{"id":"0e461725-c88b-46f1-8b04-ea80be862d7a","arxiv_id":"2501.13655","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The law of an ergodic McKean-Vlasov diffusion converges exponentially fast to the law of its linearization around the unique invariant measure, enabling simplified long-time inference.","lead":"This paper proves that the probability law of a nonlinear interacting-particle process gets exponentially close to that of a simpler linear process, once the interaction is frozen at the equilibrium distribution. This gives a formal license to use the simpler process for long-time tasks such as parameter estimation and computing diffusion coefficients.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 2.6's LSI constant is off by a factor of 4 under the paper's own definition (2.2), invalidating the explicit exponential bound in Theorem 2.7.","rationale":"The paper's central claim is the exponential closeness bound. On R^d the proof relies wholly on Lemma 2.6 to supply the time-uniform LSI constant Lambda for the Grönwall argument. I tested Lemma 2.6 in the exactly solvable OU case and found a factor-of-4 discrepancy with definition (2.2): the sharp LSI constant of N(0,1) is 2, not 1/2. The claimed constant is not merely non-sharp; it is not an admissible LSI constant, so the inequality I >= (4/Lambda) H used in (2.5) is invalid. The same pattern of factor error appears in the torus Lemma 2.18, where the Holley-Stroock multiplicative factor is underestimated. These errors do not necessarily destroy the qualitative exponential convergence, since the argument can likely be repaired with the correct constants, but they invalidate the quantitative bounds stated in Theorems 2.7 and 2.19 and their corollaries. The reader's weakest assumption concerned restrictive hypotheses and application gaps, not this internal inconsistency; hence I disagree with the reader's identification of the load-bearing point. The appropriate verdict remains CONDITIONAL: the paper should be accepted only after the LSI constants are corrected and the exponential rates are re-derived.","tokens_in":27825,"tokens_out":27287,"duration_ms":220395,"concrete_test":"Set V(x)=x^2/2 and W=0 with beta=1. Compute H(N(1,1)|N(0,1))=1/2 and I=1; the LSI (2.2) then forces lambda >= 2, while Lemma 2.6 gives lambda_infty=1/2. Then recompute Theorem 2.7 for an interacting example (W>0) with the corrected time-uniform constant Lambda = max{lambda0, 2/(beta(alpha+gamma))}: if the prefactor and rate in rho_Lambda(t) change materially, the theorem as stated is invalid.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Lemma 2.6 is false as stated. For the exactly solvable case V(x)=|x|^2/2, W=0, beta=1, alpha=1, gamma=0, the invariant measure is N(0,1). Under the paper's definition (2.2), H <= (lambda/4) I, the sharp LSI constant is lambda=2, since H(N(1,1)|N(0,1))=1/2 and I=1. Lemma 2.6 instead gives lambda_infty = 1/(2 beta (alpha+gamma)) = 1/2, which would imply the false inequality H <= (1/8) I. The correct limiting constant is 2/(beta(alpha+gamma)), a factor 4 larger. Consequently, the Grönwall step in Theorem 2.7 uses I >= (4/Lambda) H with Lambda too small, overstating the coefficient 2/(beta Lambda) in (2.5), so the stated exponential rate and prefactor are not justified. A similar factor error appears in Lemma 2.18 on the torus: from (2.20), sup phi / inf phi = kappa^2 Gamma^2, so Holley-Stroock should multiply the f_infty constant Gamma/(2 pi^2) by kappa^2 Gamma^2, yielding kappa^2 Gamma^3/(2 pi^2), not the stated kappa Gamma^2/(2 pi^2).","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the long-time comparison between the McKean SDE (1.2) and the linear diffusion (1.5) obtained by freezing the convolution at the invariant density f∞. Under convexity assumptions on R^d and a uniqueness/weak-interaction assumption on T^d, it claims exponential-in-time bounds on the relative entropy H(μt|νt) and the Wasserstein distance W2(μt,νt), with proofs based on relative-entropy estimates, LSIs, and Grönwall arguments. It then applies the linearization to construct a 'linearized' MLE for the McKean SDE (Theorem 3.3, a.s. convergence of the estimator) and to recover the diffusive-mean field CLT and invariance principle for the linearized process (Theorems 3.9 and 3.10), with numerical illustrations. I did not find a circularity in the MLE argument: the proof uses the independent L1 convergence of ft to f∞ from [41] and does not presuppose the exponential-closeness theorem.","tokens_in":28101,"tokens_out":25870,"duration_ms":231102,"significance":"If the quantitative claims hold, the paper's central message—that an ergodic McKean dynamics can be replaced at long times by a Markov diffusion with the same invariant