{"id":"86d8175e-d27e-498d-b647-62f6700863c6","arxiv_id":"2504.15898","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"McKean-Vlasov SDEs with pure-jump Lévy noise are shown to have stationary distributions under Lyapunov conditions, multiple such distributions under local dissipativity, and a unique one with exponential convergence under small interaction strength.","lead":"This paper proves conditions for existence, uniqueness, and multiplicity of stationary distributions for McKean-Vlasov stochastic differential equations driven by pure-jump Lévy processes. It also gives examples where such equations admit one, two, or three stationary distributions, and establishes exponential convergence to the equilibrium in the unique case.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1 and Theorem 1.7 rest on the unproved standing assumption that (1.5) is strongly well-posed; because this premise is not listed among (A1)-(A6), the central claims are incomplete as stated.","rationale":"The reader's weakest-assumption analysis identifies the same load-bearing concern that I would: strong well-posedness of (1.5) is assumed rather than proved and is not among the numbered hypotheses. This matters because the fixed-point formulation for stationary distributions and the nonlinear semigroup used in the uniqueness/convergence theorem are only meaningful once the nonlinear SDE has a well-defined solution flow. The concern is external to the main fixed-point technique, but it is genuine: if the cited [2, Theorem 1] does not cover the assumptions used here, then Theorems 1.1 and 1.7 are conditional on an unverified premise. I do not see a reason to move the verdict: the paper is otherwise technically careful, and the issue can be fixed by making strong well-posedness an explicit hypothesis or by supplying the missing verification. Thus the appropriate verdict remains the same as the reader's CONDITIONAL assessment, with no adjustment to the recommendation.","tokens_in":37818,"tokens_out":12480,"duration_ms":123089,"concrete_test":"Verify whether the hypotheses of [2, Theorem 1] are implied by Assumptions (A1)-(A6) together with the Lévy condition (1.6). In particular, check whether [2] requires one-sided Lipschitz or monotonicity of b(x, μ) in x uniformly in μ, and whether the radial Lyapunov bound (1.7) alone supplies it. If the cited theorem's conditions are not met by the examples and general framework of the paper, add strong well-posedness as an explicit assumption (A0) to Theorems 1.1 and 1.7, or supply a proof of strong well-posedness for the pure-jump case with heavy-tailed noise.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's strongest claims, especially Theorem 1.7, depend on the nonlinear semigroup P*_t and the identification of stationary distributions with fixed points of μ ↦ π_μ. Both require the McKean-Vlasov SDE (1.5) to be well-posed, yet the paper only states after (1.5) that strong well-posedness is assumed 'under suitable conditions' and cites the authors' preprint [2, Theorem 1]. This premise is not included in the numbered assumptions, and it is used in an essential way: the proof of Theorem 1.1 converts a fixed point of μ ↦ π_μ into a stationary distribution of (1.5) by invoking weak uniqueness of (1.5), and the proof of Theorem 1.7 uses Duhamel comparisons between frozen and nonlinear flows through the semigroup P*_t. If [2, Theorem 1] does not apply under exactly the conditions (A1)-(A6) used in this paper, then the existence and uniqueness theorems are conditional on an external, self-cited result that is not independently verified here. The convergence bound (1.18) for every initial μ ∈ P_β inherits the same difficulty, since it quantifies the nonlinear semigroup P*_t. This is not an internal inconsistency in the fixed-point argument, but it makes the hypotheses of the central theorems incomplete as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies stationary distributions of McKean-Vlasov SDEs driven by pure-jump Lévy processes. It proves, under Lyapunov-type and continuity assumptions, the existence of at least one stationary distribution via Schauder's fixed point theorem (Theorem 1.1), the existence of multiple stationary distributions under a locally dissipative condition (Theorem 1.3), and, when the measure-dependence constant K1 is sufficiently small, uniqueness of the stationary distribution together with exponential convergence in weighted total variation distance (Theorem 1.7). The appendix provides an alternative uniqueness result in the L1-Wasserstein distance and gives explicit conditions for the ergodicity of the frozen SDE. Two examples exhibit one, two, or three stationary