{"id":"492f37c8-8e9f-4d27-ba16-30c51993a2c0","arxiv_id":"2607.28078","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"McKean–Vlasov SDEs are strongly well-posed under distribution-dependent Lyapunov growth and a hybrid Perron–Nagumo condition allowing a non-integrable singularity at time zero.","lead":"The paper proves strong existence and pathwise uniqueness for McKean–Vlasov SDEs under a distribution-dependent Lyapunov condition plus a hybrid Perron–Nagumo increment bound that may blow up at t=0. It gives a usable well-posedness test beyond Lipschitz, Osgood, and monotonicity, with a non-classical existence route via absorbed local weak solutions and a restricted Yamada–Watanabe step.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified beyond the paper's own explicit scope limitation","rationale":"The paper's central claim is a sufficient criterion, not a claim of unrestricted pathwise uniqueness. The Lyapunov localization is both the enabling device and the explicit scope of uniqueness; the reader already identified this accurately. The existence route (absorbed one-step weak solutions → Euler limit → tightness across nested domains → restricted YW) is carefully set up so that the limiting weak solution lands in KV, after which Lemma 3.3 upgrades it. Residual medium correctness risk remains only because the multi-step martingale-identity and selection arguments in Lemmas 3.1–3.2 were not line-checked here, which matches the reader's assessment and does not warrant changing ACCEPT. The example in §4 cleanly separates (H4) from Lipschitz/Osgood/monotonicity, supporting the novelty claim. No stronger load-bearing concern presents itself.","tokens_in":24674,"tokens_out":497,"duration_ms":9817,"concrete_test":"Independently re-derive the integral inequality (3.15) for ζ from (3.13) under (H4), then verify that the unique solution of x'=F(t,x), x(0)=0 is identically zero (using the C1 upper function asserted in (H4) and the comparison propositions cited from [17]). If this step holds, the uniqueness half of Theorem 3.4 is secure inside KV.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest_assumption correctly notes that pathwise uniqueness in Theorem 3.4 holds only inside the Lyapunov-admissible class KV (solutions with sup_t V(t,L_X(t))<∞), not among all strong solutions. This is not a hidden gap: the abstract, the statement of Theorem 3.4, and the comparison argument in part (i) (which uses the finite bound M to close the L2 estimates and the ODE comparison for ζ) all state the restriction explicitly. The existence construction produces a solution inside KV, so the restricted Yamada–Watanabe lemma (Lemma 3.3) applies as written. No internal inconsistency or unstated assumption that would undermine the central claim as formulated was found.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper proves strong existence and pathwise uniqueness for McKean–Vlasov SDEs (2.1) under a distribution-dependent Lyapunov package (H1)–(H3) together with a hybrid Perron–Nagumo increment condition (H4) that allows a non-integrable singular weight u'/u near t=0. Pathwise uniqueness is established inside the class of strong solutions with uniformly bounded integrated Lyapunov functional (3.11). Existence is obtained by truncating coefficients on nested domains Dk, building absorbed one-step weak solutions, concatenating them (Lemma 3.1), passing to absorbed Euler limits (Lemma 3.2), removing absorption by tightness and Borel–Cantelli, and upgrading the resulting compatible weak solution to a strong solution via a restricted Yamada–Watanabe theorem (Lemma 3.3). An explicit one-dimensional example satisfies (H1)–(H4) while violating Lipschitz, Osgood, and standard monotonicity conditions.","tokens_in":24775,"tokens_out":1239,"duration_ms":43055,"significance":"The work usefully extends classical Perron–Nagumo uniqueness criteria from ordinary SDEs to the McKean–Vlasov setting and pairs them with Lyapunov localization in the measure variable. The existence route—absorbed local weak solutions plus tightness, rather than path-space truncation-and-patching—is a genuine alternative that addresses the self-consistency obstruction that arises when one truncates the state of a distribution-dependent equation. The restricted Yamada–Watanabe lemma (Lemma 3.3) is carefully adapted to the Lyapunov-admissible class KV. The example in Section 4 is concrete and cleanly separates (H4) from Lipschitz, Osgood, and monotonicity. If the arguments hold as written, the criterion is a solid addition to the well-posedness literature for mean-field SDEs.","major_comments":[{"comment":"End of the proof of Theorem 3.4 (existence half): compatibility of the limiting weak solution (X̃,B̃) is asserted in a single short paragraph by noting that the filtration is the augmentation of the one generated by (X̃,B̃). Lemma 3.3 makes compatibility a load-bearing hypothesis for the upgrade to a strong solution. A brief but explicit verification that future Brownian increments remain independent of F̃t (or a pointer to the precise statement in Carmona–Delarue used) would make this step checkable without external reconstruction.","section":"Theorem 3.4 proof, existence half"},{"comment":"Lemma 3.2 and the subsequent global limit: after Skorokhod, the indicator 1{r<θ̃k} tends to 1 almost surely, and the martingale problem is identified