{"id":"99db5425-c41c-4449-9eab-95907c90b2f6","arxiv_id":"2509.09274","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For contractive McKean-Vlasov SDEs with superlinear drift and diffusion, the projected Euler and backward Euler schemes converge in mean square at rate 1/2 uniformly in time.","lead":"This paper proves uniform-in-time mean-square convergence rates for a general class of one-step numerical methods for McKean-Vlasov SDEs with superlinear coefficients, and derives explicit order-1/2 rates for the projected Euler and backward Euler schemes. A generalist might read it because accurately simulating distribution-dependent SDEs over long times underpins mean-field games, particle methods, and Bayesian sampling.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 5.1 misstates Fournier–Guillin by a power-of-moment factor; the propagation-of-chaos theorem and hence the long-time convergence rates are not justified as written.","rationale":"The paper’s central claim is a two-step composition: uniform-in-time propagation of chaos (Theorem 3.1) plus uniform-in-time one-step discretization error (Theorem 3.3), combined in Theorem 3.4 and specialized to projected/backward Euler in Theorems 3.9/3.14. The weakest link in this chain is not the contractive gap K1>2K2 itself, which is an explicit, checkable hypothesis that the examples satisfy, but the technical correctness of the propagation-of-chaos proof. That proof relies on a quoted empirical-measure lemma whose displayed version has an incorrect moment scaling. Since the same flawed factor appears in Eq. (5.7) and in the definition of Υ(N), the proof of Theorem 3.1 is incomplete as written, and the subsequent long-time convergence results inherit the gap. The issue is concrete and verifiable, and an independent check against Fournier–Guillin settles it. It is not a fatal conceptual objection: with the corrected W_l(μ)^2 factor and with l chosen as 2q0, the uniform moment bounds appear sufficient to restore the argument. For that reason the appropriate status remains conditional rather than reject or accept: the main idea is sound but the manuscript needs a corrected Lemma 5.1 and a re-verification of the constants before the proof can be trusted. This agrees in direction with the reader’s conditional verdict, though the reader’s identified weakest assumption was the K1>2K2 condition rather than the lemma misstatement, so the agreement is partial.","tokens_in":58299,"tokens_out":20185,"duration_ms":223799,"concrete_test":"Check Lemma 5.1 against the cited Fournier–Guillin Theorem 1. For r=2, l=4, d=3 and μ_a = 0.5(δ_{-a}+δ_a), the left side E W_2^2(μ_N, μ_a) scales as a^2 for fixed N, while the displayed factor W_4(μ_a)^{2/4}=W_4(μ_a)^{1/2} scales as a^{1/2}; hence the inequality cannot hold for all a. After verifying the correct factor W_l(μ)^2 = (E|X|^l)^{2/l}, recompute (5.7)–(5.8) and Theorem 3.1 with the corrected Υ(N). If the resulting propagation-of-chaos constant is finite uniformly in t under Assumptions 2.1–2.4 for l=2q0, the long-time rates survive; if not, Theorem 3.1 and its corollaries fail.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 5.1, Lemma 5.1 claims E W_r^r(μ_N, μ) ≤ C W_l(μ)^{r/l} (N^{-1/2}+N^{-(l-r)/l}) (with log or N^{-r/d} variants). This is dimensionally wrong: if μ scales by c, then W_r^r scales as c^r, whereas W_l(μ)^{r/l} scales as c^{r/l}; the correct Fournier–Guillin coefficient is W_l(μ)^r, equivalently M_l(μ)^{r/l} where M_l is the l-th moment. The proof of Theorem 3.1 then uses the incorrect form in (5.7)–(5.8): it bounds E W_2^2(μ_t^{X_j}, μ_t^N) by C (E|X_t^j|^l)^{2/l^2} η(N), and defines Υ(N) with (E|X0|^l+1)^{2/l^2}. This propagation-of-chaos estimate is the term feeding Theorems 3.4, 3.9 and 3.14, so the central long-time convergence theorems are not proved as written. The defect is repairable—replacing the moment exponent by 2/l and checking that the resulting constant is finite under Assumptions 2.1–2.4 (e.g., with l=2q0) restores the argument—but this repair is currently absent.