measure—is useful and goes beyond the heuristic uses in [51] and [22]. The paper provides explicit rates, treats both R^d and T^d, adds a confining potential to the torus analysis, and gives a clean identifiability condition for the MLE application. The entropy-based proof architecture is transparent, and the numerical experiments usefully illustrate the applications. However, the explicit constants in the two main linearization theorems are not reliable as stated: Lemma 2.6 has a factor-4 error in the LSI constant, Lemma 2.18 has a missing factor in the Holley–Stroock step, and the torus exponential-convergence propositions require positivity of the rates ζ and η, which is not part of the standing assumptions. These issues can be repaired, but they affect the quantitative content of the central theorems.","major_comments":[{"comment":"The LSI constant in Lemma 2.6 is inconsistent with the convention (2.2). For the exactly solvable case V(x)=|x|^2/2, W=0, β=1, α=1, γ=0, the invariant measure is N(0,1). Taking μ=N(1,1) gives H(μ|ν)=1/2 and I(μ|ν)=1, so the sharp constant in (2.2) is λ=2. Lemma 2.6 instead gives Λ=1/2, which would imply the false bound H(μ|ν)≤I(μ|ν)/8. The correct asymptotic constant is 2/(β(α+γ)), a factor 4 larger. Since (2.5) uses I≥(4/Λ)H, the case split, rate, and prefactor in Theorem 2.7 and the Wasserstein bound in Corollary 2.8 are not justified as stated. The proof can be repaired by correcting the constant, but all quantitative claims in Section 2.2 need to be revisited.","section":"§2.2, Lemma 2.6 and Theorem 2.7"},{"comment":"The Holley–Stroock application after (2.20) drops a factor. From (2.20), sup φt / inf φt = κ^2Γ^2. Given the LSI constant Γ/(2π^2) for μ∞ from Lemma 2.12, Holley–Stroock applied to νt = φt μ∞ gives Ξ = κ^2Γ^3/(2π^2), not κΓ^2/(2π^2). Similarly, the bound for ψt in (2.27) gives sup ψt / inf ψt of order κ^2Γ^2 e^{Ci/ai}/(1-(Ci/ai)e^{Ci/ai}), so the stated constant ~Ξi has the same missing factor and also misses the exponential factor in the numerator. These constants propagate into Theorem 2.19 and Corollary 2.20. The qualitative exponential-closeness conclusion survives, but the explicit torus constants are wrong.","section":"§2.3.2, Lemma 2.18 and Theorem 2.19"},{"comment":"The statements that ft converges to f∞ exponentially in L2 and in relative entropy are not valid under Assumption 2.11 alone, because the rates ζ in (2.11) and η in (2.14) can be negative, e.g., for large ∥∇V∥∞ or ∥∆W∥∞. The proofs produce d/dt ≤ −ζ∥ft−f∞∥^2 and d/dt ≤ −ηH(ft|f∞), which only give useful bounds when ζ,η>0. Remark 2.16 acknowledges that positivity is a high-temperature phenomenon, but the formal statements of Propositions 2.14, 2.15, and Corollary 2.17, and hence Theorem 2.19, do not include it. The torus main theorem should either assume ζ,η>0 explicitly or be formulated under a smallness condition that guarantees them.","section":"§2.3.1, Propositions 2.14–2.15 and Corollary 2.17"}],"minor_comments":[{"comment":"In the estimate for I_T^(2), the factor e^{−αt/2} is written outside the time integral; the correct bound has the exponential inside the integral, and the convergence to zero then follows from the uniform moment bound (2.4).","section":"§3.1.1, proof of Lemma 3.5"},{"comment":"Theorem 3.3 proves strong consistency (almost sure convergence of ~θT to θ0), not asymptotic unbiasedness in the usual sense of convergence of expectations. The terminology 'asymptotically unbiased' appears in the abstract, introduction, and theorem statement; please align the wording.","section":"Theorem 3.3 and abstract"},{"comment":"If the torus has side length 1, the optimal Poincaré constant is 1/(4π^2) in the inequality ∥δ∥^2≤C∥∇δ∥^2, not 4/π^2; the constant 4/π^2 is valid but not optimal, so the phrase 'optimal constant' and the resulting rate ζ in (2.11) should be corrected or clarified.","section":"§2.3, Proposition 2.14"},{"comment":"The numerical experiments in Section 3.2.2 use the deterministic initial condition δ0, which does not satisfy the lower bound in (2.17). Please clarify that those simulations verify the CLT result and are not intended as numerical evidence for the torus entropy estimates of Theorems 2.19–2.20.","section":"§3.2, numerical experiments"},{"comment":"There are several typos: 'dynanics' in Section 1, 'Poison equation' should be 'Poisson equation' in Section 3.2, and 'Preperint' in reference [12].","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper has two concrete numerical-factor errors in the central estimates (Lemma 2.6 and Lemma 2.18) and a missing positivity condition for the torus convergence rates. These are not signs of a fundamentally flawed approach: the entropy/Grönwall architecture is sound, the qualitative exponential-closeness claims are likely correct after local corrections, and the applications are sensible. I recommend major revision rather than rejection. The authors should double-check whether the same convention mismatch affects constants imported from [43] and [33], and should state all positivity/smallness assumptions in the main theorems rather than in remarks."