{"id":"dac37f72-c83b-4422-a050-84664d2c309a","arxiv_id":"2411.14090","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A general theorem and applications show exponential ergodicity in W1 for McKean-Vlasov SDEs with distribution-dependent noise and only partially dissipative drift, from any initial distribution.","lead":"This paper proves that a broad class of McKean-Vlasov stochastic differential equations, where both the drift and the noise can depend on the law of the solution, still converge exponentially fast to a unique steady state in the Wasserstein-1 distance. The result covers previously excluded cases such as distribution-dependent diffusion coefficients, kinetic (second-order) systems, and α-stable Lévy noise, and it gives a general template for proving such convergence.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Brownian argument is coherent; the load-bearing step is the imported uniform W1-contraction of decoupled SDEs in the kinetic and α-stable cases.","rationale":"The reader's verdict ACCEPT seems right. The reader's weakest_assumption already identifies the uniform decoupled contraction as the most load-bearing point, and I agree. My stress test did not find an internal error in the Grönwall estimates or in the Wψ coupling argument of Lemma 3.5; those are self-contained and correct modulo the cited ψ''≤0 fact from [12]. The phase-transition Example 3.3 does not undermine Theorem 3.1 because its drift does not satisfy the dissipativity part of (A1). The α-stable time-change Lemma 5.1 is plausible, and the moment condition E√S_t<∞ for α>1 is correctly used. Thus the only way the central claim fails is if the imported contraction theorems do not apply with uniform constants. That is a referee-checkable external dependency, not a reason to reject. Hence the verdict should remain UNCHANGED.","tokens_in":20362,"tokens_out":21944,"duration_ms":200631,"concrete_test":"Write out the hypotheses of [23, Theorem 5] and [18, Theorem 1.3] in the notation of §3.2 and §4.1, substitute the scaled parameters (u=1/(2γ), K=2γK1I, Lg=2γ(K1+Lb), and σ(μ)≡constant in x) and confirm: (a) the stated smallness condition coincides exactly with the theorem's hypothesis; (b) the produced constants c0,λ0 are bounded uniformly in μ using only δ, K1, Lb, γ, R, and the bounds in (A2)/(A3). If both checks pass, the paper's central claim is supported; if either fails, the corresponding theorem must be weakened or re-proved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 2's Theorem 2.1 reduces the problem to three hypotheses: uniform contraction of the decoupled family (2.2), a G(t)-bound between different frozen measures, and an H(t)-bound comparing the original flow with the frozen one. The paper proves the G and H bounds by Grönwall arguments in §§3–4, and those steps are internally consistent. What is not proved in this paper is the first hypothesis. In the Brownian non-degenerate case it is proved as Lemma 3.5. In the kinetic Brownian case it is imported from [23, Theorem 5] after the σ(μ)-rescaling in (3.14); in the α-stable cases it is imported from [18, Theorem 1.3] and [14, Theorem 1.1]. The constants c0,λ0 must be independent of μ, and the smallness conditions (3.13) and L_b²γ⁻²<3K1/4 must be exactly the hypotheses of those theorems. If either import has an extra condition—e.g., a bound linking the friction γ, Lipschitz constant L_b and dissipativity K1 that is stricter than stated, or a condition on the Lévy measure—the contraction (3.20)/(4.4) can fail even when the original DDSDE is ergodic, and Theorem 2.1(i) provides no alternative route. This is not an internal inconsistency, but it is the single point on which the central claim hinges outside the Brownian case.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies exponential ergodicity in the L1-Wasserstein distance for McKean-Vlasov SDEs whose drift and noise coefficients may both depend on the law of the solution. The main abstract result, Theorem 2.1, reduces the problem to three ingredients: a uniform W1-contraction for the decoupled SDE family with frozen distribution, a G(t)-bound between invariant measures of two frozen systems, and an H(t)-bound comparing the original flow with the flow frozen at the target invariant measure. The paper then verifies these ingredients for non-degenerate and kinetic Brownian cases and for non-degenerate and kinetic alpha-stable noise, under partially dissipative drift assumptions. It also gives a one-dimensional example showing that a large distribution-coupling constant can destroy existence of stationary measures.","tokens_in":20567,"tokens_out":31438,"duration_ms":285703,"significance":"If correct, the results are a genuine advance: they extend exponential W1-ergodicity from initial Dirac laws to arbitrary P1 initial