{"id":"b82009c6-0710-49f1-b1a3-398b46bc8893","arxiv_id":"2412.01070","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Lévy-driven McKean-Vlasov SDEs are strongly well-posed under weak monotonicity and weak coercivity, with quantitative propagation of chaos.","lead":"This paper proves that a broad family of jump-driven equations describing interacting particles has a unique solution, even when forces are only one-sided Lipschitz. It also quantifies how a finite swarm of particles approaches the infinite-particle limit.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The interlacing construction in the proof of Theorem 2.1 is invalid: the claimed global solution X_t = Y_t + ∫∫ g_s(Y_s,z)N(ds,dz) does not satisfy (2.1) for state-dependent coefficients, so the proof of Theorem 1.1 collapses.","rationale":"I read the paper as a well-posedness result for Lévy-driven McKean–Vlasov SDEs under weak monotonicity/coercivity, with proofs via a classical SDE theorem, interlacing, and Banach fixed point. The central claim (Theorem 1.1) depends on Theorem 2.1, whose proof uses the interlacing technique. The apparent error is in that interlacing step: the constructed process after a big jump is the original small-jump process plus a constant jump contribution, not a solution of the small-jump SDE restarted from the post-jump state. This is not a matter of constants or regularity; the resulting process fails the defining SDE. A simple pure-jump example within (H1)-(H3) exhibits the failure. Because the decoupled SDEs in the McKean–Vlasov proof have state-dependent b,f,g (e.g., Remark 1.2), the flaw is not avoided. I acknowledge the fixed-point and moment arguments are otherwise carefully developed, and the statement may be true, but the presented proof does not establish it. Therefore I recommend rejection of the current version until Theorem 2.1's interlacing construction is corrected or replaced.","tokens_in":34461,"tokens_out":15997,"duration_ms":136204,"concrete_test":"Verify Theorem 2.1's construction on the scalar SDE dX_t = X_{t-} N(dt,dz) with V={1}, ν({1})=1, X0=1, β=1. Here b=f=0, g(x,z)=x, and (H1)-(H3) hold. The paper's interlacing formula gives X_t = 1 + N_t (since Y_t=1 and each jump contributes g(Y_{σ_i-},1)=1), while the unique solution obtained by iterating the jump map is X_t = 2^{N_t}. At t after two jumps the two disagree (3 vs 4), so the construction in the proof of Theorem 2.1 is not a solution of (2.1).","verdict_should_be":"REJECT","load_bearing_attack":"The proof of Theorem 2.1 claims that the solution of (2.1) can be written as X_t = Y_t + ∫_0^t∫_V g_s(Y_s,z)N(ds,dz), where Y solves the small-jump SDE (2.5). This is the standard interlacing formula only when g is independent of the state (or the system is linear in a very special sense). In general, after a big jump at σ_k the process must restart the small-jump dynamics from the post-jump state X_{σ_k}=X_{σ_k-}+g_{σ_k}(X_{σ_k-},p_k), so between jumps one needs dX_t = b_t(X_t)dt + ∫_U f_t(X_{t-},z)\\tilde N(dt,dz), not dX_t = b_t(Y_t)dt + ∫_U f_t(Y_{t-},z)\\tilde N(dt,dz). The displayed construction keeps the original Y_t plus constants, changing the drift and jump coefficients to be evaluated at Y_t, not X_t. A concrete failure within the assumptions: take d=1, b=f=0, V={1}, ν({1})=1, g(x,z)=x, X0=1, β=1. The true strong solution is X_t=2^{N_t}; the paper's formula with Y_t=1 gives X_t=1+N_t, which differs already after the second jump (4 vs 3). This is not a technical gap but a wrong equation: the constructed process does not solve (2.1). Since Theorem 1.1 and the common-noise extension prove well-posedness by applying Theorem 2.1 to the decoupled SDEs, the central claim is unsupported by the presented argument.