{"id":"7bc5bc6c-63da-4126-b7bc-c7b6e3a3915b","arxiv_id":"2509.02249","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Conditional McKean-Vlasov jump diffusions are exponentially contractive in law, and the contraction rate improves as jump noise intensity grows.","lead":"This paper proves that the random distribution of a jump-diffusion process with both private and common jump noise converges exponentially fast to a unique equilibrium, under a partial dissipation condition. It provides the first ergodicity result for this class of conditional McKean-Vlasov jump diffusions, using a new reflection coupling tailored to small jumps.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The exponential contraction in Theorem 1.2 depends on (H2), a uniform-in-time conditional propagation-of-chaos bound that is only supported by a self-citation; if (H2) fails, the J_i terms in Proposition 3.2 do not vanish and (1.7) is not obtained.","rationale":"The paper's central claim is a Wasserstein contraction of the conditional-law flow. The proof proceeds by an n-particle approximation: finite-time conditional propagation of chaos (Prop. 2.4), a uniform-in-time estimate via an asymptotic reflection coupling (Prop. 3.2), and then n→∞. The uniform-in-time step is the crux, and it is exactly where (H2) enters: the bound (1.5) controls the drift discrepancy between the true conditional law and the leave-one-out empirical measure. If that discrepancy is not uniform in time, the φ(n) term in (3.4) becomes an O(1) error, and taking n→∞ in the proof of Theorem 1.2 does not yield the exponential decay (1.7). The paper's only support for (H2) is a pointer to [6, Lemma 4.1] in Remark 1.1; because the lemma is not stated, a reader cannot verify that examples satisfying (H1)-(H3) also satisfy (H2). Thus the practical content of the theorem depends on a strong, unverified hypothesis. I also note the suspicious term λ3 eµ^n_t(|·|) in (3.6), which the natural add-subtract argument would replace with λ3∥Z∥_1; this appears to be a typo rather than a fundamental flaw, but it should be corrected. The reader's CONDITIONAL verdict is appropriate: the main construction is novel and the proof is plausible, but (H2) must be independently verified and the displayed estimate (3.6) cleaned up before the result can be fully trusted.","tokens_in":29709,"tokens_out":28429,"duration_ms":292850,"concrete_test":"Retrieve [6, Lemma 4.1] and verify directly that its hypotheses imply (1.5) for the system (2.4) under (H1), with sup_{t≥0} E|b(X^i_t,μ^i_t)-b(X^i_t,eµ^{n,-i}_t)| ≤ φ(n), φ(n)→0. If the lemma does not yield a rate uniform in t, construct a one-dimensional drift satisfying (H1)-(H3) whose conditional law μ_t has sup_t E|X_t|<∞ but sup_t E W1(μ_t,eµ^{n,-i}_t) does not converge to 0; such an example would show (H2) is exactly the unproved premise needed for (1.7).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The main theorem (Theorem 1.2) is proved by sandwiching W1(Lμ_t,L\\barμ_t) between empirical measures of n-particle systems and then taking n→∞. The only place the paper controls the difference between the true conditional law μ^i_t and the leave-one-out empirical measure eµ^{n,-i}_t is assumption (H2), eq. (1.5). This bound is used in Proposition 3.2 to treat the J_i terms in (3.6) and to pass from the differential inequality to (3.12). Without a uniform-in-time rate, the term (1/n)Σ_i E J_i need not vanish as n→∞, and the Gronwall argument leaves an O(1) error that survives the limit in the proof of Theorem 1.2; the exponential contraction (1.7) is then not reached. The paper does not prove (H2); Remark 1.1 only states that 'some sufficiencies are furnished in [6, Lemma 4.1]', a self-citation whose content is not reproduced. Since (H2) is effectively a uniform-in-time conditional propagation-of-chaos property, it is a strong, load-bearing hypothesis. Separately, the display (3.6) contains a term λ3 eµ^n_t(|·|) whose derivation is not shown; the standard add/subtract argument would give λ3∥Z^{n,ε}_t∥_1 instead. This looks like a typo rather than a fatal flaw, but it reinforces that the proof of Proposition 3.2 needs careful checking.