{"id":"2c3be55a-1819-4efa-838e-30c67f5e755b","arxiv_id":"2501.01139","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Partial resetting of d-dimensional Levy flights is solved: propagator, stationary distribution, moments, tails, and a Brownian-only dynamical phase transition are derived and compared with total resetting.","lead":"Partial resetting shrinks a random process toward a fixed point by a random factor at random times, instead of jumping straight back to it. This paper derives the probability distributions, stationary states, and large-time behavior of Levy flights and Brownian motion under partial resetting in any number of dimensions, and shows how they differ from total resetting.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central spline representation (20)-(21) and the large-time expansion (36) are proved only in companion arXiv:2412.15626, not in this paper; the d≥2 claims therefore rest on an unverified external proof, though the paper's internal consistency and d=1 numerics give no sign of error.","rationale":"The reader's weakest-assumption analysis matches my reading: every substantive new formula in this paper is asserted to follow from a proof placed in a companion paper by four of the same authors. The spline representation is the load-bearing structure: stationary state, moments, tails, boundedness, and the phase-transition statement all reduce to it. The paper itself contains no independent derivation of (20)-(21), and the numerical support covers d=1 only, so there is a real verification gap for the d-dimensional claims. I do not see an internal inconsistency: the formulas reduce correctly to total-resetting limits as m→0, the Brownian stationary expression matches the previously published d=1 result, and the Monte Carlo comparison in Fig. 2 agrees. Therefore the appropriate disposition is unchanged: conditional acceptance pending independent verification of the companion proofs or an independent check of the spline identity. The concrete algebraic substitution test would settle whether the core representation is correct without relying on the companion's authority.","tokens_in":11404,"tokens_out":22201,"duration_ms":209750,"concrete_test":"Independently verify the companion's central identity: substitute eqs. (20)-(21) into the renewal equation (17), take the Fourier transform in x and Laplace transform in t, and show algebraically that the result equals eq. (11) for arbitrary d≥1, using only the recurrence (18)-(19) for Pn. If the substitution reproduces (11), the spline representation is established without reliance on the companion; if it does not, the representation and all downstream conclusions are in question.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The main claim is the spline representation pr(x,t)=∫p0(x,u)μt(du), eqs. (20)-(21), and everything derived from it: the stationary distribution (22), fractional moments (29), tail asymptotics (27)-(28), the Brownian large-deviation result (36), and boundedness at x=0 (43). The paper explicitly defers the proof of (20)-(21) to companion arXiv:2412.15626 ('We do not report here the proof'), defers (36) to the same companion ('This result is proved in 30'), and gives only a d=1 Monte Carlo check (Fig. 2) at one parameter set. The in-paper derivations are otherwise consistency checks: m→0 recovers total-resetting formulas, and eq. (24) is said to match Ref. 19. Thus if the companion proof of the support property of the splines Pn (used for (43)) or of the Stein-method estimate behind (36) contains a gap, the paper's distinctive conclusions—particularly the claimed boundedness of PSR versus divergence of total resetting at x=0, and the c-independent Brownian phase transition—are unsupported. This is a verification dependency rather than a demonstrated mathematical error; the formulas are internally consistent and the d=1 numerics agree.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":11706,"tokens_out":47609,"duration_ms":368947,"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":[{"comment":"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.","section":"§V, Eq. (22)"},{"comment":"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.","section":"§V, Eq. (22)"}],"minor_comments":[{"comment":"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.","section":"§IV, Eq. (21)"},{"comment":"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).","section":"Table I"},{"comment":"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.","section":"Fig. 4 caption"},{"comment":"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.","section":"§V.B, Eq. (29)"},{"comment":"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.","section":"Figs. 2, 4, 5"}],"recommendation":"major_revision","confidential_remarks":"The manuscript's main results are proven in a companion paper arXiv:2412.15626 by four of the six authors. The editor may wish to confirm that [30] is publicly available in its final form and that the present paper has a clear independent contribution beyond the companion. The current manuscript would be strengthened by making the key lemmas of [30] accessible to referees, either by including a sketch or by having the companion paper under explicit review. The issue in Eq. (22) is central and should be fixed before acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does what it says: it gives a clean comparative picture of partial versus total resetting for isotropic α-stable processes in arbitrary dimension. The genuinely new pieces are the spline representation of the propagator, the fractional moments (29), the tail asymptotics (26)-(28), the Brownian large-deviation function (38), and the boundedness contrast at x=0. These go beyond the one-dimensional propagator in ref. [19], and the m→0 limits correctly reproduce total-resetting results. The table at the end is useful. The d=1 Monte Carlo checks in Figs. 2, 3, and 5 agree with the spline solution, and the simulations sample exact resetting times, which avoids discretization artifacts. Credit where due: the presentation is clear and the authors are explicit about what is proved where.