distributions. The proofs are detailed and combine Krylov-Bogoliubov arguments, fixed-point methods, coupling estimates, and Duhamel-type comparisons.","tokens_in":38081,"tokens_out":13820,"duration_ms":118173,"significance":"If the results are correct, they provide a systematic treatment of existence, uniqueness, and multiplicity of stationary distributions for distribution-dependent SDEs with jumps, extending a line of work previously developed mainly for Brownian McKean-Vlasov SDEs. The total-variation uniqueness result with an explicit threshold K* and exponential convergence is a meaningful contribution, as is the verification of the abstract assumptions in concrete examples. The paper is written with care and contains many explicit computations. However, two load-bearing points need attention: the unproved standing assumption on strong well-posedness of the nonlinear SDE, and a gap between uniqueness in the invariant set P_{M*}^{β*} and the claimed uniqueness in the full class P_{β*}.","major_comments":[{"comment":"The paper assumes, rather than proves, that the McKean-Vlasov SDE (1.5) is strongly well-posed, citing the authors' preprint [2, Theorem 1]. This premise is load-bearing: the proof of Theorem 1.1 converts a fixed point of μ ↦ π_μ into a stationary distribution of (1.5) by invoking weak uniqueness of (1.5), and the proof of Theorem 1.7 uses the nonlinear semigroup (P*_t) and the decoupled equation (4.11), both of which require the well-posedness of (1.5). Since this standing assumption is not included in the numbered hypotheses (A1)-(A6) and is not proved or independently verified in the present paper, the main theorems are conditional on an external result whose hypotheses are not checked. Please either prove well-posedness under (A1)-(A6) or state it as an explicit numbered assumption with a proof or a fully verified reference.","section":"Section 1.2, text following Eq. (1.5)"},{"comment":"The contraction argument establishes uniqueness of fixed points of the map Λ within the invariant set P_{M*}^{β*}, but the theorem claims uniqueness of a stationary distribution within the larger class P_{β*}. No argument is given that every stationary distribution in P_{β*} necessarily lies in P_{M*}^{β*}; without such an argument, a second stationary distribution with β*-th moment larger than M* is not excluded by the proof. This can be repaired by combining the fixed-point equation μ = π_μ with the moment bound in Lemma 2.1 to obtain a uniform bound, but this step is missing and should be added.","section":"Section 4, proof of Theorem 1.7"},{"comment":"The notation M* is used both for the moment threshold in Lemma 2.2 and for the threshold appearing in Assumption (A4) and Theorem 1.3. These appear to be different constants, and the proof of Theorem 1.3 requires both. The paper should distinguish the two quantities clearly, for example by writing M^* for the Lemma 2.2 constant and M_* for the (A4) threshold, to avoid ambiguity in the statements of Theorem 1.3.","section":"Section 2, Lemma 2.2 and Assumption (A4)"}],"minor_comments":[{"comment":"The statement of Theorem 5.1 contains a typo: it reads 'Assume that (H1) and (H2) hold, and that (1.6) is satisfised with β∗ ∈ [1, 2]. hold.' The extra word 'hold' should be deleted, and 'satisfised' should be 'satisfied'.","section":"Section 5.1, Theorem 5.1"},{"comment":"The phrase 'with regarding to' in the abstract should be 'regarding' or 'with regard to'; there are also a few other grammatical slips throughout the paper that should be corrected in a final editorial pass.","section":"Abstract and Introduction"},{"comment":"The verification of Assumptions (A2) and (A31) in Example 1.5 is delegated to [37, Theorem 7.4]; since these assumptions are central to the example, it would be helpful to state briefly why the cited theorem applies to the specific drift in (1.10).","section":"Example 1.5"}],"recommendation":"major_revision","confidential_remarks":"The main concern is the manuscript's dependence on the authors' own preprint [2] for the strong well-posedness of the nonlinear SDE. This is not a circularity, but it makes the hypotheses of the main theorems incomplete as written. If [2] is not published or not independently verifiable, the authors should include a proof of the well-posedness statement under their assumptions. The second gap concerning uniqueness within P_{β*} rather than P_{M*}^{β*} is also fixable. The paper is otherwise a substantial and careful contribution, and I would be willing to look at a revised version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"What you should know: this is a competent, carefully-written paper that does something genuinely new. It carries the Schauder fixed-point approach for stationary distributions of McKean-Vlasov SDEs from Brownian noise over to pure-jump Lévy noise, and it provides the first uniqueness-and-exponential-convergence result in weighted total variation for such systems. The proofs are detailed, the examples are concrete, and the authors are transparent about what their assumptions do. Credit where due: the jump-noise case is not a trivial rewrite. The smoothed Lyapunov functions V_{ε,β,y} and the splitting of large and small jumps show real technical care. The multiplicity examples, adapted from Zhang and Tugaut, are adapted with enough modification to be informative.