laws in settings where the noise coefficient itself depends on the law, which was left open by previous work. The Brownian non-degenerate case is essentially self-contained, with a clean reflection-coupling proof of the decoupled contraction in Lemma 3.5 and a coherent Grönwall argument for the H-bound. The time-change proof of Lemma 5.1 for alpha-stable noise is also a useful technical contribution. The main risk is that the kinetic and stable applications rely on imported uniform contraction theorems, so the verification of their hypotheses is load-bearing.","major_comments":[{"comment":"The central hypothesis of Theorem 2.1 - uniform W1-contraction of the decoupled family with constants independent of the frozen parameter - is imported from [23, Theorem 5] and [14, Theorem 1.1] rather than proved in the paper. The manuscript should state precisely the hypotheses of these theorems and verify them for the rescaled drift b-bar in (3.14) and (4.12). In particular, the derivation from condition (3.13) to the inequality L_g u gamma^{-2} < L_K/(2 L_g) is compressed, and neither the role of the dissipativity radius R in (3.16)-(3.19) nor the independence of c0, lambda0 from the frozen measure mu is demonstrated. Because the G- and H-bounds in Theorem 2.1 cannot replace this contraction step, any mismatch between the cited theorems' hypotheses and (A2) would invalidate Theorems 3.6 and 4.2 even when the original DDSDE is ergodic.","section":"Sections 3.2 and 4.2, Eqs. (3.20) and (4.4)"},{"comment":"The proof invokes [14, Theorem 1.1] for the decoupled kinetic stable system (4.12) and then states that the remaining estimates are repetitions of the proof of Theorem 4.1 with K1 replaced by Lb+2. This repetition is not entirely immediate: the H(t)-estimate in Theorem 4.1 uses the dissipative structure of (A3), whereas the kinetic system only supplies the one-sided estimate 2<Delta y, Delta x> + 2<-gamma Delta y + Delta b, Delta y> <= (Lb+2)(...)+ kappa W1^2. Although that estimate is written out, the passage from it to the analogue of (4.8) for the kinetic model should be given in detail, and the smallness condition L_b^2 gamma^{-2} < 3/4 K1 should be checked explicitly against the hypotheses of [14] after the sigma-scaling. As written, the proof delegates too much to a claimed repetition at a load-bearing point.","section":"Section 4.2, proof of Theorem 4.2"}],"minor_comments":[{"comment":"The first inequality for d|X_t^mu - Y_t^mu| suppresses the second-order Itô-Tanaka term and the local-time term; for t<tau the inequality is valid after using (A1)(iii), but the derivation should be expanded to avoid confusion.","section":"Proof of Lemma 3.5"},{"comment":"In the statement, the last display uses W instead of W1; it should read W1(P*_t eta, nu) <= c e^{-lambda t} W1(eta, nu).","section":"Theorem 2.1(ii)"},{"comment":"The piecewise definition of phi contains 'r>2r0' in the last line; the variable should be v, so the line should read 'v > 2r0'.","section":"Corollary 3.4"},{"comment":"In the definition of delta2, the expression 't> log c0 / lambda' appears to have a typo for lambda0, and there is an extra closing brace in the displayed formula; these should be corrected.","section":"Proof of Theorem 4.1"},{"comment":"There are several small typos, including 'adimits' for 'admits' in the proof of Theorem 4.1 and 'Word Scientific' for 'World Scientific' in reference [22]; these should be fixed in the final version.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"For the editor: the paper is technically serious and the main ideas are sound, but the revision should contain a precise statement of the imported contraction theorems in [23], [14], and [18] together with a point-by-point verification of their hypotheses for the rescaled systems. I do not recommend rejection, since the gaps appear local and fixable, but the external-contraction step is the single point on which the kinetic and stable applications hinge."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does real work: it proves a general perturbation/fixed-point theorem for W1-exponential ergodicity of McKean–Vlasov SDEs, then applies it to cases with distribution-dependent noise and arbitrary initial laws in P1. That genuinely improves [28], which was restricted to Dirac initial measures, and the earlier distribution-free-noise results. Theorem 2.1 is a clean contraction argument, and Lemma 5.1 (the time-change estimate for α-stable noise) is a nice reusable tool. The Brownian non-degenerate case is essentially self-contained: Lemma 3.5 supplies the decoupled contraction via reflection coupling with the auxiliary ψ-distance, and the Grönwall steps in Theorem 3.1 control the measure-dependence correctly. Example 3.3 on phase transition is a thoughtful addition.