{"id":"952bc825-c371-4907-9aeb-f08f85b31b5e","arxiv_id":"2412.15528","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Pullback measure attractors exist uniquely for non-autonomous McKean-Vlasov stochastic delay lattice systems, are singleton and exponentially mixing under stronger damping, and are upper semicontinuous as distribution dependence vanishes.","lead":"Randomly forced lattice equations whose interactions depend on the distribution of the solution are shown to have well-defined long-time statistical attractors, including when the dynamics has memory delay. The paper also proves these attractors become a single exponentially mixing invariant measure under stronger damping, and that they converge as the distribution dependence is removed.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 5.3's compactness claim is false: the set Zδ is bounded and finite-support but not equicontinuous, so tightness and hence Theorem 5.1 are not established.","rationale":"The reader's weakest_assumption was the large damping condition (2.20), which is an explicit scope hypothesis rather than a proof gap. The reader did note a minor tail-bound slip in Lemma 5.3, but the more serious defect is the false compactness of the set Zδ used in the tightness argument. Because the claimed Arzelà-Ascoli compactness is invalid, the proof of D-pullback asymptotic compactness is incomplete, and without it Theorem 5.1 does not follow as written. This is a correctness concern rather than a matter of consensus or parameter choice. It may be repairable by adding uniform time-regularity estimates for solution segments, but those estimates are not present in the paper. Since the verdict CONDITIONAL already signals that the proof needs correction, I do not change the verdict; however, the required condition should now include a correct tightness proof, not only the minor Lemma 5.3 slip identified by the reader.","tokens_in":50914,"tokens_out":15031,"duration_ms":122046,"concrete_test":"Exhibit the counterexample sequence w_n(t) = sin(nt)e_0, with m = 1 so that the nonzero coordinate lies in |i| ≤ 2m and R = 1. Each w_n belongs to the set Zδ as defined in Lemma 5.3, but ‖w_n − w_m‖Cr ≥ |sin(nt_0) − sin(mt_0)| for some t0 where the difference is large, so no uniformly convergent subsequence exists. This disproves the claimed compactness of Zδ. To settle whether tightness can be repaired, derive a uniform modulus of continuity for the solution segments, e.g. E sup_{|h|≤δ} ‖u(t+h) − u(t)‖^2 ≤ C δ^α for some α > 0, and redo Lemma 5.3 using compact sets of the form {‖w‖Cr ≤ R, finite support, uniform equicontinuity}.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Lemma 5.3, the tightness proof rests on the claim that Zδ = {w ∈ Cr : ‖w‖Cr ≤ R(δ,τ), wi(t) = 0 for |i| > 2m} is compact in Cr. The proof argues via Arzelà-Ascoli that each individual w ∈ Zδ has uniformly continuous coordinates, but equicontinuity must be uniform over the entire family. It is not: the sequence w_n(t) = sin(nt)e_0 (padded with zeros, with m large enough so that the support lies in |i| ≤ 2m) satisfies ‖w_n‖Cr ≤ 1 and has finite support, yet no subsequence converges in C([-r,0], ℓ2) because w_n ↛ w uniformly; indeed the family is not equicontinuous. Thus Zδ is not compact, and the covering argument in (5.18)–(5.20) fails to establish tightness of {Lu_τ(·, τ−t_n, v_n)}. This is load-bearing because D-pullback asymptotic compactness (Definition 2.3) and Theorem 5.1 depend directly on Lemma 5.3. No alternative time-regularity estimate, such as a Kolmogorov-type bound on E‖u(t+h) − u(t)‖^2, is provided, so the missing equicontinuity cannot be supplied from the existing estimates. The reader's flagged tail-bound slip in Lemma 5.3 is real but secondary; the invalid compactness argument is the central gap.