\n\nThe soft spots are real but not fatal. The biggest is flagged in your stress-test note: the paper simply assumes strong well-posedness of the nonlinear SDE, citing the authors' own preprint [2], and this assumption is not listed among (A1)-(A6). The fixed-point characterization and the nonlinear semigroup P*_t used in Theorem 1.7 do depend on it. I think the concern lands. It does not wreck the paper—the authors state the assumption explicitly and point to the exact external theorem—but as written, the main theorems are conditional on a premise that is not fully specified in the hypotheses. A referee should ask for this to be cleaned up, either by adding a numbered assumption that includes the needed well-posedness conditions or by proving that (A1)-(A6) imply them. The other caveats are milder: the threshold K* is non-explicit, and (A5)-(A6) are heavy and not always easy to verify in applications. The paper does provide verification routes in the appendix, which helps. I would not describe any of this as a load-bearing flaw; the central argument is sound and the conditions are stated honestly.\n\nBottom line: this is a paper for researchers working on McKean-Vlasov equations or Lévy-driven processes. It deserves a serious referee. Send it out; expect a request for revision on the well-posedness issue, but this is a genuine contribution.","headline":"Solid, honest extension of the Brownian fixed-point machinery to jump noise; the load-bearing caveat is the standing strong well-posedness assumption, which sits outside the numbered hypotheses.","tokens_in":38652,"tokens_out":1638,"would_cite":true,"duration_ms":17355,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H10","60J76","60G10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Weak mean-field coupling forces a unique stationary law for Lévy-driven McKean–Vlasov SDEs.","keywords":["McKean-Vlasov SDE","Lévy noise","stationary distribution","fixed point theorem","weighted total variation distance","phase transition","multiplicity","exponential ergodicity"],"falsifier":"Construct an explicit Lévy-driven McKean–Vlasov SDE satisfying Assumptions (A1), (A2), (A3), (A5), and (A6) for which one can compute the measure-dependence constant $K_1$ in (1.17) and exhibit two distinct stationary distributions while $K_1 \\le K^*$; this would directly contradict Theorem 1.7. A more feasible test: for the double-well potential example (1.21), compute $K_1$ numerically as the Lipschitz constant of $\\mu \\mapsto b(\\cdot, \\mu)$ in the weighted total variation norm and compare it with a numerically estimated value of $K^*$ obtained from the contraction factor in the proof.","tokens_in":37581,"feed_emoji":"📈","tokens_out":3310,"duration_ms":31468,"temperature":0.7,"pith_summary":"This paper asks when a McKean–Vlasov stochastic differential equation driven by pure-jump Lévy noise has one, several, or exactly one stationary distribution. The authors prove that under a Lyapunov condition and continuity of the drift in the measure variable, a stationary distribution always exists; under an additional locally dissipative condition, several distinct stationary distributions can coexist; and crucially, if the interaction intensity is small enough, the stationary distribution is unique and every initial law converges to it exponentially fast in a weighted total variation distance. This matters because it gives the first general criteria for phase transitions and uniqueness in the jump-noise setting, where the geometry of the state space is replaced by the structure of the Lévy measure.","feed_headline":"Tiny mean-field coupling forces a unique stationary law","feed_subtitle":"New fixed-point criteria decide existence, uniqueness, and multiplicity for McKean–Vlasov SDEs with Lévy jumps.","key_machinery":"The central object is the fixed-point map $\\Lambda: \\mathcal{P}_{\\beta_*} \\to \\mathcal{P}_{\\beta_*}$, $\\Lambda(\\mu) = \\pi_\\mu$, where $\\pi_\\mu$ is the unique invariant probability measure of the linear (frozen) SDE $dY_t = b(Y_t, \\mu) \\, dt + dZ_t$. Fixed points of $\\Lambda$ are exactly the stationary distributions of the original nonlinear SDE. Existence and multiplicity rely on Schauder's fixed point theorem applied to $\\Lambda$ on a compact convex subset of a Banach space of signed measures with a Kantorovich–Rubinstein metric; uniqueness is proved by showing $\\Lambda$ is a contraction in the weighted total variation norm, using the Duhamel formula $P^{\\mu_1}_t \\psi - P^{\\mu_2}_t \\psi = \\int_0^t P^{\\mu_2}_{t-s} (L^{\\mu_1} - L^{\\mu_2}) P^{\\mu_1}_s \\psi \\, ds$ together with the gradient estimate of Assumption (A6) to control the sensitivity to the measure argument.","core_discovery":"The central result, Theorem 1.7, states that under Assumptions (A1)–(A3) and (A5)–(A6) there is a positive constant $K^*$ such that whenever the measure-dependence constant $K_1$ in (1.17) satisfies $K_1 \\le K^*$, the McKean–Vlasov SDE (1.5) has a unique stationary distribution $\\pi \\in \\mathcal{P}_{\\beta_*}$, and the nonlinear semigroup $(P^*_t)$ converges to $\\pi$ with the explicit rate $\\|P^*_t\\mu - \\pi\\|_U \\le C e^{-\\lambda t}\\|\\mu - \\pi\\|_U$ for all initial measures $\\mu \\in \\mathcal{P}_\\beta$. Existence of at least one stationary distribution is obtained through Schauder's fixed point theorem applied to the map $\\Lambda(\\mu) = \\pi_\\mu$, where $\\pi_\\mu$ is the unique invariant measure of the frozen SDE (1.8); multiplicity is obtained by showing that the fixed-point map preserves small balls centered at distinct points $y_i$, yielding at least $k$ distinct stationary measures when the centers are sufficiently far apart. The uniqueness proof is the technical heart: it uses Duhamel's formula and gradient estimates for jump SDEs to show that $\\Lambda$ is a contraction in the weighted total variation distance whenever $K_1$ is below the threshold $K^*$.","pith_inferences":["The threshold $K^*$ likely scales with the spectral gap of the frozen SDE and the strength of the gradient estimate; an explicit formula for $K^*$ in terms of the constants in (A1), (A5), and (A6) would let practitioners check uniqueness without solving the fixed-point equation.","For symmetric $\\alpha$-stable noise, Corollary 1.9 supplies a workable form of (A6), so the uniqueness theorem applies directly whenever (A1), (A3), (A5), and one-sided Lipschitzness hold; one could numerically test the threshold by computing $K_1$ for concrete potentials such as the double-well example.","The paper's fixed-point approach may transfer to other nonlocal noise models, such as tempered stable or compound Poisson processes, as long as the associated frozen SDE admits a unique invariant measure with suitable moment and coupling estimates.","A natural testable extension is to determine whether the number of stationary distributions in the multiple regime of Example 1.5 is exactly three or could be larger; the current argument gives at least three, and the phase-transition literature suggests the count is governed by the number of local minima of the effective potential."],"forward_implications":["If the small-interaction condition $K_1 \\le K^*$ holds, the nonlinear semigroup is exponentially ergodic in a weighted total variation distance, giving quantitative convergence rates for the law of the process to its unique equilibrium.","The threshold $K^*$ provides a concrete criterion for phase transition: increasing the strength of the mean-field interaction beyond the threshold can destroy uniqueness and create multiple stationary states.","The examples with two and three stationary distributions show that the theory covers genuine non-uniqueness, not just uniqueness regimes, and they supply explicit families of Lévy-driven McKean–Vlasov SDEs where the number of equilibria is known exactly.","The uniqueness theorem also extends to the $L^1$-Wasserstein setting via the appendix, yielding a companion criterion for uniqueness when the drift is continuous in the measure under the Wasserstein distance.","The results pave the way for studying local convergence near each equilibrium in the multiple-stationary regime, a problem the authors explicitly leave open."],"supporting_citations":[{"why":"Supplies the strong well-posedness of the McKean–Vlasov SDE with jumps under monotonicity, a premise the paper assumes rather than proves.","marker":"[2]"},{"why":"Provides the Schauder fixed-point framework for existence and non-uniqueness of stationary distributions for distribution dependent SDEs, which the authors adapt to the Lévy-noise setting.","marker":"[39]"},{"why":"Establishes exponential ergodicity for SDEs and McKean–Vlasov processes with Lévy noise, used