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core idea is good and mostly right: freezing the law at the invariant measure gives a linear diffusion that is exponentially close to the McKean process, and the entropy–Grönwall framework is the right tool. The torus extension with a confining potential and the linearized MLE are genuinely new, and the diffusive-limit section gives a clean re-derivation of known results. The paper deserves a serious referee, but it is not ready as is.\n\nThe main problem is concrete and checkable. Lemma 2.6 states an LSI constant that is off by a factor of 4 under the paper's own definition (2.2). Take V(x)=x^2/2, W=0, β=1, α=1. The invariant measure is N(0,1). For μ=N(1,1) and ν=N(0,1), H=1/2 and I=1, so the sharp constant is λ=2. Lemma 2.6 gives λ∞=1/(2β(α+γ))=1/2, which would imply H≤(1/8)I, false. The same factor error appears in Lemma 2.18: from the bounds on φ and ψ the Holley–Stroock ratio is κ^2Γ^2, not κΓ, and the denominator in ~Ξ is inverted. These contants feed directly into Theorems 2.7 and 2.19, so the explicit exponential rates and prefactors as printed are not justified. The qualitative exponential closeness almost certainly survives with corrected constants—this is a repair issue, not a fatal one—but the paper needs a real revision, not a tweak.\n\nThe other soft spots are the ones the authors mostly admit. The numerics run deterministic initial conditions, which violate the torus condition (2.17), and the bistable example sits outside the convexity assumptions, with only a conjecture covering it. Remark 3.4 explicitly omits the propagation-of-chaos step, so the MLE theory is for a single McKean path, not for actual particle-system data. The diffusive-limit section is a re-derivation, fine as an application but not a new hammer.\n\nWho gets value: anyone working on long-time behavior of McKean–Vlasov dynamics, inference for mean-field models, or homogenization of interacting diffusions. The methodology is worth engaging with, but only after the constants are corrected. A careful referee should ask for the Gaussian example to be run through Lemma 2.6 and for Lemma 2.18 to be re-derived from the displayed bounds. Send it to review; just do not take the printed rates at face value.","headline":"The linearization idea is sound and useful, but the LSI constants in Lemma 2.6 and 2.18 are wrong by factors, so the printed rates need correction before the theorems are usable.","tokens_in":28671,"tokens_out":5000,"would_cite":false,"duration_ms":45540,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["26D10","35Q70","35Q83","60J60","82C22"],"pacs":[],"model":"deepseek-v4-flash","headline":"A nonlinear McKean diffusion can be replaced, at long times, by a linear diffusion: their laws converge exponentially fast in relative entropy and Wasserstein distance.","keywords":["McKean–Vlasov PDE","linearization","logarithmic Sobolev inequality","relative entropy","interacting particle systems","invariant measure","maximum likelihood estimation","diffusive-mean field limit"],"falsifier":"Take the Desai–Zwanzig model on the torus below the phase transition with a deterministic initial condition (a point mass, as used in the numerics) and compute numerically the relative entropy between the nonlinear and linearized laws; if it does not decay exponentially, the assumption that initial densities are bounded away from zero is essential. Alternatively, run the linearized MLE on a single nonstationary path with a non-affine drift and check whether the estimator deviates from the true parameter at a rate slower than the theory predicts.","tokens_in":27611,"feed_emoji":"🎲","tokens_out":5672,"duration_ms":46212,"temperature":0.7,"pith_summary":"McKean stochastic differential equations describe the mean-field limit of many weakly interacting particles, but their law appears inside the drift, making the dynamics nonlinear and nonlocal. This paper shows that when such a system has a unique ergodic invariant measure, the long-time behavior is faithfully reproduced by a much simpler linear diffusion obtained by pinning the law at the invariant density. The main results quantify this: the relative entropy and Wasserstein distance between the nonlinear process and its linearization decay exponentially in time, on both the whole space and the torus. Because the linear process is an ordinary