{"id":"3f42512e-9306-42f7-bfbc-46ebd031974b","arxiv_id":"2412.13430","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For fully coupled multi-scale McKean-Vlasov equations with irregular coefficients, the slow process converges to an explicit averaged equation and the fast process's law converges to an averaged invariant distribution, with rates.","lead":"This paper proves a new averaging theorem for two-timescale stochastic systems in which the slow and fast components both depend on each other's probability distributions. It identifies the fast motion's limiting distribution, a result the authors say is new even for classical multi-scale stochastic equations.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.3 defines its limiting objects through the unique invariant measure of the frozen McKean-Vlasov equation (1.16), but Remark 1.4(i) postpones the uniqueness proof; without it the averaged coefficients and the fast limit are not well-defined.","rationale":"The reader's weakest_assumption identifies the missing uniqueness proof, and this is indeed the most load-bearing issue. Theorem 1.3's target equation and the fast-motion limit are defined through zeta^{x,mu}; if this invariant measure is not unique, the theorem's statement is ill-posed. This is structurally distinct from the convergence proof: the approximation scheme may select one invariant measure, but the claimed limit is 'the' unique one. The citation to [51] likely provides the needed argument, but since it is not included in the present article, conditional acceptance is appropriate. The secondary concerns mentioned by the reader, such as the mutual reference between Lemma 6.4 and Lemma 6.5 and the recursive constants in Theorem 6.1, appear manageable: the convergence part of Lemma 6.4 does not depend on Lemma 6.5, and the uniform-in-n estimates are obtained by the indicated contraction argument. No internal inconsistency was found beyond the missing uniqueness proof. The verdict remains unchanged, conditional on supplying the uniqueness argument for the frozen invariant measure.","tokens_in":77696,"tokens_out":8329,"duration_ms":72757,"concrete_test":"Verify uniqueness of the invariant measure of (1.16) under (H1)-(H2) by writing out the adaptation of [51, Theorem 3.1]: define the map mu -> zeta^{x,mu} via the stationary McKean-Vlasov equation, and check that the weighted total variation contraction constant is strictly less than 1 using (1.21) with the stated small-kappa condition and V = 1 + |y|^p. If the contraction argument goes through, the concern is resolved. If a missing step or a counterexample is found, for instance a double-well potential within (H1)-(H2) admitting two stationary measures, then Theorem 1.3 must be reformulated to specify which invariant measure is selected.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central statement Theorem 1.3 and the definitions (1.15)-(1.17) are formulated in terms of the unique invariant measure zeta^{x,mu} of the frozen McKean-Vlasov equation (1.16). Remark 1.4(i) explicitly states that the uniqueness proof is omitted and postponed to another work, citing [51, Theorem 3.1]. This is not a cosmetic gap: Lemma 6.4 only constructs an invariant measure zeta^{x,mu} as a limit along the Picard sequence mu_n and proves that it satisfies the stationary equation; it does not show that this limit is independent of the approximating sequence or that no other invariant measure exists. The contraction estimate (6.25) and the regularity result Lemma 6.5 rely on the smallness of kappa in (1.21), which is exactly the mechanism that should yield uniqueness, but the verification is absent. If (1.16) admits multiple invariant measures, the phase-transition phenomenon acknowledged in Remark 1.2, then bar b, bar sigma, and tilde zeta in (1.15)-(1.17) depend on the choice of zeta, the averaged equation (1.14) is ambiguous, and the convergence claims in Theorem 1.3 do not have a well-defined target. Thus the main theorem is conditionally well-posed only if the postponed uniqueness argument is supplied.