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies one-dimensional conditional McKean-Vlasov jump diffusions driven by rotationally invariant Lévy noise with both idiosyncratic and common jumps. The main result, Theorem 1.2, states that under hypotheses (H1)–(H3) and for sufficiently small mean-field interaction λ3, the law of the conditional distribution flow is exponentially contractive in the Wasserstein distance W1, thus yielding a unique invariant probability measure for the measure-valued process. The proof proceeds through a stochastic interacting particle system, an asymptotic coupling by reflection adapted to jump noise, a finite-time conditional propagation of chaos, and a uniform-in-time conditional propagation-of-chaos estimate obtained under (H2). The paper also shows that the conditional distribution flow solves a stochastic Fokker–Planck equation driven by a Poisson random measure and that larger jump intensities improve the exponential rate.","tokens_in":30088,"tokens_out":16998,"duration_ms":177114,"significance":"If the main theorem is correct, it is a meaningful extension of ergodicity results for conditional McKean-Vlasov equations from Brownian common noise to Lévy jump noise, with a coupling construction that treats small jumps by reflection and large jumps synchronously. The paper gives explicit constant dependence and an example showing that both idiosyncratic and common jump intensities accelerate convergence. The proof strategy is original in this setting and the finite-time conditional PoC results are useful in themselves. However, the central uniform-in-time estimate rests on the strong hypothesis (H2), which is not established for the jump setting and is only supported by a self-citation, and the proof of Proposition 3.2 contains an inequality that appears false as written. These issues are load-bearing and prevent the present version from being accepted.","major_comments":[{"comment":"The standard add/subtract argument in (3.6) does not produce the term λ3 eμ^n_t(|·|). After adding and subtracting b(X^i_t,eμ^n_t) and b(X^{i,n,ε}_t,eμ^n_t), the measure-dependent part is bounded by λ3 W1(eμ^n_t,bμ^{n,ε}_t), and the natural coupling gives W1(eμ^n_t,bμ^{n,ε}_t) ≤ (1/n)Σ_j |X^j_t-X^{j,n,ε}_t| = ∥Z^{n,ε}_t∥_1. Replacing this by λ3 eμ^n_t(|·|) is not justified and is false: eμ^n_t(|·|) has expectation O(1) in general, while ∥Z^{n,ε}_t∥_1 is the quantity that can be absorbed into the λ0 term. The later bound, 'λ3Eeμ^n_t(|·|) + 1/n Σ_i EJ_i ≤ λ3E∥Z^{n,ε}_t∥_1 + ...', is also invalid as written; for example, if all X^j_t=x and X^{j,n,ε}_t=0, then λ3E eμ^n_t(|·|)=λ3|x| whereas the right-hand side is O(|x|/n). This error directly affects Proposition 3.2 and hence Theorem 1.2. If this is a typo, it must be corrected in (3.6), (3.9), and the display before (3.12), and the Gronwall","section":"Section 3, Eq. (3.6) and subsequent display before (3.12)"},{"comment":"Hypothesis (H2) is a uniform-in-time conditional propagation-of-chaos bound for the drift: max_i sup_t E|b(X^i_t,μ^i_t)-b(X^i_t,eμ^{n,-i}_t)| ≤ φ(n). This is the only place in Proposition 3.2 where the difference between the true conditional law and the leave-one-out empirical measure is controlled, and it is essential for the J_i terms to vanish as n→∞. If (H2) fails, the O(φ(n)) term in (3.4) does not disappear and the contraction (1.7) is not obtained. The paper does not prove (H2); Remark 1.1 merely states that 'some sufficiencies are furnished in [6, Lemma 4.1]', a self-citation whose content is not reproduced. Since [6] concerns Brownian common noise, it is not immediate that those sufficiencies apply to the jump setting of the present paper. The authors should either prove (H2) for a class of examples satisfying (H1)–(H3), or reproduce the relevant lemma and verify its hypotheses","section":"Assumption (H2), Eq. (1.5)"},{"comment":"The distance function f in (3.10) is only C^1, and under the stated hypotheses on g_* its