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies strong well-posedness and propagation of chaos for McKean-Vlasov SDEs driven by Lévy processes under weak monotonicity and weak coercivity. The main result, Theorem 1.1, claims existence and uniqueness of a strong solution to (1.1) with moment bounds, using a Banach fixed point on the law of the decoupled SDE; Theorem 1.3 gives quantitative weak and strong propagation of chaos; Section 4 extends these to common noise. The paper first proves a classical time-inhomogeneous SDE result (Theorem 2.1) via a Picard iteration for the small-jump part and an interlacing argument for big jumps.","tokens_in":34793,"tokens_out":10990,"duration_ms":98920,"significance":"If the results were established, they would fill a recognized gap in the literature: strong well-posedness of Lévy-driven SDEs under one-sided Lipschitz and weak coercivity conditions, and the corresponding McKean-Vlasov well-posedness under L^β-Wasserstein Lipschitz assumptions. The paper is clearly organized, self-contained, and gives explicit examples showing the scope of the assumptions. There is no circularity and no fitted parameters. However, the central proof is invalid as written, so the claimed results are not currently established.","major_comments":[{"comment":"The interlacing construction is incorrect. After the first big jump at σ1, the true solution must restart the small-jump dynamics from the post-jump position X_{σ1}=X_{σ1-}+g_{σ1}(X_{σ1-},p1), but the paper sets X_t = Y_t + g_{σ1-}(Y_{σ1-},p1) for σ1<t<σ2, where Y solves (2.5) from the original initial value. The displayed representation X_t = Y_t + ∫_0^t∫_V g_s(Y_s,z)N(ds,dz) is not a solution of (2.1) for state-dependent coefficients. Concretely, under the assumptions of Theorem 2.1, take d=1, b=f=0, V={1}, ν({1})=1, g(x,z)=x, X0=1, β=1. The true solution is X_t=2^{N_t}, while the paper's formula gives 1+N_t, and these differ already after the second jump. This invalidates the existence proof of Theorem 2.1 and therefore also the applications of Theorem 2.1 in Theorem 1.1 and Section 4.","section":"§2, Proof of Theorem 2.1"},{"comment":"The contraction proof for the map Φ depends on the same invalid interlacing representation. The estimate (3.8) is stated for t∈[σ_n,σ_{n+1}) and the jump update (3.9) uses X_{σ_n}=X_{σ_n-}+g(X_{σ_n-}, μ_{σ_n}, p_n). These formulas are compatible only with a construction in which the process is restarted after every big jump; the paper does not provide such a construction, and its displayed global formula is inconsistent with it. Consequently, the inequalities (3.6)-(3.7) do not establish that the map defined in (3.3) is contractive for the actual solution process.","section":"§3.1, Eqs. (3.8)-(3.10)"},{"comment":"Theorem 4.1 is proved by invoking Theorem 2.1 for the SDE (4.6) with random coefficients, with the statement that Theorem 2.1 'is still available' in that setting. This extension is not proved, and the hypotheses (H1)-(H3) in Section 2 are formulated for deterministic coefficients. A justification is needed for the measurability and pathwise arguments when the frozen measure μ_t is random, especially because the comment in Remark 4.2 explicitly notes that the interlacing trick is not usable for random coefficients.","section":"§4.1, Proof of Theorem 4.1"}],"minor_comments":[{"comment":"Strong existence is only sketched with the words 'more or less standard'; since Theorem 2.1 relies on it, the authors should either provide the full argument or cite a statement that exactly covers the present setting.","section":"§2.2, Proof of Proposition 2.4"},{"comment":"The exponent in (1.13) is written |X^{i,n}_t - X^i_t|^{p q_1}, but in the proof the supremum is taken with |Q^i_t|^{p q_1}; the notation should be made consistent and the supremum should be over s, not t.","section":"Theorem 1.3 and its proof"},{"comment":"There are several typographical issues: 'cádlág' should be 'càdlàg', 'it’s' should be 'its', 'mean-filed' should be 'mean-field', and in Remark 1.2(ii) 'rigorous than' should probably be 'stronger than'.","section":"Throughout"},{"comment":"The estimate uses ν((1+|·|^β)1_V(·)), while condition (1.2) is stated with 1∨|·|^β; the comparison should be made explicit so that the constants in (3.2) are clearly finite.","section":"Eq. (3.2)"}],"recommendation":"major_revision","confidential_remarks":"The reported interlacing error is in the core proof, so the current version should not be accepted. I recommend major revision rather than outright rejection because the underlying theorems may be true and the proof could in principle be repaired by a correct restart-after-jump interlacing argument; however, the repair is substantial and affects Section 2, the fixed point argument in Section 3, and the common-noise extension in Section 4. I would be willing to review a revised version that contains such a corrected argument."