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a non-autonomous approximation method for the fully coupled multi-scale McKean-Vlasov system (1.1), where the coefficients of both the slow and fast equations depend on the distributions of both components. The main theorem (Theorem 1.3) identifies the averaged coefficients in (1.15) through the invariant measure of the frozen McKean-Vlasov equation (1.16), proves convergence in law of the slow process with rate ε^{α/2}, characterizes the limit of the fast law by (1.17) with rate ε^{α/2}+e^{-γ0 t/ε}, and, under an extra regularity assumption on σ, proves strong convergence of the slow process with rate ε^{α∧1}. The method replaces the non-linear system by a sequence of linear non-autonomous systems, analyzed via Kolmogorov equations on Wasserstein space, Poisson equations with parameters, and a mollification procedure on the space of measures.","tokens_in":77992,"tokens_out":13732,"duration_ms":114277,"significance":"If the results are correct, the paper is a substantial advance: it addresses an open problem recorded in [3, Section 7] for fully coupled multi-scale McKean-Vlasov systems, introduces a technique that avoids mean-field PDEs, and provides what appears to be the first characterization of the fast-motion limit for such systems, with rates matching the classical Itô SDE case. The paper contains extensive original technical machinery, including optimal regularity estimates for backward and forward Kolmogorov equations on Wasserstein space, Poisson-equation analysis for measure-dependent coefficients, and quantitative mollification with explicit rates. However, the central theorem is conditional on an unproved uniqueness statement, and one auxiliary pair of lemmas relies on a mutual reference for a key convergence argument. These gaps must be repaired before the claims can be accepted.","major_comments":[{"comment":"Theorem 1.3 and definitions (1.15)-(1.17) are stated in terms of 'the unique invariant measure' ζ^{x,μ} of the frozen McKean-Vlasov equation (1.16). Remark 1.4(i) explicitly postpones the proof of uniqueness, citing [51, Theorem 3.1] and saying the property is not needed in the proof. This is not a cosmetic omission: Lemma 6.4 only constructs an invariant measure as a cluster point of the approximating sequence and verifies the stationary equation, but it does not show that the limit is independent of the approximating sequence or that no other invariant measure exists. Since (H2) includes the small-κ condition (1.21), which is exactly the mechanism that should yield a contraction in the weighted total variation metric, the missing argument is likely repairable; but as it stands, the averaged coefficients \\bar b, \\bar σ in (1.15) and the fast limit \\tilde ζ in (1.17) are not rigorously well-defined objects. The theorem is therefore conditional on a postponed result.","section":"Section 1.3, Remark 1.4(i), Lemma 6.4"},{"comment":"The proof of Lemma 6.4 invokes Lemma 6.5 for the α-Hölder regularity of \\bar f(x)=∫ f(y)ζ^{x,μ}(dy), in order to control a ρ_V term by ρ_{α,M}(μ_{n-1},μ). Lemma 6.5, in turn, proves the regularity of averaged coefficients by an approximation argument whose convergence step is justified by the phrase 'by exactly the same procedure as in Lemma 6.4'. This creates a mutual reference: the convergence of the fixed-μ approximating measures ζ̂^{x,μ}_n to ζ^{x,μ}, which is the backbone of Lemma 6.5, is not proved in the text, and it is precisely the type of contraction/uniqueness argument that would also supply the missing uniqueness for (1.16). As written, the two lemmas do not independently establish their claims.","section":"Lemmas 6.4 and 6.5"}],"minor_comments":[{"comment":"The sentence 'since we will not need this property in our proof' is misleading: the theorem statement itself uses uniqueness to define the limit objects, so the property is needed for well-posedness even if it is not used directly in the convergence estimates.","section":"Remark 1.4(i)"},{"comment":"The definition of the spaces C^{(2,α)}_b and C^{(2,β)}_p does not explicitly state that all lower-order functional derivatives are bounded; the proofs, e.g. in Lemma 3.2, rely on boundedness of the first-order derivative. This should be clarified.","section":"Notation section"},{"comment":"The mollification of V on P_2(R^{d1}×R^{d2}) is defined by convolving only the first marginal with ρ_n^2, but Lemma 4.4 is stated for functions on P_2(R^d). The adaptation to product measures and the justification of the resulting estimates need at least a brief explanation.","section":"Section 5.3, proof of Theorem 5.1(iii)"},{"comment":"There are occasional typographical errors, for example 'excepted' where 'expected' is meant in Section 1.1(iii).","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper is technically deep and likely within the scope of the journal if the authors can supply the deferred uniqueness proof for the frozen McKean-Vlasov equation and resolve the mutual reference between Lemmas 6.4 and 6.5. The current version is not acceptable as is, because Theorem 1.3's limiting objects are not well-defined without uniqueness, and the regularity of the averaged coefficients depends on an unproved convergence argument."