second derivative f''(r) = -g'_*(r)e^{-g_*(r)} is unbounded as r↓0 (e.g., in Example 3.4, g'_*(r) ~ r^{θ-1} with θ∈(0,1)). The proof of Proposition 3.2 applies Itô's formula to f(|Z^{i,n,ε}_t|) and uses the pointwise inequality f(r+δ)+f(r-δ)-2f(r) ≤ f''(r)δ^2. No justification is given for the Itô formula in this non-C^2 case. Although this is likely repairable by a standard mollification argument or by citing a generalized Itô formula, it is a technical gap in a central proof and should be addressed explicitly.","section":"Proposition 3.2 and function f in (3.10)"}],"minor_comments":[{"comment":"There are several typos, e.g., 'Poission' in Section 2.1, 'Theorem1.2' missing a space, and in Lemma 2.7 the display contains 'X^{N,N,ε}_t' where 'n' is intended.","section":"Throughout"},{"comment":"The notation in the proof of statement (ii) is sometimes heavy; the definitions of the five Γ^{j,ε}_i terms are clear but could be displayed more uniformly.","section":"Section 2, Lemma 2.7"},{"comment":"The two systems with initial data X_0 and \\bar X_0 are denoted with the same symbol X^i_t in places, which is confusing. Use \\bar X^i_t consistently in (3.16) and thereafter.","section":"Section 3, proof of Theorem 1.2"},{"comment":"The discussion of the coupling construction is helpful, but the comparison with [6,30] could state explicitly that the present paper uses a threshold on jump sizes rather than on Brownian increments.","section":"Remark 1.4"},{"comment":"It would be useful to state explicitly that the constants C_θ, C_1 are positive and that the monotonicity of Λ_1,Λ_2 in |σ|,|σ_0| is what drives the 'noise enhances convergence' statement.","section":"Example 3.4"}],"recommendation":"major_revision","confidential_remarks":"The paper has a solid overall strategy and the asymptotic coupling construction for jump noise is interesting. My main concern besides the specific technical error in (3.6) is the heavy reliance on the unproved uniform-in-time condition (H2), which is only supported by a self-citation. The authors should be asked to either prove H2 in the Lévy setting or clearly delimit the class of examples where the theorem applies. The typo in (3.6) is likely fixable, but as written it invalidates the central estimate of Proposition 3.2."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear —,\n\nThe thing to know: this is a genuine extension of [6,30] from Brownian common/idiosyncratic noise to rotationally invariant jump noise, proving exponential W1-contraction for the conditional distribution flow of a one-dimensional conditional McKean-Vlasov jump diffusion. The coupling construction—reflection for small jumps, synchronous for large jumps, with a threshold tied to the distance—is new and the paper works through the machinery carefully. Proposition 3.2, the radial-process inequality, is the heart, and it is plausible.\n\nWhat it does well: it sets up the SFPE for the conditional law driven by Poisson noise, proves finite-time conditional PoC under weaker moment assumptions than prior work, and shows the particle-system approximation is tight enough to pass to the limit. The main theorem is honestly stated with the small-interaction condition λ3 < λ*_3.\n\nSoft spots, in order:\n\n1. H2 is load-bearing and essentially a uniform-in-time conditional propagation-of-chaos estimate. The paper cites [6, Lemma 4.1] for sufficiencies, but does not reproduce or prove it. If H2 fails, the J_i terms in Proposition 3.2 do not vanish and the contraction in (1.7) does not follow. This makes the main result conditional on an assumption that is arguably as hard as the ergodicity itself. It is not fatal—many theorems are conditional—but a referee should ask for more detail or a more primitive condition.\n\n2. In the display above (3.6) the term λ3 eµ^n_t(|·|) appears where the add/subtract argument gives λ3∥Z^{n,ε}_t∥_1. I agree with the stress-test that this is probably a typo, but it sits inside the key estimate and should be corrected.