagator and its stationary measure, and discuss in detail their asymptotic behavior. Interestingly, while approaching to stationarity, a dynamical phase transition occurs for the Brownian motion, but not for L\'evy flights. We also analyze the behavior of the process around the resetting position and find significant differences between PSR and total resetting.

Figures

Figures reproduced from arXiv: 2501.01139 by the authors.

Figure 1
Figure 1. FIG. 1. Comparison between total resetting (yellow line) and [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Comparison in log scale between Monte Carlo [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Brownian motion with PSR in [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Large deviation function versus rescaled variable [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Propagator of L´evy flight with PSR in the case [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Figure showing boundedness of the propagator. The parameters of the simulations are [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.