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Candid take: this is a substantial paper, possibly the right solution to a real open problem, but Theorem 1.3 currently rests on a uniqueness statement the authors explicitly postpone. That is a load-bearing gap, not a cosmetic one.\n\nWhat is new: the fully coupled multiscale McKean-Vlasov problem, with fast coefficients depending on slow state and both marginals, was open. The non-autonomous approximation method is a genuine idea: turn the nonlinear system into a sequence of linear non-autonomous SDEs, prove averaging for those, then pass to the limit. The averaged coefficients (1.15) and the fast-law limit (1.17) have a structure nobody expected, with integrals over the slow law. The rates are sharp and match the classical case; the fast-motion limit appears to be new even for classical two-scale SDEs. The PDE machinery (Kolmogorov equations on Wasserstein space, Poisson equation, mollification) is developed in impressive detail.\n\nSoft spots. The main one: Theorem 1.3 and the definitions (1.15)-(1.17) are stated for 'the unique invariant measure' zeta^{x,mu} of the frozen McKean-Vlasov equation (1.16). Remark 1.4(i) says the uniqueness proof is omitted and postponed to another work. But without uniqueness, the averaged coefficients and the fast limit are not well-defined. Lemma 6.4 only produces an invariant measure as a limit along a Picard sequence; it does not prove the limit is independent of the sequence or that there are no others. The small-kappa condition (1.21) is the mechanism that should imply uniqueness, but the verification is absent. So the main theorem is conditional on a theorem that does not yet exist.\n\nA second, smaller issue: there is a mutual reference around the n to infinity step. Lemma 6.4 invokes Lemma 6.5 to get Holder regularity of the averaged coefficients, while Lemma 6.5's proof uses the convergence of invariant measures that Lemma 6.4 establishes. The dependency might be disentangled by a simultaneous induction, but as written it is circular.\n\nAlso, the paper relies on well-posedness and uniqueness for the original McKean-Vlasov system, citing [11,23]; that is standard and not a concern. The citation pattern is fine.\n\nBottom line: this is a serious paper that deserves a serious referee. The referee should demand the missing uniqueness proof, or at least a clear reduction to [51, Theorem 3.1] with all conditions verified. The mutual reference needs to be untangled. If those are fixed, the result is likely publishable in a top journal. Address: major revision, conditional acceptance.","headline":"Load-bearing uniqueness claim postponed; otherwise a serious and novel contribution to fully coupled multiscale McKean-Vlasov averaging.","tokens_in":78529,"tokens_out":3373,"would_cite":true,"duration_ms":33736,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60F15","60H50","70K70","70K65"],"pacs":[],"model":"deepseek-v4-flash","headline":"As the time-scale separation vanishes, a fully coupled multiscale McKean-Vlasov system converges to an averaged equation whose coefficients mix frozen invariant measures over the slow law, with explicit rates.","keywords":["multi-scale McKean-Vlasov SDEs","averaging principle","non-autonomous approximation","invariant measure","Wasserstein space","Kolmogorov equation","convergence rates","Hölder continuous coefficients"],"falsifier":"Construct a system satisfying (H1)-(H2) for which the frozen equation (1.16) has two distinct invariant measures for the same parameters; then the averaged coefficients in (1.15) are not uniquely determined and the equations (1.14) and (1.17) are ambiguous. A direct route is to take a double-well confining potential and an interaction term strong enough to violate the small-κ condition in (1.21), producing the known phase-transition multiplicity for McKean-Vlasov invariant