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the non-autonomous McKean-Vlasov stochastic delay lattice system (1.1) on Z. It proves well-posedness of strong and weak solutions, defines the non-autonomous dynamical system on probability measures induced by solution segments, and claims existence and uniqueness of a D-pullback measure attractor under a large-damping condition (2.20). It then derives singleton attractors and exponential mixing under an additional condition (6.1), and upper semicontinuity of attractors as the distribution dependence vanishes. The proofs rely on uniform tail estimates, fourth-moment estimates, and an Arzelà-Ascoli/tightness argument in C([-r,0],ℓ2).","tokens_in":51158,"tokens_out":12155,"duration_ms":107924,"significance":"If the main existence theorem is established, the paper would provide a useful framework for McKean-Vlasov stochastic delay lattice systems, with explicit checkable assumptions and an explicit exponential mixing rate. The strengths include the detailed well-posedness argument, the explicit large-damping conditions (2.20) and (6.1), the absence of fitted parameters, and the clear statement of upper semicontinuity under the convergence hypotheses (7.1)-(7.2). However, the central tightness proof in Lemma 5.3 contains a false compactness assertion, so Theorem 5.1 and the subsequent results are not established as written.","major_comments":[{"comment":"The set Zδ = {w ∈ Cr : ||w||_Cr ≤ R(δ,τ), w_i(t)=0 for |i|>2m} is claimed to be compact in Cr, but this is false. Arzelà-Ascoli requires equicontinuity uniform over the whole family; a uniform sup-norm bound and finite support do not imply it. The sequence w_n(t)=sin(nt)e_0 (with m fixed large enough) lies in Zδ and has no convergent subsequence in C([-r,0],ℓ2), since the family is not equicontinuous. Pointwise uniform continuity of each w is not enough. Consequently the covering argument (5.18)-(5.20) does not establish tightness of {Lu_τ(·,τ-t_n,v_n)}, and D-pullback asymptotic compactness (Definition 2.3), hence Theorem 5.1, is not proved. A uniform-in-n modulus-of-continuity estimate for the solution segments would be needed to repair this gap.","section":"Lemma 5.3"},{"comment":"The tail estimate (5.16) controls sum_{|i|≥2m0} |u_i|^2, whereas (5.17) bounds ||θ_{m0}u_τ||^2. The cutoff θ_{m0} also charges the annulus m0 ≤ |i| < 2m0, so (5.17) does not follow from (5.16) as written. One should apply Lemma 4.3 at a level that controls the m0-tail, or replace θ_{m0} by θ_{2m0} and adjust the finite-support set accordingly. This is a local slip, but it occurs inside the same lemma that contains the compactness gap.","section":"Eqs. (5.16)-(5.17)"},{"comment":"In deriving the fourth-moment estimate, the term 12∫ e^{2εs}||κ(s)||^2 E||u(s)||^2 ds is absorbed into a multiple of ∫ e^{2εs} E||u(s)||^4 ds without justification. The displayed inequality does not follow from (2.15) as printed; a Hölder/Young estimate controlling ∫ e^{2εs} E||u||^2 over [τ-t,τ] is needed. Since (4.26) is the fourth-moment estimate behind Lemma 5.2 and Theorem 5.1, this also requires correction.","section":"Lemma 4.4, Eq. (4.30)"}],"minor_comments":[{"comment":"The conclusion 'υ ∈ P4(H)' should read 'υ ∈ P4(Cr)'.","section":"Lemma 5.3 (end)"},{"comment":"The phrase 'Following (2.20) and Lemma 4.3' should refer to Lemma 4.2, since the preceding estimate uses the uniform L2 bound from Lemma 4.2.","section":"Lemma 4.3 (proof)"},{"comment":"The notation Φ(τ,t) in the final sentence should be Φ(t,τ), and equation (5.8) should be P*_{τ,τ+t}, not P*_{τ,t}, for consistency with the definition (5.1).","section":"Corollary 5.1"},{"comment":"In the definition after (7.4), the operator σ~ : R × ℓ2 × ℓ2 → L(ℓ2,ℓ2) is written as σ~ε, which appears to be a typo; the distribution-independent operator should be σ~.","section":"Section 7"},{"comment":"The proof contains a term (g_ϵ(t)-g(t))dt even though the system has a common forcing g; this term is zero and should be removed for clarity.","section":"Lemma 7.1 (proof)"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is a straightforward extension of [49] to McKean-Vlasov stochastic delay lattice systems. What's genuinely new: pullback measure attractors for non-autonomous distribution-dependent delay lattice systems on ℓ2, plus singleton attractor, exponential mixing, and upper semicontinuity. The proof machinery follows the standard template: well-posedness via Picard iteration, L2/L4 uniform estimates, absorbing sets, then contraction. The estimates are detailed and mostly credible; the authors clearly know the literature and the delay terms add real technical work. If the main theorem held, this would be a useful contribution.