using uniform integrability from (H2) and the Lyapunov bound (3.29). Because (H1) only gives local boundedness, it would help to record explicitly that the limiting integrands b(r,X̃(r),μ̃(r)) and σ belong to L1/L2 on [0,T] via the growth (3.8) and Fatou, so that the stochastic integral is well-defined as an Itô integral (not merely a local martingale). This is implicit but load-bearing for Definition 2.1.","section":"Lemma 3.2; Theorem 3.4(ii)"}],"minor_comments":[{"comment":"Definition 2.2 presupposes that the one-sided derivatives D± exist everywhere on [0,t0]. A one-line remark that upper/lower functions are taken in the class where these derivatives exist (as in the cited ODE work) would avoid a pedantic objection.","section":"Definition 2.2"},{"comment":"In (H4) the constant α is defined with the factor 4α1(T∨1)u(T)/(1−4α2(T∨1)). Remark 3.5 correctly notes that the signed-drift variant removes the T∨1 factor; cross-referencing that remark already in the statement of (H4) would help the reader track the constants.","section":"(H4); Remark 3.5"},{"comment":"Figure 1 is helpful conceptually but is only sketched in text. If the journal allows, a simple diagram of the absorbed paths versus the limiting path would clarify the “law-based approximation” slogan.","section":"Figure 1 caption"},{"comment":"Typos/notation: “ana priori” in the Introduction; occasional spacing in operator names (e.g., LX(t)); and the date line “July 31, 2026” looks like a placeholder.","section":"Introduction; throughout"},{"comment":"The example verifies (H3) by a one-line appeal to Young and boundedness. Expanding the computation of LV for V(t,x,μ)=|x|2+μ(|·|2) by a few lines would make the example fully self-contained.","section":"Section 4"}],"recommendation":"minor_revision","confidential_remarks":"The central claims appear sound and the uniqueness restriction to KV is stated explicitly throughout; I did not find a hidden circularity or an unstated assumption that breaks Theorem 3.4 as formulated. The existence half is long and technical; the two major comments are requests for tighter write-up of load-bearing justifications rather than evidence of error. Fit for a solid probability journal is good. Self-citation to the authors’ ODE paper [17] and Lyapunov MVSDE paper [18] is appropriate and not excessive."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The punchline is a usable sufficient condition: distribution-dependent Lyapunov localization plus a hybrid Perron–Nagumo increment that allows a non-integrable singularity at t=0. Pathwise uniqueness holds inside the class of strong solutions with uniformly bounded integrated Lyapunov functional; existence produces one such solution via absorbed Euler concatenations, tightness, and a restricted Kurtz Yamada–Watanabe map.\n\nWhat is actually new is the existence path. Classical pathwise truncation breaks self-consistency of the law in the McKean–Vlasov setting, and the path-space truncation used by Ren–Tang–Wang and by Liu–Ma does not fit a pure Perron–Nagumo modulus. The authors instead freeze the law on mesh intervals, build absorbed one-step weak solutions, select and concatenate (Lemma 3.1), pass to an absorbed Euler limit (Lemma 3.2), then lift by a restricted YW theorem (Lemma 3.3). That construction is carefully written and different from the usual patching arguments. Uniqueness is the expected L2 + ODE comparison under (H4), with the singular weight handled by \theta = I/u and upper/lower functions exactly as in their earlier ODE work; the a-priori bound M closes the estimates cleanly. The example in Section 4 is concrete and separates the criterion from Lipschitz, Osgood, and monotonicity.\n\nThe soft spot is exactly the one the paper states: uniqueness is only inside KV. If two strong solutions with infinite Lyapunov mass exist, the theorem is silent. That is not a hidden gap; abstract, theorem statement, and proof all flag it. Residual risk is ordinary multi-step weak-convergence bookkeeping (selection, martingale identities, J1 tightness of the indicators); I did not line-check every identity, but nothing looks load-bearing-broken. Citations are appropriate; self-cites supply prior comparison lemmas, not circular restatements.\n\nThis is for people who already work on non-Lipschitz or mean-field SDEs and need a broader sufficient package. It deserves a serious referee. I would send it out.","headline":"Solid sufficient well-posedness criterion for MVSDEs; the existence route is the real novelty, uniqueness is cleanly restricted to the Lyapunov class.","tokens_in":25454,"tokens_out":513,"would_cite":true,"duration_ms":9877,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H10","60H20"],"pacs":[],"model":"grok-4.5","headline":"McKean–Vlasov SDEs are strongly well-posed under a distribution-dependent Lyapunov condition plus a hybrid Perron–Nagumo increment bound that allows a non-integrable singularity at time zero.","keywords":["McKean–Vlasov SDE","pathwise uniqueness","strong existence","Perron-type condition","Nagumo-type condition","distribution-dependent Lyapunov function","Yamada–Watanabe","mean-field SDE"],"falsifier":"Either exhibit two distinct strong solutions of the example SDE that both keep the integrated Lyapunov functional finite, or produce coefficients satisfying (H1)–(H4) for which no strong solution with finite Lyapunov mass exists