measures.","tokens_in":77482,"feed_emoji":"🎲","tokens_out":7210,"duration_ms":63356,"temperature":0.7,"pith_summary":"The paper settles, for irregular coefficients, the averaging problem for a two-scale mean-field stochastic system in which the slow and fast components and their distributions all interact. It proves that as the time-scale ratio ε tends to zero the slow process converges to a McKean-Vlasov equation whose drift and diffusion are averaged against a frozen invariant measure, and it identifies the limiting law of the fast process as the mixture of those invariant measures under the law of the slow limit. The rates are $ε^{{α/2}}$ for convergence in law and ε^α in L² for the slow process, where α is the Hölder exponent in the slow variable. The method replaces the nonlinear system by a sequence of linear but non-autonomous two-scale systems, avoiding mean-field PDEs and showing that strong convergence follows from weak convergence.","feed_headline":"Fast motion's invariant law pins down the two-scale mean-field limit","feed_subtitle":"A fully coupled McKean-Vlasov system averages to a limit whose coefficients mix frozen invariant measures, with sharp rates.","key_machinery":"The central mechanism is the non-autonomous approximation method: at each step n the nonlinear system is replaced by a linear non-autonomous two-scale system whose coefficients depend only on the distributions produced by the previous step, so each approximation is a classical SDE. The analysis of these linear systems rests on Kolmogorov equations on Wasserstein space, in particular a forward Kolmogorov equation on the product space P2(Rd1 × Rd2) whose solution decays exponentially in time, together with Poisson-equation estimates and a mollifying approximation on P2 with explicit rates. These estimates control the limit of the fast law and the coefficient differences between successive approximations, and the final limit is obtained by letting the approximation index go to infinity before ε goes to zero.","core_discovery":"Theorem 1.3 is the central claim: under non-degeneracy (H1) and dissipativity (H2), with coefficients Hölder-α in the slow space variable and Hölder-β in the fast variable and its law, the slow process X^ε_t converges to the McKean-Vlasov equation (1.14) with averaged coefficients defined in (1.15), and the law of the fast process converges to the mixture E $ζ^{{\\bar X_t, L\\bar X_t}}$ defined by (1.17). A distinctive feature is that the frozen equation (1.16) is itself a McKean-Vlasov equation, because the fast coefficients depend on the fast law, and the averaged coefficients contain an integral over the slow law of the frozen invariant measures; the paper argues that the formally guessed frozen equation without this integral gives the wrong limit. The paper also claims that the fast-law characterization is new even for classical two-scale Itô SDEs, and that the proof derives strong convergence directly from weak convergence by viewing the nonlinear system as a linear non-autonomous one.","pith_inferences":["Editorial extension: the proof's mechanism of freezing the distributions to convert a nonlinear system into a linear non-autonomous one suggests that, in other fully coupled mean-field systems where only weak convergence is available, strong convergence may follow by the same freeze-then-average step; the paper does not state this extension.","Editorial extension: since the rates are independent of β, one would expect the qualitative averaging limit to persist under very rough fast coefficients; the estimates support this but do not take β to zero.","Editorial extension: the mixture formula for the fast law may offer a route to propagation-of-chaos statements for finite-particle two-scale systems, where the empirical fast measure should converge to the same mixture; this is not treated in the paper."],"forward_implications":["For any fixed horizon, the law of the slow process converges to the law of the averaged McKean-Vlasov equation at rate ε^{α/2} for test functions in C^{(2,α)}_b(P2(Rd1)).","Under the extra regularity assumption on σ, the slow process itself converges in mean square at rate ε^α.","For every fixed t>0, the law of the fast process converges to the mixture E ζ^{\\bar X_t,L\\bar X_t} at rate ε^{α/2}+e^{-γ_0 t/ε}, with constants independent of time.","The convergence rates depend only on the Hölder exponent α in the slow variables and not