\n\nThe problem is the tightness argument in Lemma 5.3. The authors define Zδ as the set of Cr functions with norm ≤ R and support in |i|≤2m, and claim it is compact by Arzelà-Ascoli. That is false. The set is bounded and finite-support, but it is not equicontinuous uniformly over the family. The sequence w_n(t)=sin(nt)e_0 (padded with zeros) lies in Zδ for any m≥1 but has no uniformly convergent subsequence. The Arzelà-Ascoli argument in the proof only shows each individual function is uniformly continuous; it does not provide a modulus of continuity valid for the whole set. This is not a minor slip, because the compactness of Zδ is what carries the tightness of {Luτ(·,τ−t_n,v_n)}. Without it, the covering argument in (5.18)-(5.20) collapses.\n\nCould it be repaired? Probably, but with additional work. One would need a uniform-in-time regularity estimate for the solution segments, e.g., a Kolmogorov-type bound on E||u(t+h)−u(t)||^2 decaying in h, or some compact embedding. No such estimate appears in the paper. The tail-bound index slip the reader noticed (using (5.16) with θ_{m0} while the bound starts at 2m0) is real but secondary; the compactness failure is the load-bearing issue.\n\nThe rest of the paper is in decent shape. The large-damping condition (2.20) is explicit and used consistently; the mixing and semicontinuity sections are conditional on the attractor existence. Some typos and small notational glitches, but nothing else that looks fatal.\n\nWho is this for? Specialists in stochastic lattice dynamics or McKean-Vlasov equations. They would get a clear template for this class of systems, but they should not take Theorem 5.1 at face value yet. I'd send it to a referee, because the core idea is sound and the gap is identifiable and possibly fixable. But the authors need to address the compactness issue before it is citable.","headline":"Useful extension with a real gap: the tightness proof in Lemma 5.3 does not establish compactness, so the main attractor theorem is currently unproved.","tokens_in":51774,"tokens_out":3787,"would_cite":false,"duration_ms":32859,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H15","37L55","37L30","35R60"],"pacs":[],"model":"deepseek-v4-flash","headline":"Under large damping, the law of the solution segment of a McKean-Vlasov stochastic delay lattice system has a unique pullback measure attractor; extra damping makes it a singleton and gives exponential mixing.","keywords":["McKean-Vlasov equation","pullback measure attractor","stochastic delay lattice system","upper semicontinuity","tail-ends estimate","invariant measure","exponential mixing","Wasserstein distance"],"falsifier":"Set $g=\\kappa=\\chi=0$ and take $f_i(t,u,v,\\mu)=a v_i$ with constant $a>\\lambda$, choosing the constants so that (2.20) fails while (H1)-(H3) still hold. The deterministic delay equation $\\dot u=-\\lambda u+a u(t-r)$ has characteristic equation $z+\\lambda-a e^{-zr}=0$; since this function is negative at $z=0$ and tends to $+\\infty$, there is a root with positive real part, so $\\mathbb{E}\\|u(t)\\|^2$ grows exponentially. That growth directly contradicts the $D$-pullback absorbing estimate of Lemma 4.1 and shows the claimed conclusion depends on the explicit large-damping threshold.","tokens_in":50604,"feed_emoji":"🎲","tokens_out":8044,"duration_ms":74335,"temperature":0.7,"pith_summary":"This paper studies the long-term statistical behavior of an infinite lattice of stochastic equations in which each coordinate depends on its own past value and on the probability distribution of its current value. The authors prove that, under a large-damping condition, the family of probability laws of solution segments defines a non-autonomous dynamical system with a unique pullback measure attractor: a compact, invariant, time-dependent family of distributions that attracts every suitably bounded family of initial laws. With a stronger damping condition, the attractor collapses to a single law, and in the time-homogeneous case there is a unique invariant measure to which every other law converges exponentially in Wasserstein distance. The paper also shows that the attractors are upper semi-continuous as the distribution-dependent coefficients converge to distribution-independent ones. The significance is that even though McKean-Vlasov