\n\nThe soft spot is exactly what the authors themselves flag: equations (20)-(21) and (36) are stated as results with proofs deferred to companion arXiv:2412.15626, written by four of the same authors. The boundedness conclusion (43) relies directly on the support property of the splines, and the Brownian LDF relies on a Stein-method estimate in the companion. None of this is derived in the manuscript. That is a large external dependency for a paper whose central claims all rest on it. The reader's circularity burden of 4/10 is fair — no fitted parameters, no invented entities, but the novelty is 'as described in our companion.' Another limitation: the numerical validation is d=1 only, and the Monte Carlo histograms have no error bars. These are addressable issues rather than demonstrated errors. The formulas are internally consistent and reduce properly to known limits, so I see no sign that the companion proof is wrong.\n\nWho gets value from this? Physicists working on stochastic resetting who want the d-dimensional statements and the phase-transition comparison; mathematicians may prefer the companion. For a physics journal, this is a legitimate results paper, but the referee should be asked to check the companion or the authors should include at least a proof sketch of the spline representation.\n\nRecommendation: send it to peer review. It deserves a serious referee, and the conditional verdict is the right one — the paper is solid if the companion holds up.","headline":"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.","tokens_in":12217,"tokens_out":1003,"would_cite":true,"duration_ms":12062,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60G51","60G52","60J65","60F10"],"pacs":["05.40.-a","02.50.-r","05.10.Gg"],"model":"deepseek-v4-flash","headline":"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…","keywords":["partial stochastic resetting","Lévy flights","α-stable processes","spline representation","stationary distribution","dynamical phase transition","Brownian motion","boundedness"],"falsifier":"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).","tokens_in":11182,"feed_emoji":"🎲","tokens_out":6480,"duration_ms":52034,"temperature":0.7,"pith_summary":"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.","feed_headline":"Partial resetting tames the origin divergence of Lévy flights","feed_subtitle":"Same tails and same Brownian phase transition, but a finite peak instead of an infinite one at the reset point.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the proof of the spline representation (20)–(21) and the large-time expansion (36) on which all results in this paper depend.","marker":"[30]"},{"why":"Provided the one-dimensional PSR propagator and stationary distribution in Fox H form that the present paper generalizes to d dimensions and analyzes.","marker":"[19]"},{"why":"Gives the total resetting propagator (15) and the unboundedness at x = 0 for d ≥ 2, the main contrast object for the boundedness result.","marker":"[34]"},{"why":"Established the total-resetting Brownian large deviation function and dynamical phase transition, which the paper shows is identical for partial resetting.","marker":"[40]"},{"why":"Introduced the pantograph Fokker-Planck equation for proportional (partial) resetting, extended here to arbitrary dimensions.","marker":"[17]"},{"why":"Provides the scaling properties of stable Lévy processes used to derive the renewal solution and the spline representation.","marker":"[37]"},{"why":"Supplies the large-|x| tail p_0(x,t) ~ t |x|^{-d-α} used to argue the absence of a dynamical phase transition for α < 2.","marker":"[42]"}],"fun_headline_variants":["Partial resetting stops Lévy origin blow-up","Lévy flights: partial resetting keeps origin finite","No origin divergence under partial resetting","Partial resetting: Brownian transition, Lévy no","Partial resetting: same tail, new finite peak"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Partial resetting stops Lévy origin blow-up","Lévy flights: partial resetting keeps origin finite","No origin divergence under partial resetting","Partial resetting: Brownian transition, Lévy no","Partial resetting: same tail, new finite peak"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000194,"raw_usage":{"total_tokens":1353,"prompt_tokens":943,"completion_tokens":410,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":559,"completion_tokens_details":{"reasoning_tokens":336}},"tokens_in":559,"tokens_out":410,"duration_ms":3938,"temperature":1.0,"reasoning_tokens":336,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:34:29.468683+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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).","supporting_citations":[{"cited_title":"Stationary states for stable processes with partial resetting","cited_arxiv_id":"2412.15626","evidence_quote":"Supplies the proof of the spline representation (20)–(21) and the large-time expansion (36) on which all results in this paper depend."},{"cited_title":"Di Bello , author A","cited_arxiv_id":null,"evidence_quote":"Provided the one-dimensional PSR propagator and stationary distribution in Fox H form that the present paper generalizes to d dimensions and analyzes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the total resetting propagator (15) and the unboundedness at x = 0 for d ≥ 2, the main contrast object for the boundedness result."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Established the total-resetting Brownian large deviation function and dynamical phase transition, which the paper shows is identical for partial resetting."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the scaling properties of stable Lévy processes used to derive the renewal solution and the spline representation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the large-|x| tail p_0(x,t) ~ t |x|^{-d-α} used to argue the absence of a dynamical phase transition for α < 2."}],"review_version":1}