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a general framework for the long-time strong convergence of one-step time discretizations of McKean–Vlasov SDEs with superlinear growth coefficients. It first establishes an infinite-horizon propagation-of-chaos estimate for the interacting particle system under a contractive monotonicity condition with quantitative gap K1 > 2K2, together with uniform moment bounds. It then proves a general one-step error theorem (Theorem 3.3) that reduces uniform-in-time mean-square convergence to local weak and strong error checks plus uniform numerical moments. The framework is applied to the projected Euler scheme and to the backward Euler scheme, yielding a uniform mean-square rate of order 1/2 in the time step, up to the propagation-of-chaos term in N. Numerical experiments for two scalar McKean–Vlasov equations are reported as supporting evidence.","tokens_in":58624,"tokens_out":23458,"duration_ms":236593,"significance":"If the proofs are completed, the paper gives a useful unified route to long-time strong convergence for non-globally Lipschitz McKean–Vlasov SDEs. The general theorem is a genuine structural contribution: it separates the propagation-of-chaos argument from the numerical stability/moment argument and reduces each application to finite checks. The two worked schemes, projected Euler and backward Euler, are natural and the obtained rate 1/2 is consistent with the expected strong order. The paper is also honest about the assumptions, including the explicit K1 > 2K2 gap that makes the propagation of chaos uniform in time. However, as written there are several load-bearing gaps: the empirical-measure lemma is misstated, the strict hypothesis of the general theorem excludes the two applications, and the moment bookkeeping in the proof of Theorem 3.3 is incomplete and internally inconsistent with the stated exponents. These issues are repairable, but they affect the central claims and require a major revision.","major_comments":[{"comment":"Lemma 5.1 misstates the Fournier–Guillin empirical-measure estimate. The standard bound is E W_r^r(μ_N, μ) ≤ C_{r,l,d} W_l(μ)^r times the N-dependent factor, not W_l(μ)^{r/l}. The manuscript writes W_l(μ)^{r/l} in (5.1), and this erroneous exponent is propagated into Eq. (5.7), where E W_2^2(μ_t^{X_j}, μ_t^N) is bounded by C_{d,l} (E|X_t^j|^l)^{2/l^2} η(N), and into the definition of Υ(N) in Theorem 3.1. The correct coefficient is W_l(μ)^2, equivalently (E|X_t^j|^l)^{2/l}. Since Theorem 3.1 feeds Theorems 3.4, 3.9 and 3.14, the propagation-of-chaos estimates and all downstream rates are not justified as written. The defect is repairable by replacing 2/l^2 with 2/l under the finite l-th moment available from Assumptions 2.3-2.4, but the repair is absent.","section":"Section 5.1, Lemma 5.1 and Eqs. (5.7)-(5.8)"},{"comment":"The general theorem requires q2 ∈ (1/2, q1 - 1/2), i.e. q2 < q1 - 1/2. However the projected Euler scheme has q1 = 3/2 and q2 = 1 (Theorem 3.7, (3.21)-(3.22)), and the backward Euler scheme has the same values (Theorem 3.12, (3.29)-(3.30)). Thus q2 = q1 - 1/2, not q2 < q1 - 1/2. The proof of Theorem 3.3 uses only q1 ≥ q2 + 1/2 (see the line after (5.48)), so the strict inequality is not needed. As stated, Theorems 3.8 and 3.13 cannot be obtained from Theorem 3.3; the hypothesis should be weakened to q2 ≤ q1 - 1/2 (or q1 ≥ q2 + 1/2 with q2 > 1/2).","section":"Section 3.2, Theorem 3.3 (H2) and Theorems 3.7/3.12"},{"comment":"The moment exponent λ in Theorem 3.3 is not justified by the displayed estimates. After Young's inequality, the forcing term contains E[(1+|X_k|^{2κ-2}+|Y_k|^{2κ-2})^{1/2}(1+|Y_k|^{2η2})] h^{2q2}. Expanding this produces terms |Y_k|^{κ-1+2η2} and |X_k|^{κ-1}|Y_k|^{2η2}; bounding the latter by Cauchy–Schwarz/Hölder requires moments of |Y_k| up to order 4η2 and initial moments of order 4η2 η3, not just the advertised (κ−1/2+η2)η3. The concrete exponents in Theorems 3.8 and 3.13 (18κ+22 and 17κ+23) are also inconsistent with the formula for λ when η3=2, which is the value forced by the moment bounds in Theorems 3.6 and 3.11. The phrase 'combining (3.5), (3.12), and (5.49)' is therefore not sufficient as written. A complete Hölder/Young bookkeeping with the correct λ, or an explicit restriction on κ/q0 ensuring the needed moments, must