{"id":"55322d4b-26a2-4b6d-8e25-abe71a3bffcd","arxiv_id":"2501.18208","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Under additional regularity, the L2 distance between the Nmin-Nmax branching-Moran particle system and its Feynman-Kac semigroup is bounded linearly in time, and uniformly with error of order 1/sqrt(Nmin).","lead":"This preprint improves bounds on how closely a particle model mixing branching and Moran-type resampling approximates its underlying Markov average process. It shows that, under extra regularity, the error can be kept small uniformly over time, which matters for numerical methods that simulate rare events and quasi-stationary distributions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 4's Theorem 4 depends on an unproved coupling of the particle system to independent reflected Brownian motions; the domination inequality and the independence used in Lemma 5 are asserted but not constructed, so the Nmin^{-1} bound is unsupported.","rationale":"The central claim of the paper is the linear/uniform-in-time L2 bound in Theorem 1 and its application to killed Brownian motion in Theorem 4. The proof of Theorem 1 contains a compressed step: estimate (6) is imported from the authors' earlier paper [8] with only the statement 'This is obtained via a modification of the end of the proof of Theorem 2.6 in [8]'. I did not make this the primary concern because the estimate is plausible and could be checked against [8]; the Section 4 coupling is a genuinely new construction and is explicitly not supplied. The paper's own text admits the omission, so this is not an artifact of my reading. Lemma 5's proof begins 'Since the random variables R_i^1 have a bounded density ... and are independent', but the independence is exactly the unproved assertion. The pathwise domination is likewise asserted, and without it the conversion from (13) to the Nmin^{-1} rate collapses. Assumption 3 is also verified through this coupling. Thus the main new application in Theorem 4 is conditionally supported at best. This matches the reader's CONDITIONAL verdict, and I agree with the reader's identification of the coupling as the weakest assumption. If the authors supply the construction and proof, or explicitly restate the domination and independence as assumptions and derive Theorem 4 under them, the paper would become substantially stronger; as it stands, the main novelty is not fully demonstrated.","tokens_in":11703,"tokens_out":16252,"duration_ms":171188,"concrete_test":"Reconstruct the coupling of Section 4 in full, then verify in the minimal case Nmin=2, Nmax=3, b=1, kappa=0, D a disk: (i) the pathwise domination sum_{i=1}^{N_t} rho_D(X^i_t) >= R^1_t+R^2_t after any selection event that removes an unassociated particle; and (ii) the independence factorization E[e^{-uR^1_1-vR^2_1}] = E[e^{-uR^1_1}] E[e^{-vR^2_1}]. If either fails, or if the construction cannot be made explicit at the hard-killing/no-resampling transition, Theorem 4's bound does not follow from the stated assumptions.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 4's proof of Theorem 4 hinges on a coupling between the Nmin-Nmax particle system and independent jumping reflected Brownian motions R^i on [0,a]. The paper states 'we leave the details of the construction to the reader' and later asserts that 'the R_i are independent' with the proof 'very similar to [31]'. This is load-bearing. First, the pathwise domination sum_i rho_D(X^i_t) >= sum_{i=1}^{Nmin} R^i_t is used to pass from (13), E[1/(sqrt(N_t) mhat_t(h))] <= E[C sqrt(Nmax) / sum_i rho_D(X^i_t)], to the C sqrt(Nmax)/Nmin bound; without a proof of the domination, this passage is unsupported. Second, Lemma 5 computes E[1/sum R^i_1] <= C/Nmin using the product form integral_0^infty L(t)^{Nmin} dt, which requires exactly the asserted independence of the R^i. The rules for reassigning the R^i after selection and resampling events introduce common event times and label choices; independence is not a formal consequence of the one-dimensional dynamics and must be proved. Third, the verification of Assumption 3 also uses the coupling: the paper deduces tau_n -> +infty from the impossibility that all R^i accumulate at 0, which again relies on the asserted independence. The transition rules are also incompletely specified: in the hard-killing/no-resampling case, the text says a reflected Brownian motion is associated to a 'new particle, not already associated', although no new particle is created. If the coupling or the independence fails, the main new bound in Theorem 4 does not follow.