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper tries to fill a real gap—strong well-posedness for Lévy-driven McKean-Vlasov SDEs under one-sided Lipschitz drift—and the literature review is careful. But the proof of Theorem 2.1, the classical SDE result that everything else builds on, has a fatal error in the interlacing construction. The constructed process X_t = Y_t + ∫ g_s(Y_s,z)N(ds,dz) does not solve (2.1) when g depends on the state. After a big jump at σ, the equation forces the small-jump dynamics to restart from X_σ = X_{σ-}+g_{σ-}(X_{σ-},p), not to continue along the original Y path plus a constant.\n\nHere is a concrete failure inside their assumptions: d=1, b=f=0, V={1}, ν({1})=1, g(x,z)=x, X0=1. The true strong solution is X_t=2^{N_t}; their formula gives 1+N_t. Already after the second jump these differ (4 vs 3). The constructed path does not satisfy the SDE, so the existence/uniqueness statement in Theorem 2.1 is not proved.\n\nThis is not a minor gap. The McKean-Vlasov Theorem 1.1 is obtained by applying Theorem 2.1 to decoupled SDEs, so its proof collapses too. The common-noise extension has the same problem. I suspect the theorems are true and the interlacing argument can be fixed by restarting Y after each big jump, but that is a substantial rewrite of Section 2, not a one-line correction.\n\nWhat the paper does well: the Picard iteration for the small-jump SDE (Proposition 2.4) is detailed, the moment estimates under weak coercivity are careful, and the contraction argument for the measure variable is an interesting way to handle the one-sided Lipschitz drift. The propagation of chaos part is competently done, though largely standard once the well-posedness is granted.\n\nOther soft spots are minor: strong existence in Proposition 2.4 is dismissed as 'more or less standard,' and the proof of Theorem 1.3 is compressed. The global weak coercivity (A2) is the structurally weak assumption, but it is benign for the applications the authors cite.\n\nVerdict: this paper should not be accepted as is. It deserves a serious referee because the gap is real and the intended result is useful, but the referee will need to send it back for a corrected interlacing construction. If the authors fix Section 2, the rest of the paper may stand.\n\nBest.","headline":"The gap-filling goal is right, but the interlacing proof of Theorem 2.1 is wrong, so the main existence/uniqueness results are unsupported as written.","tokens_in":35380,"tokens_out":6418,"would_cite":false,"duration_ms":55015,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60G51","60J25","60J76"],"pacs":[],"model":"deepseek-v4-flash","headline":"Lévy-driven McKean-Vlasov SDEs are strongly well-posed under weak monotonicity and weak coercivity, with quantitative propagation of chaos and common-noise extensions.","keywords":["McKean-Vlasov SDE","Lévy process","weak monotonicity","weak coercivity","strong well-posedness","propagation of chaos","common noise","Banach fixed point theorem"],"falsifier":"Compute the contraction constant of the map $\\Phi$ under the weighted metric $W_{\\beta,\\gamma}$ for the explicit non-Lipschitz example in Remark 1.2(i), letting $\\gamma\\to\\infty$. The theorem asserts the map becomes a contraction for large $\\gamma$; if the constant never drops below 1, the Banach fixed-point step that carries the proof fails.","tokens_in":34212,"feed_emoji":"🎲","tokens_out":15805,"duration_ms":132949,"temperature":0.7,"pith_summary":"This paper proves that Lévy-driven McKean-Vlasov stochastic differential equations are strongly well-posed under weak monotonicity and weak coercivity, a setting in which the drift and small-jump coefficients may fail to be globally Lipschitz in the state variable. The main result, Theorem 1.1, gives existence and pathwise uniqueness of the solution whenever the initial law has finite $\\beta$-th moment with $\\beta\\in[1,2]$, together with the moment estimate $E|X_t|^\\beta \\le C_T(1+E|X_0|^\\beta)$. The paper also quantifies weak and strong propagation of chaos for the interacting particle system, with explicit Wasserstein convergence rates, and extends both well-posedness and conditional propagation of chaos to equations with common noise. The contribution fills a known gap: jump-type McKean-Vlasov SDEs under one-sided Lipschitz drift previously had strong well-posedness results only under stronger Lipschitz or growth conditions.","feed_headline":"One-sided Lipschitz drift yields unique Lévy mean-field