Markov diffusion with the same invariant measure, it can be used for statistical inference and for homogenization limits, and the paper proves that a maximum-likelihood estimator built from the linear likelihood remains asymptotically unbiased.","feed_headline":"A nonlinear SDE's law runs exponentially close to a linear one","feed_subtitle":"Replacing the law by the invariant measure provably reproduces long-time statistics, estimation, and homogenization limits.","key_machinery":"The engine of the proof is the entropy-production estimate $H(\\mu_t|\\nu_t) + \\frac{1}{2\\beta}\\int_0^t I(\\mu_s|\\nu_s)\\,ds \\le H(\\mu_0|\\nu_0) + \\frac{\\beta}{2}\\int_0^t \\int |\\nabla W * (\\mu_s - \\mu_\\infty)|^2 \\mu_s\\,dx\\,ds$, taken from entropy estimates for Fokker–Planck equations, combined with a logarithmic Sobolev inequality (LSI, $H\\le \\frac{\\lambda}{4}I$) for the Gibbs measure and for the time-marginals of both processes. The LSI converts the relative Fisher information into relative entropy, while the exponential $L^1$ relaxation of $f_t$ to $f_\\infty$, which follows from convexity on $\\mathbb{R}^d$ or from a new nonlinear-LSI argument on the torus, bounds the convolution error. Grönwall's inequality then yields the explicit exponential rates.","core_discovery":"The central discovery is the exponential closeness theorem: under uniform convexity of $V$ and $W$ on $\\mathbb{R}^d$ (or an H-stability/smallness condition on the torus with a confining potential), the law $\\mu_t$ of the McKean process $X_t$ and the law $\\nu_t$ of the linearized process $Y_t$, defined by $dY_t = -\\nabla V(Y_t)\\,dt - (\\nabla W * f_\\infty)(Y_t)\\,dt + \\sqrt{2\\beta^{-1}}\\,dB_t$, satisfy $H(\\mu_t|\\nu_t) \\le \\rho_\\Lambda(t)$ with $\\rho_\\Lambda(t)$ decaying exponentially, and $W_2(\\mu_t,\\nu_t) \\le \\sqrt{\\Lambda \\rho_\\Lambda(t)}$. On $\\mathbb{R}^d$ it also proves that $\\mathbb{E}[|X_t - Y_t|^2]$ decays at rate $e^{-\\alpha t/2}$ through a coupling argument. As applications, the linearized maximum-likelihood estimator is asymptotically unbiased, and the diffusive-mean field central limit theorem holds for the linearized process with the same covariance matrix $D = \\mathbb{E}_{\\phi_\\infty}[(I+\\nabla\\Phi)(I+\\nabla\\Phi)^\\top]$ as for the nonlinear process.","pith_inferences":["A practical diagnostic follows from the theorem: if a numerical simulation of the particle system shows that relative entropy to the linearized law does not decay exponentially, the system is likely in a multi-stable or phase-transition regime where linearization around a single invariant measure is not valid.","The linearized MLE requires only one trajectory, whereas the nonlinear likelihood needs the full mean-field expectation; this gives a concrete reduction in data requirements for real interacting-particle systems.","The torus smallness condition $C_i/a_i < W(1)$ involving the Lambert function could be tested numerically; finding the threshold where convergence breaks down would delimit the scope of the nonlinear LSI proof.","The same entropy-plus-LSI mechanism might extend to kinetic (underdamped) McKean equations and to multiple invariant measures selected by basin of attraction, but each extension needs new estimates beyond this paper's assumptions."],"forward_implications":["Long-time properties of the nonlinear process, such as asymptotic variances, first-passage quantities, and invariant functionals, can be computed from the simpler linearized Markov process.","The linearized MLE $\\tilde\\theta_T$, evaluated along a single path of the nonlinear SDE, is consistent, so parameter inference does not require observing the full particle system or the time-dependent expectation $\\mathbb{E}[X_t]$.","In the diffusive-mean field limit on the torus, the central limit theorem and invariance principle hold for the linearized process with the same diffusion matrix as the nonlinear process, confirming commutativity of the two limits in the unique-steady-state regime.","The explicit rates, depending on convexity parameters, inverse temperature, and LSI constants, tell the user in practice when the linearization is accurate at a given time horizon."],"supporting_citations":[{"why":"Provides the key entropy estimate (Lemma 3.1) and the Grönwall lemma (Lemma A.1) used to prove the exponential relative-entropy convergence in Theorems 2.7 and 2.19.","marker":"[39]"},{"why":"Establishes the exponential L1 convergence of the density to f∞ and uniform moment bounds under the convexity assumptions on R^d, giving the base rate in equation (2.3).","marker":"[41]"},{"why":"Supplies time-uniform log-Sobolev inequalities for the nonlinear and linear semigroups, used in Lemma 2.6 to obtain the LSI constant Λ for the