in the appendix to verify the contraction condition (1.16).","marker":"[20]"},{"why":"Yan's perturbation theorem for semigroups of linear operators is invoked in the Duhamel formula that transfers contractivity from the frozen semigroup to the nonlinear semigroup.","marker":"[38]"},{"why":"Provides the Bismut–Elworthy–Li type derivative formula and gradient estimates for SDEs driven by $\\alpha$-stable processes, used to verify Assumption (A6).","marker":"[41]"},{"why":"Delivers refined basic couplings and Wasserstein-type contraction results for Lévy-driven SDEs, employed in the appendix for the $L^1$-Wasserstein uniqueness theorem.","marker":"[24]"},{"why":"Introduces the nonlinear semigroup formalism for distribution dependent SDEs, used to formulate the exponential convergence statement (1.18).","marker":"[33]"},{"why":"Provides the phase-transition analysis for a class of time-inhomogeneous diffusions, used in Example 1.5 to deduce the existence of three stationary distributions.","marker":"[22]"}],"fun_headline_variants":["Coupling threshold decides unique or multiple stationary laws","Fixed-point maps count stationary laws for jump McKean-Vlasov","Schauder fixed point and Duhamel contraction give stationary laws","Levy-driven McKean-Vlasov: threshold for unique stationary state","All stationary laws for Levy-driven McKean-Vlasov at a glance"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper assumes, without proof in this work, that the McKean–Vlasov SDE (1.5) is strongly well-posed; if that well-posedness fails, the fixed-point formulation and the nonlinear semigroup used in the main theorems are not defined.","fun_headline_variants_meta":{"raw":{"variants":["Coupling threshold decides unique or multiple stationary laws","Fixed-point maps count stationary laws for jump McKean-Vlasov","Schauder fixed point and Duhamel contraction give stationary laws","Levy-driven McKean-Vlasov: threshold for unique stationary state","All stationary laws for Levy-driven McKean-Vlasov at a glance"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000905,"raw_usage":{"total_tokens":3933,"prompt_tokens":1022,"completion_tokens":2911,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":638,"completion_tokens_details":{"reasoning_tokens":2819}},"tokens_in":638,"tokens_out":2911,"duration_ms":20014,"temperature":1.0,"reasoning_tokens":2819,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T11:15:15.853875+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct an explicit Lévy-driven McKean–Vlasov SDE satisfying Assumptions (A1), (A2), (A3), (A5), and (A6) for which one can compute the measure-dependence constant $K_1$ in (1.17) and exhibit two distinct stationary distributions while $K_1 \\le K^*$; this would directly contradict Theorem 1.7. A more feasible test: for the double-well potential example (1.21), compute $K_1$ numerically as the Lipschitz constant of $\\mu \\mapsto b(\\cdot, \\mu)$ in the weighted total variation norm and compare it with a numerically estimated value of $K^*$ obtained from the contraction factor in the proof.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Schauder fixed-point framework for existence and non-uniqueness of stationary distributions for distribution dependent SDEs, which the authors adapt to the Lévy-noise setting."},{"cited_title":"and Wang, J.: Exponential e rgodicity for SDEs and McKean-Vlasov processes with Lévy noise, Ann","cited_arxiv_id":null,"evidence_quote":"Establishes exponential ergodicity for SDEs and McKean–Vlasov processes with Lévy noise, used in the appendix to verify the contraction condition (1.16)."},{"cited_title":"1321, Springer, 1988, 89–91","cited_arxiv_id":null,"evidence_quote":"Yan's perturbation theorem for semigroups of linear operators is invoked in the Duhamel formula that transfers contractivity from the frozen semigroup to the nonlinear semigroup."},{"cited_title":"Appl., 123 (2013), 1213–1228","cited_arxiv_id":null,"evidence_quote":"Provides the Bismut–Elworthy–Li type derivative formula and gradient estimates for SDEs driven by $\\alpha$-stable processes, used to verify Assumption (A6)."},{"cited_title":"Appl., 128 (2018), 595–621","cited_arxiv_id":null,"evidence_quote":"Introduces the nonlinear semigroup formalism for distribution dependent SDEs, used to formulate the exponential convergence statement (1.18)."},{"cited_title":"and Zhang, M.: Phase transitions for a cl ass of time-inhomogeneous diffusion processes, J","cited_arxiv_id":null,"evidence_quote":"Provides the phase-transition analysis for a class of time-inhomogeneous diffusions, used in Example 1.5 to deduce the existence of three stationary distributions."}],"review_version":1}