paths","feed_subtitle":"Banach fixed-point proof yields well-posedness and propagation-of-chaos rates for jump-type mean-field equations.","key_machinery":"The load-bearing mechanism is the decoupled (distribution-frozen) SDE (3.1) combined with the Banach fixed point theorem. The classical SDE result of Section 2 handles each frozen measure path: a Picard iteration with piecewise-constant-in-time coefficients proves strong well-posedness for small jumps, and the interlacing technique superimposes big jumps at the Poisson jump times $\\sigma_n$. The contraction is proved in the weighted $L^\\beta$-Wasserstein metric $W_{\\beta,\\gamma}$; a direct Itô estimate would leave the term $|Y|^{\\beta-2}W_\\beta(\\mu,\\tilde\\mu)^2$ uncontrolled near $|Y|=0$, so the proof conditions on the intervals between big jumps and applies the estimates (3.8) and (3.10) to obtain a contraction for large $\\gamma$.","core_discovery":"The central claim is that the one-sided Lipschitz (weak monotonicity) condition on the pair $(b,f)$, together with a global weak coercivity bound and local integrability, is sufficient for the McKean-Vlasov SDE (1.1) to have a unique strong solution, and that this solution has finite $\\beta$-th moments on every finite horizon. The proof freezes the law path $\\mu$ and solves the decoupled time-inhomogeneous SDE (3.1) via Picard iteration for the small-jump part and interlacing for the big-jump part; it then shows that the map $\\Phi(\\mu)_t = \\mathcal{L}_{X^\\mu_t}$ is a contraction on the complete metric space of law paths under the weighted distance $W_{\\beta,\\gamma}(\\mu,\\tilde\\mu)=\\sup_t e^{-\\gamma t}W_\\beta(\\mu_t,\\tilde\\mu_t)$. The same framework yields the quantitative propagation-of-chaos rates of Theorem 1.3 and, after strengthening the monotonicity assumption, the common-noise well-posedness and conditional propagation of chaos in Section 4.","pith_inferences":["One testable extension is to localize (A2) to measures with bounded $\\beta$-th moment; the proof as written uses the uniform-over-$P_\\beta$ form to keep the successive approximations alive, so any localization would need a new a priori bound.","The interlacing-based contraction estimate suggests a numerical scheme for non-globally Lipschitz coefficients: iterate the piecewise-constant-coefficient small-jump SDE and overlay big jumps, and the contraction estimate may yield a convergence rate in particle number and time step.","The paper's propagation-of-chaos result assumes $f$ is independent of the measure; Remark 1.4 shows that when $b\\equiv 0$ this independence is forced by the monotonicity inequality, so dropping the assumption would require a different measure-monotonicity condition rather than a minor tweak.","Because the proof uses the $L^\\beta$-Wasserstein metric and $\\beta\\in[1,2]$, trying $\\beta<1$ with a different metric (for instance a Wasserstein-type distance with exponent below one) is a natural test of whether the uniqueness failure described in the paper is intrinsic or an artifact of the chosen topology."],"forward_implications":["Every equation in the class (1.1) satisfying (A1)-(A3) has a unique strong solution, so the standard obstacle for jump SDEs under one-sided Lipschitz drift — a missing reference rather than a missing result — is removed for the McKean-Vlasov case.","For the mean-field particle system (1.8), the empirical measure converges to the common law in $W_p$ at the rate $\\varphi_{p,\\beta,d}(n)$ from (1.12), and each particle path converges to an independent copy of the limit in a sup-norm moment sense, so simulation-based inference on finite particle systems has quantitative guarantees.","Strengthening the coercivity condition from (A2) to (1.6) upgrades the moment bound from time slices to the running supremum, $E(\\sup_{0\\le t\\le T}|X_t|^\\beta)\\le C'_T(1+E|X_0|^\\beta)$.","With common noise, the same well-posedness and a conditional version of propagation of chaos hold: conditioned on the common noise, particles become asymptotically independent and the conditional empirical measure converges with the same rate function."],"supporting_citations":[{"why":"Supplies the diffusive prototype: strong well-posedness under local weak monotonicity and global weak coercivity, which the paper adapts to jump noise.","marker":"[33]"},{"why":"Provides the interlacing technique for splicing big