linearized dynamics.","marker":"[43]"},{"why":"Gives the Holley–Stroock perturbation argument and the LSI for the invariant measure on the torus, used in Lemmas 2.12 and 2.18.","marker":"[33]"},{"why":"Proves strict convexity of the free energy on the torus under H-stability or small interaction, ensuring a unique invariant measure and the convergence-to-equilibrium framework extended here to nonzero confining potentials.","marker":"[14]"},{"why":"Provides the L2-convergence argument for a noisy consensus model on the torus that inspires Proposition 2.14.","marker":"[17]"},{"why":"Contains the Talagrand transportation inequality linking relative entropy to the 2-Wasserstein distance, converting the entropy bounds into the Wasserstein estimates in Corollaries 2.8 and 2.20.","marker":"[48]"},{"why":"Establishes the diffusive-mean field limit and the homogenization formula with the Poisson equation, which is reproduced for the linearized process in Section 3.2.","marker":"[22]"}],"fun_headline_variants":["McKean SDE laws converge exponentially to linearized ones","Exponential convergence of nonlinear McKean processes to linear","Linearized McKean SDEs match long-time behavior exponentially","Exponential closeness between nonlinear and linear McKean laws","McKean SDEs linearized: exponential law convergence, applications"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on the system having a unique invariant measure and on conditions that force exponentially fast relaxation to it—uniform convexity on the whole space, and on the torus, initial densities bounded away from zero together with a smallness bound on the interaction.","fun_headline_variants_meta":{"raw":{"variants":["McKean SDE laws converge exponentially to linearized ones","Exponential convergence of nonlinear McKean processes to linear","Linearized McKean SDEs match long-time behavior exponentially","Exponential closeness between nonlinear and linear McKean laws","McKean SDEs linearized: exponential law convergence, applications"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000675,"raw_usage":{"total_tokens":3094,"prompt_tokens":992,"completion_tokens":2102,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":608,"completion_tokens_details":{"reasoning_tokens":2015}},"tokens_in":608,"tokens_out":2102,"duration_ms":13421,"temperature":1.0,"reasoning_tokens":2015,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T15:44:51.727690+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the Desai–Zwanzig model on the torus below the phase transition with a deterministic initial condition (a point mass, as used in the numerics) and compute numerically the relative entropy between the nonlinear and linearized laws; if it does not decay exponentially, the assumption that initial densities are bounded away from zero is essential. Alternatively, run the linearized MLE on a single nonstationary path with a non-affine drift and check whether the estimator deviates from the true parameter at a rate slower than the theory predicts.","supporting_citations":[{"cited_title":"Lacker and L","cited_arxiv_id":null,"evidence_quote":"Provides the key entropy estimate (Lemma 3.1) and the Grönwall lemma (Lemma A.1) used to prove the exponential relative-entropy convergence in Theorems 2.7 and 2.19."},{"cited_title":"Malrieu, Logarithmic Sobolev inequalities for some nonlinear PDE’s, Stochastic Process","cited_arxiv_id":null,"evidence_quote":"Establishes the exponential L1 convergence of the density to f∞ and uniform moment bounds under the convexity assumptions on R^d, giving the base rate in equation (2.3)."},{"cited_title":"Guillin, P","cited_arxiv_id":null,"evidence_quote":"Gives the Holley–Stroock perturbation argument and the LSI for the invariant measure on the torus, used in Lemmas 2.12 and 2.18."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves strict convexity of the free energy on the torus under H-stability or small interaction, ensuring a unique invariant measure and the convergence-to-equilibrium framework extended here to nonzero confining potentials."},{"cited_title":"Chazelle, Q","cited_arxiv_id":null,"evidence_quote":"Provides the L2-convergence argument for a noisy consensus model on the torus that inspires Proposition 2.14."},{"cited_title":"Otto and C","cited_arxiv_id":null,"evidence_quote":"Contains the Talagrand transportation inequality linking relative entropy to the 2-Wasserstein distance, converting the entropy bounds into the Wasserstein estimates in Corollaries 2.8 and 2.20."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the diffusive-mean field limit and the homogenization formula with the Poisson equation, which is reproduced for the linearized process in Section 3.2."}],"review_version":1}