equations do not generate Markov semigroups, the system still has a well-defined statistical long-time limit with exponential mixing.","feed_headline":"Noisy delay lattices get one statistical long-term attractor","feed_subtitle":"Under strong damping, solution laws converge and mix exponentially; the mean-field limit preserves the attractor.","key_machinery":"The central object is the law-evolution semigroup $P^*_{\\tau,t}\\mu = \\mathcal{L}u_t(\\cdot,\\tau,\\mu)$, which is well defined by weak uniqueness, and the non-autonomous dynamical system $\\Phi(t,\\tau)=P^*_{\\tau,\\tau+t}$ acting on probability measures rather than paths. The argument runs through four mechanisms: Itô-formula weighted energy estimates with exponential weights $e^{\\varepsilon(s-\\tau)}$; uniform tail-end estimates with cutoff functions $\\theta_n$ to overcome the non-compactness of $\\ell^2$; Vitali convergence to prove continuity of $\\Phi$ on bounded sets; and Arzelà-Ascoli on finite-dimensional truncations to prove tightness of the solution-segment laws. The large-damping inequalities (2.20) and (6.1) are what make dissipation dominate the delayed, distribution-dependent, and noise terms in every estimate.","core_discovery":"The central claim is that the solution-segment laws of the non-autonomous McKean-Vlasov stochastic delay lattice system (2.16), regarded as a non-autonomous dynamical system $\\Phi$ on $\\mathcal{P}_4(C_r)$ with $C_r=C([-r,0],\\ell^2)$, admit a unique $\\mathcal{D}$-pullback measure attractor $\\mathcal{A}=\\{\\mathcal{A}(\\tau)\\}$ under hypotheses (H1)-(H3), the large-damping condition (2.20), and the integrability conditions (2.22)-(2.23). This means there is a compact invariant family of probability measures on delay segments that attracts all bounded initial-law families in $\\mathcal{D}$. Under the additional condition (6.1), the attractor is a singleton $\\{\\mu(\\tau)\\}$; in the autonomous case this yields a unique invariant measure $\\mu\\in\\mathcal{P}_4(C_r)$ satisfying $W_2(P^*_{0,t}\\upsilon,\\mu)\\le \\tilde{c}_3^{1/2}e^{-\\frac12\\varepsilon(t-r)}W_2(\\upsilon,\\mu)$ for all $t\\ge r$. Finally, Theorem 7.1 establishes upper semi-continuity: as the distribution-dependent coefficients approach distribution-independent ones, the corresponding attractors $\\mathcal{A}_\\varepsilon(\\tau)$ converge to $\\mathcal{A}(\\tau)$ in Hausdorff semi-distance.","pith_inferences":["The explicit threshold in (2.20) depends on the BDG constant $c_1$, so the numerical meaning of 'large damping' is not canonical: any improvement in the BDG constant changes the quantitative condition without changing the qualitative result.","The upper semicontinuity proof gives no rate, but combining the exponential contraction of Lemma 6.1 with the $O(\\varepsilon)$ solution convergence of Lemma 7.1 should yield an explicit Wasserstein rate of convergence of the coupled invariant measures to the uncoupled one.","The tail-end and tightness strategy is modular enough that the same pullback measure attractor theory should extend to weighted $\\ell^2$ spaces or to stronger nonlinear drifts, provided the corresponding weighted energy estimates are available."],"forward_implications":["For any initial-law family in $\\mathcal{D}$, the segment laws $\\mathcal{L}u_\\tau(\\cdot,\\tau-t,\\mu_{\\tau-t})$ converge to $\\mathcal{A}(\\tau)$ as $t\\to\\infty$, giving the system a well-defined statistical steady state despite the absence of a Markov semigroup.","With condition (6.1), distinct initial distributions are pulled together exponentially: the Wasserstein-2 distance between two solution-segment laws decays like $e^{-\\frac12\\varepsilon(t-r)}$, so the attractor consists of one point.","In the autonomous case this yields a unique invariant measure $\\mu\\in\\mathcal{P}_4(C_r)$, and the exponential contraction implies exponential mixing in $W_2$ for every initial law with finite second moment.","Under the $\\varepsilon$-scaling (7.1)-(7.2), the attractors of the distribution-dependent system upper-semicontinuously approach the attractor of the distribution-independent system; in the autonomous singleton case, the invariant measures themselves converge as $\\varepsilon\\to0^+$."],"supporting_citations":[{"why":"Supplies the abstract existence