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the binary branching model with Moran type interactions (BBMMI), specifically the Nmin-Nmax process introduced in the authors' earlier paper [8]. Its main claim is an L2 error bound for the difference between the normalized empirical measure mhat_T and the normalized Feynman-Kac semigroup m0-hat(Q_T)/m0-hat(Q_T 1_E), stated as Theorem 1: under Assumptions 1-3 the error is at most C times the sum of alpha_{T-s-1}(f) E[1/(sqrt{N_s} mhat_s(h))]. In Remark 3 this is converted into a linear-in-T bound and, when alpha is summable, a uniform-in-time O(1/sqrt{Nmin}) bound. Section 4 specializes to Brownian motion with drift killed at the boundary of a C^2 domain and claims the improved estimate C sqrt{Nmax}/Nmin plus an initial-condition term. The paper also contains two counterexamples showing that linear or uniform bounds can fail without the stated assumptions.","tokens_in":12074,"tokens_out":4526,"duration_ms":46749,"significance":"If the proofs were complete, the result would be a genuine improvement over the exponential-in-T bound of [8] and would extend uniform-in-time particle approximation bounds for Moran-type and Fleming-Viot-type particle systems, including an optimal O(1/sqrt{Nmin}) rate in a Brownian setting. The paper is clearly written and gives useful structural decomposition of the error into a semigroup stability term alpha and a particle-sampling term. It also provides explicit counterexamples that delineate the necessity of the hypotheses. However, the central proof currently rests on several substantial unproved ingredients: the one-step inequality (6) is said to follow by a modification of a proof in [8], and the coupling in Section 4 is described but its key domination, independence, and regularity properties are left to the reader. These omissions are load-bearing, so the significance is conditional until those details are supplied.","major_comments":[{"comment":"The proof of Theorem 1 begins with inequality (6), which is introduced with the sentence 'This is obtained via a modification of the end of the proof of Theorem 2.6 in [8].' No modification is shown. Since every later bound in Theorem 1 and Remark 3 is built on (6), this is a load-bearing step and must be proved, or at least quoted with a precise derivation that the reader can verify. A reference to an unpublished 'modification' is not sufficient for a journal proof.","section":"Section 3, inequality (6)"},{"comment":"The proof of Theorem 4 hinges on a coupling between the Nmin-Nmax particle system and independent jumping reflected Brownian motions R^i on [0,a] that satisfies sum_i rho_D(X^i_t) >= sum_{i=1}^{Nmin} R^i_t for all t. The text states 'The construction of such a process follows similar ideas to those presented in [31] and so we leave the details of the construction to the reader' and later asserts that the R^i are independent 'the proof is very similar to the one developed in [31]'. This is not an acceptable proof for the main new estimate: the pathwise domination is used to pass from (13) to the C sqrt{Nmax}/Nmin bound, and the independence is used in Lemma 5. The transition rules are also incompletely specified; in particular, in the hard-killing/no-resampling case the paper says a reflected Brownian motion 'is associated to a new particle, not already associated to a Brownian motion', although in that case no new particle is created. The coupling must be constructed in detail and the independence property proved.","section":"Section 4, construction of the coupling"},{"comment":"Lemma 5 assumes without proof that the random variables R^i_1 have a bounded density f_R with respect to Lebesgue measure on [0,a] and that they are independent. The product-form identity E[1/sum_i R^i_1] = integral_0^infinity L(t)^{Nmin} dt requires exactly this independence, and the subsequent bound requires the boundedness and regularity of the density. Neither property follows from the informal description of the R^i process, especially after repeated jumps to 0 and reassociation at selection and resampling events. A proof or a precise citation for these properties must be provided.","section":"Section 4, Lemma 5"},{"comment":"The verification of Assumption 3 in Section 4 also depends on the unproved coupling: the paper argues that if the event time tau_n had a finite limit, then the distance to the boundary of the particle system would accumulate at 0, and hence the set of R^i would accumulate at 0, 'which is not possible (by independence of the processes R^i)'. Since the independence of the R^i is not established, this argument is currently unsupported. The same issue affects the claimed validity of Assumption 3 and hence the applicability of Theorem 