be provided.","section":"Section 5.3, proof of Theorem 3.3, Eqs. (5.45)-(5.48) and Eq. (3.13)"},{"comment":"The proof reduces the sup over j to a single index by asserting that the joint sequence (X_k^{j,N}, Y_k^{j,N}) is identically distributed. This exchangeability of the one-step scheme is not a consequence of assumptions (H1)-(H3); it requires the local map Ψ^{j,N} in (3.6) to be symmetric with respect to the particle index. The projected and backward Euler schemes have this property, but the general theorem is stated for arbitrary one-step methods satisfying only (H2)-(H3). The theorem should either include an explicit symmetry/exchangeability condition or be restricted to schemes for which the asserted joint exchangeability holds.","section":"Section 3.2, Theorem 3.3 and proof around (5.39)"}],"minor_comments":[{"comment":"The statements allow any 2 < l ≤ q*, but Assumption 2.3 only guarantees E|X0|^{2q0} < ∞. Since the propagation-of-chaos constant contains (E|X0|^l+1)^{2/l} (after the correction to Lemma 5.1), the range should be restricted to l ≤ 2q0, or an additional finite l-th moment assumption should be stated. This does not affect the rate, since one can choose e.g. l=4 for d≤4 with l≠4, but the current 'any' is formally incorrect.","section":"Theorems 3.1, 3.4, 3.9, 3.14"},{"comment":"The implicit backward Euler scheme requires existence and uniqueness of the algebraic equation defining X̂_{k+1}^{i,N}. This follows from the strong monotonicity in Assumption 2.1, but it is not stated or proved. A short remark would make the scheme well-posed.","section":"Section 3.4, Eq. (3.25)"},{"comment":"The numerical experiments use T=16 and report rates visually from log-log plots. Quantitative slope estimates, confidence intervals, or a table of rates would make the validation more conclusive. Also, the theoretical claims are uniform in time, so presenting errors at a longer terminal time (e.g., T=64 or T=100) would better match the infinite-horizon statement.","section":"Section 4"},{"comment":"The definition of λ is given after Eq. (5.50) in the proof but is used in the theorem statement before the proof. The statement should define λ explicitly in the theorem, and the proof should verify that the same λ satisfies all moment requirements, including the cross terms discussed in the major comments.","section":"Theorem 3.3, Eq. (3.13)"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a timely and important problem, and the overall framework appears sound and likely repairable. The main obstacle is not the strategy but the rigor of the moment estimates and the exact hypotheses of the general theorem. I would encourage the editor to request a careful revision rather than reject, because the Lemma 5.1 exponent, the strict-inequality issue, and the λ bookkeeping are all localized fixes. The referee should ask the authors to provide a complete proof of Theorem 3.3 with all moment exponents tracked, and to align the statements of Theorems 3.8 and 3.13 with that proof."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing you should know: this paper is a serious candidate for the long-time strong convergence analysis of one-step methods for superlinear MV-SDEs, but as written the central propagation-of-chaos theorem rests on an incorrect statement of Fournier–Guillin. The error is a missing power of the moment functional; it propagates into the main convergence theorems. I think it is repairable, but it is not a typo in a footnote—it is load-bearing.\n\nWhat is actually new: a general one-step framework (Theorems 3.3-3.4) that reduces long-time mean-square convergence to local weak/strong error bounds plus uniform moment bounds, and explicit order-1/2 rates for projected Euler and backward Euler under non-globally Lipschitz drift. The long-time propagation of chaos (Theorem 3.1) under the contractive gap K1>2K2 is also a genuine extension of finite-time results. The framework is clean and reusable, and the moment-bound lemmas for the two schemes are worked out in detail. That is real substance.