jumps onto a small-jump solution, used in the proof of Theorem 2.1.","marker":"[3]"},{"why":"Gives the Lipschitz $L^\\beta$-Wasserstein well-posedness and propagation-of-chaos results that Theorem 1.1 weakens, and the $\\beta\\in(0,1)$ non-uniqueness remark.","marker":"[6]"},{"why":"Supplies the empirical-measure Wasserstein convergence rate $\\varphi_{p,\\beta,d}(n)$ used in Theorems 1.3 and 4.3.","marker":"[14]"},{"why":"Supplies the stochastic Gronwall inequality used for the moment estimates and the common-noise argument.","marker":"[38]"},{"why":"Provides the framework for common noise and the definition of conditional propagation of chaos used in Section 4.","marker":"[5]"},{"why":"Documents the long-missing reference problem for jump SDEs under one-sided Lipschitz drift and serves as the comparison baseline under $L^2$-Wasserstein assumptions.","marker":"[31]"}],"fun_headline_variants":["Weak monotonicity gives unique Lévy mean-field paths","Lévy jump SDEs: well-posed under one-sided Lipschitz","Propagation of chaos for monotone Lévy McKean-Vlasov SDEs","Interlacing + fixed point: Lévy McKean-Vlasov well-posedness","Common noise included: monotone Lévy mean-field SDEs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the global weak coercivity bound (A2), which must hold uniformly over every law in the class $P_\\beta(\\mathbb{R}^d)$; it is what keeps the proof's successive approximations from exploding before the fixed-point argument can run.","fun_headline_variants_meta":{"raw":{"variants":["Weak monotonicity gives unique Lévy mean-field paths","Lévy jump SDEs: well-posed under one-sided Lipschitz","Propagation of chaos for monotone Lévy McKean-Vlasov SDEs","Interlacing + fixed point: Lévy McKean-Vlasov well-posedness","Common noise included: monotone Lévy mean-field SDEs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000271,"raw_usage":{"total_tokens":1703,"prompt_tokens":1091,"completion_tokens":612,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":707,"completion_tokens_details":{"reasoning_tokens":508}},"tokens_in":707,"tokens_out":612,"duration_ms":5616,"temperature":1.0,"reasoning_tokens":508,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T04:42:10.219821+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the contraction constant of the map $\\Phi$ under the weighted metric $W_{\\beta,\\gamma}$ for the explicit non-Lipschitz example in Remark 1.2(i), letting $\\gamma\\to\\infty$. The theorem asserts the map becomes a contraction for large $\\gamma$; if the constant never drops below 1, the Banach fixed-point step that carries the proof fails.","supporting_citations":[{"cited_title":"and Röckner, M.: A Concise Course on Stochastic Partial Differential Equatio ns, Springer, Berlin, 2007","cited_arxiv_id":null,"evidence_quote":"Supplies the diffusive prototype: strong well-posedness under local weak monotonicity and global weak coercivity, which the paper adapts to jump noise."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the interlacing technique for splicing big jumps onto a small-jump solution, used in the proof of Theorem 2.1."},{"cited_title":"and Guillin, A.: On the rate of convergence in Wasserstein distance of the empirical measure, Probab","cited_arxiv_id":null,"evidence_quote":"Supplies the empirical-measure Wasserstein convergence rate $\\varphi_{p,\\beta,d}(n)$ used in Theorems 1.3 and 4.3."},{"cited_title":"and Zhang, X.: Ergodicity of stochastic differen tial equations with jumps and singular coeffi- cients, Ann","cited_arxiv_id":null,"evidence_quote":"Supplies the stochastic Gronwall inequality used for the moment estimates and the common-noise argument."},{"cited_title":"and Delarue, F.: Probabilistic Theory of Mean Field Games with Applications II: Mean Field Games with Common Noise and Master Equations , Springer, Cham, 2018","cited_arxiv_id":null,"evidence_quote":"Provides the framework for common noise and the definition of conditional propagation of chaos used in Section 4."},{"cited_title":"and Zangeneh, Bia n Z.: Propagation of chaos for stochastic spatially structured neuronal networks with delay driven b y jump diffusions, Ann","cited_arxiv_id":null,"evidence_quote":"Documents the long-missing reference problem for jump SDEs under one-sided Lipschitz drift and serves as the comparison baseline under $L^2$-Wasserstein assumptions."}],"review_version":1}