1 in the Brownian setting.","section":"Section 4, verification of Assumption 3"}],"minor_comments":[{"comment":"In equation (8), the denominator is written with N0 and m0, but by the Markov property at time s the displayed expression should involve N_s and m_s; otherwise the final expectation E[1/(sqrt{N_s} mhat_s(h))] does not follow. This appears to be a typographical slip, but it should be corrected because the displayed inequality is otherwise not the one being iterated.","section":"Section 3, proof of Theorem 1, equation (8)"},{"comment":"In the definition of alpha_t(f), the expression 'for all >= 0' is missing the variable t; it should read 'for all t >= 0'.","section":"Section 3, equation (4)"},{"comment":"The notation dR^i_t = dB^i_t - ||r||_infinity dt + dL^{i,0}_t - dL^{i,a}_t is used before the local time processes L^{i,0} and L^{i,a} are formally introduced; adding a sentence defining these local times immediately after the display would improve readability.","section":"Section 4, equation (14)"},{"comment":"There are several minor typographical and formatting issues, such as 'Itoˆ's formula' and the reference to 'Annales de l’Institut Henri Poincare, Probabilites et Statistiques' in the bibliography; these should be cleaned up in revision.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper leans on the authors' own prior work [8] for a key inequality and on [31] for the coupling construction. This is not circular in the sense that the central new claim does not reduce to those inputs, but the volume of unproved 'modifications' and 'details left to the reader' means the boundary between new and imported material is currently impossible to assess. I would ask the editor to insist on a complete proof of the one-step inequality and of the Section 4 coupling before considering the paper further."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main result is real: under Assumptions 1–3, Theorem 1 turns the exponential-in-T bound from [8] into a summable linear or uniform-in-time bound when h is bounded below and α_t decays. The two counter-examples in Section 3 are the right kind of sanity checks—they show the assumptions on h and on semigroup convergence are not just technical. That part of the paper reads well and is a genuine step forward for the Nmin–Nmax model.\n\nThe Brownian application in Section 4 is the soft spot, and it is load-bearing. The claimed C√Nmax/Nmin bound depends on a pathwise domination of Σρ_D(X^i_t) by a system of Nmin independent jumping reflected Brownian motions. But the construction is explicitly left to the reader, the independence is asserted without proof, and the reassignment rules after selection/resampling events are under-specified (e.g., the hard-killing/no-resampling case associates a reflected Brownian motion to a \"new particle\" that does not exist). Lemma 5 then computes E[1/ΣR^i_1] using independence and a bounded density, neither of which is established. If that coupling fails, Theorem 4 falls apart, and the verification of Assumption 3 via the R_i also collapses.\n\nThere is also a smaller gap before that: inequality (6) is quoted from the authors' prior paper with \"a modification of the end of the proof of Theorem 2.6 in [8]\" but the modification is not shown. For a bound that drives the telescoping sum, the reader has to either trust [8] or reproduce the argument from scratch. It is probably fixable, but it is not present.\n\nOn balance, the Section 3 result is plausible and worth engaging with seriously; the counter-examples are good; and the application to Brownian motion is an attractive claim, but it is not yet supported as written. The paper is not incoherent—if anything, the authors are transparent about what they are leaving out—but the omissions are in the wrong places for a final version.\n\nThis deserves a serious referee. If I were handling it, I would send it to review with a specific request: either prove the coupling and the independence in Section 4, or state them as explicit assumptions; and show the derivation of (6). With those filled, I would expect this to be a solid contribution to the numerical analysis of Feynman–Kac semigroups and interacting particle systems.","headline":"The Section 3 bound is a genuine improvement over [8] and the counter-examples are instructive, but Theorem 4 rests on an unproved coupling and the one-step estimate is imported with a missing modification.","tokens_in":12618,"tokens_out":1635,"would_cite":false,"duration_ms":17409,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["82C22","82C80","65C05","60J25","92D25","60J80"],"pacs":[],"model":"deepseek-v4-flash","headline":"The