\n\n3. The distance function f in (3.10) is only C1 on [0,∞) with possibly unbounded f'' near 0; the Itô-formula step needs an explicit smooth approximation. This is standard in the coupling literature, but here it is not supplied.\n\n4. The abstract's claim that noise intensity enhances the convergence rate is established only for the stable-like example (3.14); the word 'can' saves it, but the statement should be tied to the example.\n\nBottom line: worth a serious referee. The novelty is real, the proof is detailed, and the soft spots are addressable without changing the main claim. I would send it to review and ask for the fixes above. For an expert in McKean-Vlasov ergodicity it is a useful contribution; I'd probably cite it once H2 is made more transparent.","headline":"Solid extension of Brownian common-noise ergodicity to jump noises; the main contraction result is proved under a strong, unproved uniform-in-time propagation-of-chaos assumption (H2), so it is a conditional theorem rather than a fully self-contained one, but the proof is detailed and the novelty is real.","tokens_in":30596,"tokens_out":7231,"would_cite":true,"duration_ms":69272,"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":"For a one-dimensional conditional McKean-Vlasov jump diffusion, the flow of conditional laws contracts exponentially fast in Wasserstein distance whenever the mean-field interaction is weak; hence the measure-valued process has a unique inv","keywords":["conditional McKean-Vlasov SDE","jump diffusion","common noise","exponential ergodicity","propagation of chaos","Wasserstein contraction","asymptotic coupling by reflection","measure-valued process"],"falsifier":"Compute W1(L mu_t, L bar-mu_t) numerically for a one-dimensional linear example, e.g. b(x,mu) = -x + lambda_3 (mean(mu) - x), with symmetric alpha-stable noise for alpha in (1,2). If the ratio W1(L mu_t, L bar-mu_t)/W1(L mu_0, L bar-mu_0) does not decay exponentially for some lambda_3 below the threshold, or if increasing the common jump intensity slows instead of speeds the decay, Theorem 1.2 would be contradicted. A direct analytic check is to evaluate Lambda_1 and Lambda_2 in Example 3.4 for two different values of |sigma_0| and see whether the predicted rate ordering matches the actual exp","tokens_in":29582,"feed_emoji":"📈","tokens_out":6504,"duration_ms":67593,"temperature":0.7,"pith_summary":"The paper studies the long-time behaviour of a one-dimensional conditional McKean-Vlasov SDE: a particle whose drift depends on its own conditional law given a common noise, driven by two independent rotationally invariant pure-jump Levy processes—one idiosyncratic, one common. The object of interest is not the particle path but the conditional distribution flow, a measure-valued process solving a stochastic partial integral-differential equation driven by a Poisson random measure. The central result is exponential contraction of this flow in the Wasserstein distance between laws of conditional laws, under partially dissipative drift and a small mean-field interaction. As a consequence the measure-valued process has a unique invariant probability measure. The argument goes through a uniform-in-time conditional propagation of chaos and a new coupling construction that reflects small jumps and synchronizes large ones.","feed_headline":"Jump noise speeds exponential relaxation in mean-field systems","feed_subtitle":"Larger common and private jump intensities make the conditional law flow converge faster to its stationary measure.","key_machinery":"The asymptotic coupling by reflection is the load-bearing tool. For a small jump, the second copy's jump is multiplied by the reflection matrix Pi_{epsilon,d}(x) = I_d - 2 h_epsilon(rho(x)) n(phi(x)) tensor n(phi(x)), which in one dimension reduces to 1 - 2 h_epsilon(||x||_1), flipping the sign of the jump so the two copies move toward each other; for large jumps the coupling is synchronous. A carefully chosen distance function f built from g_*(r) = lambda_1 integral_0^r s / F_{sigma,sigma_0}(s) ds makes the radial process contract: the reflected small jumps make the jump-integral