\n\nThe soft spot: Lemma 5.1 states E W_r^r(μ_N,μ) ≤ C W_l(μ)^{r/l} (N^{-1/2}+...). The correct Fournier–Guillin coefficient is M_l(μ)^{r/l} = W_l(μ)^r, not W_l(μ)^{r/l}. Check scaling under μ↦cμ: the left side scales as c^r, the paper's bound as c^{r/l}. The error enters (5.7) and produces the Υ(N) exponent 2/l^2 instead of 2/l. Consequently Theorem 3.1 and everything that feeds on it—Theorems 3.4, 3.9, 3.14—is not proved as written. The repair is straightforward: replace the exponent and verify the moment factor is finite under Assumptions 2.1–2.4 (e.g., l=2q0). But the repair is not in the manuscript, and the authors also assert priority over existing long-time results ([3],[44]) without reconciling their own claim with those papers. Numerical experiments support the rates but lack error bars and code, so they are illustrative only.\n\nBottom line: the framework is worth engaging with, and the flaw is fixable, not fundamental. If the authors repair Lemma 5.1 and the dependent bounds, this could be a solid paper. I'd send it to a referee who can check the repair, but I would not trust the current version's headline rates without seeing the corrected proof.","headline":"A useful long-time convergence framework for superlinear McKean–Vlasov SDEs, but the propagation-of-chaos proof currently rests on a misstated empirical-measure bound.","tokens_in":59128,"tokens_out":4652,"would_cite":false,"duration_ms":41579,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65C30","60H10","60H35"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves uniform-in-time mean-square convergence at rate 1/2 for two Euler-type schemes applied to McKean–Vlasov SDEs with superlinear growth.","keywords":["McKean-Vlasov SDEs","superlinear growth","uniform-in-time strong convergence","propagation of chaos","projected Euler scheme","backward Euler scheme","mean-square convergence","one-step methods"],"falsifier":"Run an MV-SDE satisfying the paper's assumptions but with K1 = 2K2, using the projected Euler scheme with a fixed step h, and compare mean-square errors at T = 10, 10^2, 10^3. If the error grows without bound with T, the theorem's uniform-in-time claim fails; the paper's bound instead predicts an error that stays of order h^(1/2) for all T.","tokens_in":58169,"feed_emoji":"🎲","tokens_out":9289,"duration_ms":107326,"temperature":0.7,"pith_summary":"McKean–Vlasov stochastic differential equations describe the motion of a typical particle in a large interacting system, and their coefficients depend on the distribution of the solution. When those coefficients grow faster than linearly, standard Euler methods can blow up, and long-time error control has been missing. The paper shows that a broad class of one-step methods, including a projected Euler scheme and a backward Euler scheme, have global mean-square error that is uniformly of order h^(1/2) over the whole infinite time horizon, plus a propagation-of-chaos error that decays as the number of simulated particles grows. The result matters because it makes long-run simulations of superlinear mean-field systems rigorous, without global Lipschitz assumptions on the drift.","feed_headline":"Euler schemes hit rate 1/2 uniformly in time for superlinear MV-SDEs","feed_subtitle":"Long-run simulations of superlinear mean-field particle systems now carry a rigorous rate-1/2 error bound.","key_machinery":"The machinery is a one-step recursion with contraction. At each step the global error is written as the difference of two exact flows starting from the true and the numerical state, plus the one-step local error. The contractive monotonicity assumption with K1 > K2 gives an exponential decay factor e^(-(K1-K2)h) for the flow-difference term, and a modified Gronwall inequality converts a recurrence with such a factor into a uniform-in-time bound. The projection operator is the second mechanism: it keeps numerical iterates within a bounded ball of radius h^(-1/(2(kappa+2))), preventing the superlinear drift from destroying moment bounds while perturbing the solution by only a small amount per","core_discovery":"The central discovery is a general decomposition of the long-time error into a propagation-of-chaos term and a discretization term, with both controlled uniformly in time. For any one-step scheme whose local weak error