normalized empirical measure of the $N_{\\min}$–$N_{\\max}$ BBMMI particle system converges to the normalized Feynman–Kac semigroup with an error that is at most linear in time, and uniformly in time under exponential contraction.","keywords":["binary branching model with Moran interactions","Nmin-Nmax model","Feynman-Kac semigroup","many-to-one formula","L2 approximation error","quasi-stationary distribution","Moran model","Brownian motion killed at boundary"],"falsifier":"Simulate the $N_{\\min}{-}N_{\\max}$ model for Brownian motion with drift in a bounded $C^2$ domain with small $N_{\\min}$, and test whether the summed distance-to-boundary of the $N_{\\min}$ tracked particles is stochastically dominated by the sum of independent jumping reflected Brownian motions with the stated jump-to-zero rate; a violation would break the coupling and with it the proof of Theorem 4.","tokens_in":11431,"feed_emoji":"🧮","tokens_out":10207,"duration_ms":91072,"temperature":0.7,"pith_summary":"This note strengthens the error bound for the binary branching model with Moran-type interactions (BBMMI), an interacting particle system whose population size is kept between $N_{\\min}$ and $N_{\\max}$ by resampling and selection events. Its main theorem replaces the exponential-in-$T$ error bound of the earlier construction with a bound that is linear in $T$, expressed as a sum of contraction coefficients $\\alpha_t(f)$ weighted by a term involving the particle-size process. The practical stake is that the particle system can approximate Feynman–Kac semigroups—used for conditioned and killed Markov processes—over long time horizons without the error exploding. When the semigroup contracts exponentially to a quasi-stationary law and $h$ is bounded away from zero, the bound becomes uniform in time with the optimal $C_f/\\sqrt{N_{\\min}}$ rate, and the paper proves this rate for Brownian motion with drift killed on a bounded $C^2$ domain.","feed_headline":"Error bound turns exponential-in-T into linear-in-T for particles","feed_subtitle":"If the semigroup converges exponentially, the L2 error stays uniformly O(1/sqrt(Nmin)) in time.","key_machinery":"The proof's engine is the pair $(h,\\alpha_t)$: $h(x)$ records how much mass a single trajectory starting at $x$ can lose relative to the best possible starting point over one unit of time, and $\\alpha_t(f)$ records how far the normalized semigroup action on $f$ is from a chosen probability measure $\\nu_t$. Starting from the one-step estimate inherited from [8], the argument applies it to $Q_{T-s-1}f$ at each integer time $s$ and adds the $T$ pieces with Minkowski's inequality and the Markov property. For the Brownian domain result, the additional mechanism is a stochastic domination: the sum of the selected particles' distances to the boundary is dominated by the sum of $N_{\\min}$ independent jumping reflected Brownian motions on $[0,a]$, whose inverse-sum expectation is $O(1/N_{\\min})$; the paper notes the construction follows [31] and leaves its details to the reader.","core_discovery":"Theorem 1 states that under Assumptions 1, 2 and 3 there is $C>0$ such that for all $T\\ge 1$ and bounded measurable $f$, $$\\left\\|\\frac{\\hat m_0 Q_T f}{\\hat m_0 Q_T 1_E} - \\hat m_T(f)\\right\\|_2 \\le C \\sum_{s=0}^{T-1} \\alpha_{T-s-1}(f)\\, \\mathbb E\\!\\left[\\frac{1}{\\sqrt{N_s}\\,\\hat m_s(h)}\\right],$$ where $h(x)=\\inf_{t\\ge1}\\delta_x Q_t 1_E / \\|Q_{t-1}1_E\\|_\\infty$ and $\\alpha_t(f)=\\sup_x|\\delta_x Q_t f/(\\delta_x Q_t 1_E)-\\nu_t(f)|$. This replaces the exponential-in-$T$ factor from the earlier BBMMI paper. Under the extra conditions that $h$ is bounded away from zero and $\\sum_t \\alpha_t(f)<\\infty$, the right-hand side is bounded uniformly in time by $C_f/\\sqrt{N_{\\min}}$; Example 4 shows some such condition is needed, since the bound can fail when the semigroup does not converge uniformly.","pith_inferences":["If the omitted boundary-distance coupling can be constructed rigorously, the same strategy should transfer to other killed diffusions whose distance to the boundary is controlled by a one-dimensional reflected process, potentially relaxing the $C^2$ boundary assumption.","The reference measures $\\nu_t$ are free parameters; choosing them time-dependent appears to offer a route to linear-in-$T$ bounds in time-inhomogeneous settings even when exponential contraction is unavailable.","A direct numerical check is to fix $N_{\\min}$ and increase $T$: the error should plateau exactly when $\\sum_t \\alpha_t(f)$ converges, giving a practical diagnostic for how long the particle system can be trusted.","The inverse-sum lemma suggests the $O(1/N_{\\min})$ rate is tied to the reflected Brownian motions' bounded drift; replacing the drift with an unbounded one might degrade the rate, so the bounded-drift assumption is likely essential."],"forward_implications":["Under Assumptions 1–3, the $L^2$ error grows at most linearly in $T$, removing the exponential factor of the earlier BBMMI bound.","If $h$ is bounded away from zero and the coefficients $\\alpha_t(f)$ are summable, the error is at most $C_f/\\sqrt{N_{\\min}}$ uniformly in $T$.","For Brownian motion with drift killed on a bounded $C^2$ domain, the bound is $C\\sqrt{N_{\\max}}/N_{\\min}\\|f\\|_\\infty + \\mathbb E[C/(\\sqrt{N_0}\\,\\hat m_0(\\rho_D))]\\|f\\|_\\infty$, giving the $1/\\sqrt{N_{\\min}}$ rate.","In the Moran case $N_{\\min}=N_{\\max}$, uniform-in-time convergence follows without the generator and carré-du-champs regularity conditions used in earlier Moran-model results.","Example 4 shows the uniform bound genuinely needs the semigroup's normalized action to converge; without it, even $h>0$ does not save the estimate."],"supporting_citations":[{"why":"Defines the BBMMI and the $N_{\\min}{-}N_{\\max}$ model, proves the Feynman–Kac identity $\\mathbb E[\\Pi^A_T\\Pi^B_T m_T(f)] = m_0 Q_T(f)$, and supplies the exponential-in-$T$ bound whose proof Theorem 1 modifies.","marker":"[8]"},{"why":"Provides the reflected-Brownian-motion coupling for distance to the boundary in related particle systems; Theorem 4's domination coupling is asserted to follow the same ideas.","marker":"[31]"},{"why":"Supplies the quasi-stationary exponential-convergence criteria and the lower bound $\\delta_x Q_t 1_D \\ge c_0 \\rho_D(x)\\sup_y \\delta_y Q_{t-1}1_D$, used to verify the assumptions and get exponential decay of $\\alpha_t(f)$ for the Brownian case.","marker":"[3]"}],"fun_headline_variants":["Particle error bound: exponential becomes linear in time","From exponential to linear: sharper error bound for branching model","Moran branching error bound now linear, not exponential","Uniform time bound for binary branching with Moran interactions","Error bound improved: linear in T for Moran branching model"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"For the Brownian-motion application, the proof relies on a coupling between the $N_{\\min}{-}N_{\\max}$ process and independent jumping reflected Brownian motions such that the sum of boundary distances dominates their sum; the paper says the construction follows [31] and leaves it to the reader, so if that coupling fails, the uniform $1/\\sqrt{N_{\\min}}$ bound for this application is not supported.","fun_headline_variants_meta":{"raw":{"variants":["Particle error bound: exponential becomes linear in time","From exponential to linear: sharper error bound for branching model","Moran branching error bound now linear, not exponential","Uniform time bound for binary branching with Moran interactions","Error bound improved: linear in T for Moran branching model"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000179,"raw_usage":{"total_tokens":1283,"prompt_tokens":910,"completion_tokens":373,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":526,"completion_tokens_details":{"reasoning_tokens":296}},"tokens_in":526,"tokens_out":373,"duration_ms":3818,"temperature":1.0,"reasoning_tokens":296,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T00:18:19.900796+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the $N_{\\min}{-}N_{\\max}$ model for Brownian motion with drift in a bounded $C^2$ domain with small $N_{\\min}$, and test whether the summed distance-to-boundary of the $N_{\\min}$ tracked particles is stochastically dominated by the sum of independent jumping reflected Brownian motions with the stated jump-to-zero rate; a violation would break the coupling and with it the proof of Theorem 4.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the BBMMI and the $N_{\\min}{-}N_{\\max}$ model, proves the Feynman–Kac identity $\\mathbb E[\\Pi^A_T\\Pi^B_T m_T(f)] = m_0 Q_T(f)$, and supplies the exponential-in-$T$ bound whose proof Theorem 1 modifies."},{"cited_title":"Villemonais","cited_arxiv_id":null,"evidence_quote":"Provides the reflected-Brownian-motion coupling for distance to the boundary in related particle systems; Theorem 4's domination coupling is asserted to follow the same ideas."},{"cited_title":"Champagnat, K","cited_arxiv_id":null,"evidence_quote":"Supplies the quasi-stationary exponential-convergence criteria and the lower bound $\\delta_x Q_t 1_D \\ge c_0 \\rho_D(x)\\sup_y \\delta_y Q_{t-1}1_D$, used to verify the assumptions and get exponential decay of $\\alpha_t(f)$ for the Brownian case."}],"review_version":1}