terms vanish by rotational invariance, leaving a negative drift of order -lambda_0 |z|. The same construction yi","core_discovery":"The paper establishes Theorem 1.2: for the one-dimensional conditional McKean-Vlasov jump diffusion (1.1), under assumptions (H1)-(H3) and sigma, sigma_0 different from zero, there are constants C, lambda*_0, lambda*_3 > 0 such that for all t>=0 and lambda_3 in [0,lambda*_3], W1(L mu_t, L bar-mu_t) <= C e^{-lambda*_0 t} W1(L mu_0, L bar-mu_0). In words, the law of the conditional law forgets its initial condition exponentially fast, uniformly over all times, whenever the drift's dependence on the measure variable is not too strong. The proof first shows the conditional distribution flow solves a stochastic Fokker-Planck equation driven by a Poisson random measure, then proves an infinite-hor","pith_inferences":["Editorial: the same threshold argument suggests a quantitative trade-off curve for lambda*_3 versus noise intensity: stronger noise should permit stronger mean-field coupling while still preserving contraction, a relation the paper does not spell out.","Editorial: the small-jump reflection / large-jump synchronous split is a natural blueprint for numerical simulation of coupled conditional laws; one could test the predicted rate monotonicity in a stable-noise example such as the paper's Example 3.4.","Editorial: the one-dimensional restriction comes from the vanishing of the jump-integral terms; a high-dimensional extension would likely need a different geometric construction or a non-isotropic distance, since the scalar sign-flip reflection no longer suffices.","Editorial: if rotational invariance of the Levy measures is dropped, the odd integrals do not vanish, so the same proof breaks; a change-of-measure or tilting coupling might restore contraction but is beyond the paper's scope."],"forward_implications":["If the theorem holds, the conditional law flow (mu_t) has a unique invariant probability measure whenever lambda_3 is below an explicit threshold, and convergence is exponentially fast in the Wasserstein distance W1.","Larger Levy intensity in either the common or the idiosyncratic noise increases the exponential rate, because the constants Lambda_1 and Lambda_2 in assumption (H3) decrease with |sigma| and |sigma_0|.","The proof supplies an infinite-time conditional propagation of chaos under a qualitative condition (H2), requiring only a first moment rather than higher-order integrability.","The result extends exponential ergodicity from Brownian common-noise McKean-Vlasov models to pure-jump Levy-type noises.","The SFPE formulation gives a Poisson-random-measure-driven forward equation for the conditional law flow, which is the natural object for further statistical or numerical study."],"supporting_citations":[{"why":"Supplies the Brownian-common-noise one-dimensional result and Lemma 4.1, the cited sufficiency for assumption (H2); the present proof follows its propagation-of-chaos plus coupling strategy.","marker":"[6]"},{"why":"Gives the reflection/synchronous coupling template for Brownian common noise that the paper adapts to pure-jump noise by thresholding the jump size.","marker":"[30]"},{"why":"Provides the technique for taking conditional expectations of the common-noise jump integral, used to derive the stochastic Fokker-Planck equation of Proposition 2.1.","marker":"[27]"},{"why":"Supplies the exponential-ergodicity framework for McKean-Vlasov processes with Levy noise, including the f(r+delta)+f(r-delta) inequality and the role of rotational invariance.","marker":"[28]"},{"why":"Establishes strong well-posedness of the Levy-driven McKean-Vlasov SDE under weak monotonicity and coercivity, used for the non-interacting and interacting particle systems.","marker":"[5]"},{"why":"The Aldous tightness criterion used to prove that the epsilon-approximate coupled particle system has a weakly convergent subsequence whose limit is the desired coupling.","marker":"[4]"},{"why":"Provides the proof skeleton, cited in Proposition 2.3, that the conditional laws