is of order q1>1, whose local strong error is of order q2 in (1/2, q1-1/2), and whose numerical solutions have uniform moment bounds, the paper proves a global L2 bound of order h^(q2-1/2) on every grid point, uniformly in the number of steps. It verifies the hypotheses for two schemes: the projected Euler scheme, which caps the state via the projection psi(x)=min{1,h^(-1/(2(kappa+2)))|x|^(-1)}x, and the backward Euler scheme with a globally Lipschitz diffusion coefficient. Fo","pith_inferences":["A natural extension the paper leaves implicit: the same framework should cover tamed, split-step, and adaptive Euler variants, provided one can verify the local-error and moment hypotheses; the proof structure is not tied to the two schemes analyzed.","The constants in the bounds grow like 1/(K1-2K2), so for nearly critical systems the uniform-in-time guarantee is real but practically weak; the paper's condition is also a stiffness indicator.","Because the error bound is uniform in time, approximating invariant measures and long-run statistics of superlinear MV-SDEs is a direct downstream use, although the paper does not explicitly discuss invariant measures.","The dimension-dependent propagation-of-chaos rate implies that in high dimension d>4, increasing N gives only N^(-2/d) improvement; practical high-dimensional simulation will likely require variance reduction or multilevel methods rather than just larger N."],"forward_implications":["Any one-step scheme satisfying the paper's three structural hypotheses, local weak order q1>1, local strong order q2>1/2, and uniform numerical moments, inherits a uniform-in-time mean-square error of order h^(q2-1/2).","The propagation-of-chaos bound is uniform over t>=0, so interacting-particle simulations of superlinear mean-field systems remain reliable for arbitrarily long horizons provided N is large enough.","The projected Euler scheme is a practical explicit method with a rigorous long-time rate 1/2 despite superlinear drift and diffusion.","The backward Euler scheme also attains rate 1/2, but only under an extra global Lipschitz condition on the diffusion coefficient, showing an asymmetry in the hypotheses needed by the two methods.","The paper's numerical experiments on polynomial and trigonometric mean-field examples report measured error slopes close to 1/2, consistent with the theoretical rates."],"fun_headline_variants":["MV-SDE solvers: rate-1/2 error bound uniformly in time","Superlinear mean-field SDEs: long-time strong rate 1/2","One-step schemes for MV-SDEs get long-time rate 1/2","Projected & backward Euler: uniform rate 1/2 for MV-SDEs","Rate-1/2 strong convergence on infinite horizon for MV-SDEs"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"Everything rests on the contractive monotonicity gap K1 > 2K2; without it the negative feedback that makes the particle system and its discretization contract toward the nonlinear law disappears, and the uniform-in-time error bounds no longer follow.","fun_headline_variants_meta":{"raw":{"variants":["MV-SDE solvers: rate-1/2 error bound uniformly in time","Superlinear mean-field SDEs: long-time strong rate 1/2","One-step schemes for MV-SDEs get long-time rate 1/2","Projected & backward Euler: uniform rate 1/2 for MV-SDEs","Rate-1/2 strong convergence on infinite horizon for MV-SDEs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001122,"raw_usage":{"total_tokens":4478,"prompt_tokens":694,"completion_tokens":3784,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":438,"completion_tokens_details":{"reasoning_tokens":3678}},"tokens_in":438,"tokens_out":3784,"duration_ms":23553,"temperature":1.0,"reasoning_tokens":3678,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T19:23:51.752660+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run an MV-SDE satisfying the paper's assumptions but with K1 = 2K2, using the projected Euler scheme with a fixed step h, and compare mean-square errors at T = 10, 10^2, 10^3. If the error grows without bound with T, the theorem's uniform-in-time claim fails; the paper's bound instead predicts an error that stays of order h^(1/2) for all T.","supporting_citations":[],"review_version":1}