on β, the regularity with respect to the fast variables and their distributions.","Because the result includes convergence of nonlinear test functions of the fast distribution, it is strictly broader than the classical weak convergence statement for multi-scale Itô SDEs."],"supporting_citations":[{"why":"States the fully coupled averaging problem as open, the benchmark this paper resolves.","marker":"[3]"},{"why":"Supplies well-posedness for the nonlinear system and the Wasserstein-space PDE machinery used throughout.","marker":"[11]"},{"why":"Provides the fluctuation estimates and Poisson-equation technique for classical fully coupled systems that the proof adapts.","marker":"[44]"},{"why":"Gives the classical averaging rates and coefficient-regularity results that the new rates match.","marker":"[45]"},{"why":"Provides the V-exponential ergodicity used for the frozen SDEs.","marker":"[53]"},{"why":"Establishes existence of invariant measures for dissipative distribution-dependent SDEs, used for the frozen equation.","marker":"[54]"},{"why":"The paper models its deferred uniqueness proof for the frozen invariant measure on this theorem.","marker":"[51]"}],"fun_headline_variants":["Fast law's invariant measure sets the whole limit","Non-autonomous trick: fast law governs the averaged system","Open problem solved: coupled mean-field averaging","New perspective: fast motion dictates the asymptotic limit","Sharp rates for fully coupled McKean–Vlasov limits"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The limit objects are defined through the unique invariant measure of the frozen nonlinear equation (1.16), and the paper states that the proof of this uniqueness is postponed to another article, so the theorem is only as solid as that deferred uniqueness claim.","fun_headline_variants_meta":{"raw":{"variants":["Fast law's invariant measure sets the whole limit","Non-autonomous trick: fast law governs the averaged system","Open problem solved: coupled mean-field averaging","New perspective: fast motion dictates the asymptotic limit","Sharp rates for fully coupled McKean–Vlasov limits"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000291,"raw_usage":{"total_tokens":1770,"prompt_tokens":1085,"completion_tokens":685,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":701,"completion_tokens_details":{"reasoning_tokens":609}},"tokens_in":701,"tokens_out":685,"duration_ms":7369,"temperature":1.0,"reasoning_tokens":609,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T13:09:03.450793+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a system satisfying (H1)-(H2) for which the frozen equation (1.16) has two distinct invariant measures for the same parameters; then the averaged coefficients in (1.15) are not uniquely determined and the equations (1.14) and (1.17) are ambiguous. A direct route is to take a double-well confining potential and an interaction term strong enough to violate the small-κ condition in (1.21), producing the known phase-transition multiplicity for McKean-Vlasov invariant measures.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"States the fully coupled averaging problem as open, the benchmark this paper resolves."},{"cited_title":"Chaudru de Raynal and N","cited_arxiv_id":null,"evidence_quote":"Supplies well-posedness for the nonlinear system and the Wasserstein-space PDE machinery used throughout."},{"cited_title":"R¨ ockner and L","cited_arxiv_id":null,"evidence_quote":"Provides the fluctuation estimates and Poisson-equation technique for classical fully coupled systems that the proof adapts."},{"cited_title":"R¨ ockner and L","cited_arxiv_id":null,"evidence_quote":"Gives the classical averaging rates and coefficient-regularity results that the new rates match."},{"cited_title":"Xie and X","cited_arxiv_id":null,"evidence_quote":"Provides the V-exponential ergodicity used for the frozen SDEs."},{"cited_title":"Zhang: Existence and non-uniqueness of stationary distrib utions for distribution dependent SDEs","cited_arxiv_id":null,"evidence_quote":"Establishes existence of invariant measures for dissipative distribution-dependent SDEs, used for the frozen equation."},{"cited_title":"Wang: Exponential ergodicity for singular reflecting McKe an-Vlasov SDEs","cited_arxiv_id":null,"evidence_quote":"The paper models its deferred uniqueness proof for the frozen invariant measure on this theorem."}],"review_version":1}