on [0,T].","tokens_in":25496,"feed_emoji":"📐","tokens_out":967,"duration_ms":18603,"temperature":0.7,"pith_summary":"This paper gives a criterion for strong existence and pathwise uniqueness of McKean–Vlasov stochastic differential equations when the coefficients need not be Lipschitz, Osgood, or monotone. The coefficients are controlled by a distribution-dependent Lyapunov function that localizes growth, together with a hybrid Perron–Nagumo condition on increments in both the state and the law that allows a non-integrable weight near the initial time. Pathwise uniqueness is proved inside the class of strong solutions whose integrated Lyapunov functional stays finite. Existence is obtained by truncating on nested domains, building absorbed local weak solutions, passing to a global weak solution by tightness, and converting to a strong solution with a restricted Yamada–Watanabe theorem. An explicit one-dimensional example meets the new criterion while violating the classical Lipschitz, Osgood, and monotonicity conditions, so the result genuinely enlarges the known well-posedness range for mean-field SDEs.","feed_headline":"Mean-field SDEs well-posed beyond Lipschitz and Osgood","feed_subtitle":"A Lyapunov bound plus a hybrid Perron–Nagumo condition gives strong existence and uniqueness","key_machinery":"The hybrid Perron–Nagumo condition (H4): it bounds squared increments of drift and diffusion by a concave modulus plus a possibly non-integrable weight u′/u near t=0, then reduces pathwise uniqueness to an ODE comparison whose only solution through the origin is zero; existence is carried by absorbed Euler concatenations, tightness, and a restricted Yamada–Watanabe map on the Lyapunov-admissible class.","core_discovery":"Under local boundedness and continuity, a coercive integrated Lyapunov condition, and a hybrid Perron–Nagumo increment condition, a McKean–Vlasov SDE admits a strong solution whose integrated Lyapunov functional stays bounded on a fixed time horizon, and that solution is pathwise unique among all strong solutions satisfying the same Lyapunov bound.","pith_inferences":["The same Lyapunov-plus-Perron–Nagumo package may extend to McKean–Vlasov equations with jumps or path-dependent coefficients once an analogous absorbed weak-solution construction is available.","Because uniqueness is only inside the Lyapunov class, numerical schemes that preserve a discrete Lyapunov bound would automatically select the unique strong solution the theorem identifies.","The law-based absorption limit (rather than pathwise patching) could be reused for other distribution-dependent problems where truncating the state changes the measure argument of the coefficients."],"forward_implications":["Mean-field SDEs whose coefficients fail Lipschitz, Osgood, and monotonicity can still be strongly well-posed if they obey a distribution-dependent Lyapunov bound and the hybrid Perron–Nagumo increment condition.","Pathwise uniqueness is guaranteed inside the Lyapunov-admissible class even when the modulus of continuity carries a non-integrable singularity at the initial time.","Existence for McKean–Vlasov equations can be obtained by absorbed local weak solutions and tightness without classical pathwise truncation-and-patching, which breaks self-consistency of the law.","The restricted Yamada–Watanabe principle converts a compatible Lyapunov-admissible weak solution into the unique strong solution on any prescribed stochastic basis with the same initial law."],"fun_headline_variants":["Lyapunov plus hybrid Perron–Nagumo yields McKean–Vlasov well-posedness","Strong existence and uniqueness for McKean–Vlasov SDEs via Lyapunov bound","Distribution-dependent Lyapunov condition ensures pathwise unique strong solutions","McKean–Vlasov SDEs well-posed under hybrid Perron–Nagumo singularity","Coercive Lyapunov criterion beats Lipschitz, Osgood, and monotonicity gaps"],"cache_read_input_tokens":128,"weakest_assumption_plain":"Uniqueness holds only among solutions whose integrated Lyapunov functional stays finite; if two strong solutions blow that bound, the criterion says nothing.","fun_headline_variants_meta":{"raw":{"variants":["Lyapunov plus hybrid Perron–Nagumo yields McKean–Vlasov well-posedness","Strong existence and uniqueness for McKean–Vlasov SDEs via Lyapunov bound","Distribution-dependent Lyapunov condition ensures pathwise unique strong solutions","McKean–Vlasov SDEs well-posed under hybrid Perron–Nagumo singularity","Coercive Lyapunov criterion beats Lipschitz, Osgood, and monotonicity gaps"]},"model":"grok-4.5","effort":"low","cost_usd":0.003382,"raw_usage":{"total_tokens":1052,"prompt_tokens":678,"num_sources_used":0,"completion_tokens":112,"cost_in_usd_ticks":33824000,"prompt_tokens_details":{"text_tokens":678,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":262,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":678,"tokens_out":112,"duration_ms":5766,"temperature":1.0,"reasoning_tokens":262,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-31T18:28:41.816038+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Either exhibit two distinct strong solutions of the example SDE that both keep the integrated Lyapunov functional finite, or produce coefficients satisfying (H1)–(H4) for which no strong solution with finite Lyapunov mass exists on [0,T].","supporting_citations":[],"review_version":1}