and uniqueness criterion for pullback attractors that is used as Proposition 2.1.","marker":"[53]"},{"why":"Introduces the measure-attractor concept that the paper extends to distribution-dependent delay lattice systems.","marker":"[47]"},{"why":"Provides the weak-uniqueness and path-distribution-dependent SDE machinery used in Proposition 3.2.","marker":"[30]"},{"why":"Prior construction of pullback measure attractors for McKean-Vlasov reaction-diffusion equations that the present delay lattice result builds on.","marker":"[49]"},{"why":"Source of the tail-end estimate technique for non-autonomous stochastic lattice systems on unbounded index sets.","marker":"[9]"},{"why":"Baseline theory of attractors for stochastic lattice dynamical systems that sets up the compactness problem in $\\ell^2$.","marker":"[7]"},{"why":"Provides exponential-stability estimates for stochastic delay lattice systems that inform the delay and BDG estimates used here.","marker":"[61]"}],"fun_headline_variants":["Unique pullback attractor for stochastic delay lattice laws","Mean-field delay lattices: one attractor rules their laws","Stochastic delay lattices: laws converge to a single attractor","Mean-field limit preserves the unique pullback attractor","Exponential mixing and unique law attractor on delay lattices"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument collapses if the damping coefficient $\\lambda$ is not large enough to dominate the delay feedback, the noise coefficients, and the distribution coupling through the explicit inequalities (2.20) and (6.1), because all absorbing and contraction estimates rely on those inequalities.","fun_headline_variants_meta":{"raw":{"variants":["Unique pullback attractor for stochastic delay lattice laws","Mean-field delay lattices: one attractor rules their laws","Stochastic delay lattices: laws converge to a single attractor","Mean-field limit preserves the unique pullback attractor","Exponential mixing and unique law attractor on delay lattices"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000575,"raw_usage":{"total_tokens":2725,"prompt_tokens":968,"completion_tokens":1757,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":584,"completion_tokens_details":{"reasoning_tokens":1674}},"tokens_in":584,"tokens_out":1757,"duration_ms":11587,"temperature":1.0,"reasoning_tokens":1674,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T11:22:38.059120+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Set $g=\\kappa=\\chi=0$ and take $f_i(t,u,v,\\mu)=a v_i$ with constant $a>\\lambda$, choosing the constants so that (2.20) fails while (H1)-(H3) still hold. The deterministic delay equation $\\dot u=-\\lambda u+a u(t-r)$ has characteristic equation $z+\\lambda-a e^{-zr}=0$; since this function is negative at $z=0$ and tends to $+\\infty$, there is a root with positive real part, so $\\mathbb{E}\\|u(t)\\|^2$ grows exponentially. That growth directly contradicts the $D$-pullback absorbing estimate of Lemma 4.1 and shows the claimed conclusion depends on the explicit large-damping threshold.","supporting_citations":[{"cited_title":"Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J","cited_arxiv_id":null,"evidence_quote":"Supplies the abstract existence and uniqueness criterion for pullback attractors that is used as Proposition 2.1."},{"cited_title":"Schmalfuss, Long-time behaviour of the stochastic N avier-Stokes equation, Math","cited_arxiv_id":null,"evidence_quote":"Introduces the measure-attractor concept that the paper extends to distribution-dependent delay lattice systems."},{"cited_title":"Huang, M","cited_arxiv_id":null,"evidence_quote":"Provides the weak-uniqueness and path-distribution-dependent SDE machinery used in Proposition 3.2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Source of the tail-end estimate technique for non-autonomous stochastic lattice systems on unbounded index sets."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Baseline theory of attractors for stochastic lattice dynamical systems that sets up the compactness problem in $\\ell^2$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides exponential-stability estimates for stochastic delay lattice systems that inform the delay and BDG estimates used here."}],"review_version":1}