of the non-interacting particles coincide almost surely under the common noise.","marker":"[13]"}],"fun_headline_variants":["Conditional jump diffusion laws forget initial data exponentially","Jump noise accelerates relaxation in conditional mean-field systems","Reflection coupling proves exponential ergodicity for jumpy mean-field laws","Common and private jumps speed up exponential convergence of conditional laws","Partial dissipation suffices for exponential ergodicity with jump noise"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The proof leans on assumption (H2): replacing the true conditional law by the leave-one-out empirical measure of n-1 other particles makes the drift discrepancy vanish uniformly over all time, with a rate phi(n) tending to zero. If this uniformity fails, the particle approximation errors do not die out and the exponential contraction is not obtained.","fun_headline_variants_meta":{"raw":{"variants":["Conditional jump diffusion laws forget initial data exponentially","Jump noise accelerates relaxation in conditional mean-field systems","Reflection coupling proves exponential ergodicity for jumpy mean-field laws","Common and private jumps speed up exponential convergence of conditional laws","Partial dissipation suffices for exponential ergodicity with jump noise"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00049,"raw_usage":{"total_tokens":2232,"prompt_tokens":715,"completion_tokens":1517,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":459,"completion_tokens_details":{"reasoning_tokens":1436}},"tokens_in":459,"tokens_out":1517,"duration_ms":17013,"temperature":1.0,"reasoning_tokens":1436,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T11:44:03.069014+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute W1(L mu_t, L bar-mu_t) numerically for a one-dimensional linear example, e.g. b(x,mu) = -x + lambda_3 (mean(mu) - x), with symmetric alpha-stable noise for alpha in (1,2). If the ratio W1(L mu_t, L bar-mu_t)/W1(L mu_0, L bar-mu_0) does not decay exponentially for some lambda_3 below the threshold, or if increasing the common jump intensity slows instead of speeds the decay, Theorem 1.2 would be contradicted. A direct analytic check is to evaluate Lambda_1 and Lambda_2 in Example 3.4 for two different values of |sigma_0| and see whether the predicted rate ordering matches the actual exp","supporting_citations":[{"cited_title":"and Wang, J.: Long time behavior of one-dimensional McKean-Vlasov SDEs with common noise,J","cited_arxiv_id":null,"evidence_quote":"Supplies the Brownian-common-noise one-dimensional result and Lemma 4.1, the cited sufficiency for assumption (H2); the present proof follows its propagation-of-chaos plus coupling strategy."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the technique for taking conditional expectations of the common-noise jump integral, used to derive the stochastic Fokker-Planck equation of Proposition 2.1."},{"cited_title":"and Wang, J.: Exponential ergodicity for SDEs and McKean-Vlasov processes with Lévy noise,Ann","cited_arxiv_id":null,"evidence_quote":"Supplies the exponential-ergodicity framework for McKean-Vlasov processes with Levy noise, including the f(r+delta)+f(r-delta) inequality and the role of rotational invariance."},{"cited_title":"A note on L\\'evy-driven McKean-Vlasov SDEs under monotonicity","cited_arxiv_id":"2412.01070","evidence_quote":"Establishes strong well-posedness of the Levy-driven McKean-Vlasov SDE under weak monotonicity and coercivity, used for the non-interacting and interacting particle systems."},{"cited_title":"Probab., 6 (1978), 335–340","cited_arxiv_id":null,"evidence_quote":"The Aldous tightness criterion used to prove that the epsilon-approximate coupled particle system has a weakly convergent subsequence whose limit is the desired coupling."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the proof skeleton, cited in Proposition 2.3, that the conditional laws of the non-interacting particles coincide almost surely under the common noise."}],"review_version":1}