\n\nThe soft spots are concentrated where the stress-test note points. The kinetic and α-stable applications import uniform contraction results from [23, Theorem 5], [18, Theorem 1.3], and [14, Theorem 1.1]. The paper checks that the rescaled systems satisfy the hypotheses—the conditions (3.13) and L_b²γ⁻² < 3K1/4 are exactly the right shape—and the constants are claimed to be independent of the frozen μ. That is plausible and the checks look correct to me, but a referee should verify the alignment of constants with those external theorems, especially the smallness conditions on friction and dissipativity. This is a normal dependency, not a circular one, and the authors do not hide it.\n\nMinor issues: a few typos (e.g., 'W' for 'W1' in Theorem 2.1, a stray brace in the definition of δ2, 'adimits'), and the derivation of (3.11) is compressed but valid given the coupling inequality—worth a referee's eye, not a red flag.\n\nOverall the central argument holds up. The paper is honest about what is imported and what is new, and the math in the Brownian case is sound. I would take it seriously and recommend sending it to a competent referee. It is a solid contribution to the McKean–Vlasov ergodicity literature.","headline":"A genuinely useful general theorem with self-contained Brownian proofs, and a clear, checkable reliance on imported contraction results in the kinetic and stable cases.","tokens_in":21218,"tokens_out":1749,"would_cite":true,"duration_ms":18716,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H10","60J60","60G52","37A25"],"pacs":[],"model":"deepseek-v4-flash","headline":"For McKean–Vlasov SDEs with law-dependent noise and partially dissipative drifts, exponential ergodicity in $W_1$ holds from every initial law.","keywords":["exponential ergodicity","Wasserstein distance","McKean–Vlasov SDE","distribution dependent noise","partially dissipative drift","α-stable noise","kinetic Langevin system","reflection coupling"],"falsifier":"Compute the stationary equations for a one-dimensional model of the form of Example 3.3 with the distribution-coupling strength below the theorem's $\\delta_0$; if two distinct stationary densities solve the equilibrium Fokker–Planck equation, the uniqueness claim of Theorem 3.1 is false. Alternatively, for the kinetic case, exhibit a drift satisfying the partial dissipativity condition but violating the uniform decoupled contraction estimate, and check whether the original distribution-dependent SDE nevertheless has two invariant measures.","tokens_in":20057,"feed_emoji":"📉","tokens_out":6153,"duration_ms":57962,"temperature":0.7,"pith_summary":"This paper establishes that McKean–Vlasov SDEs whose drift is only partially dissipative and whose noise coefficient may itself depend on the law can still converge exponentially fast to a unique equilibrium in the $L^1$-Wasserstein distance $W_1$, for every initial law in $P_1(\\mathbb{R}^d)$. The proof works by freezing the law in the coefficients, proving the resulting classical SDE contracts uniformly in $W_1$, and then treating the law-dependence as a small perturbation. The result covers non-degenerate and multiplicative Brownian noise, degenerate second-order (kinetic) systems, and additive rotationally invariant $\\alpha$-stable noise, in both non-degenerate and kinetic settings, under a smallness condition on the strength $\\kappa$ of the distribution coupling. A sympathetic reader would care because previous results either required distribution-free noise coefficients or only proved convergence from Dirac initial laws; here the initial law is arbitrary in $P_1$.","feed_headline":"Exponential convergence to equilibrium survives law-dependent noise","feed_subtitle":"A two-step freeze-perturb proof covers Brownian, kinetic, and α-stable McKean–Vlasov SDEs from any initial law.","key_machinery":"The machinery is a fixed-point and perturbation scheme on the space of laws. First, freeze a measure $\\mu$ in the coefficients and study the decoupled SDE (2.2); reflection coupling with a specially constructed concave function $\\psi$ (the $\\psi$-Wasserstein distance $W_\\psi$, with $\\psi''\\le 0$) produces a uniform contraction $W_1((P^\\mu_t)_*\\eta_1,(P^\\mu_t)_*\\eta_2)\\le c_0e^{-\\lambda_0 t}W_1(\\eta_1,\\eta_2)$. Synchronous couplings and Gronwall estimates then control the variation of the frozen maps in the distribution variable, giving the growth functions $G(t)$ and $H(t)$. Theorem 2.1 assembles these into a Banach fixed point for $\\Gamma$ on $(P_1(\\mathbb{R}^d),W_1)$ and a one-step contraction for the semigroup $P_t^*$, yielding the exponential rate. For $\\alpha$-stable noise, a time-change lemma (Lemma 5.1) is used because the jump L\\'evy measure has infinite second moment, so a direct It\\^o-type expansion fails.","core_discovery":"The central claim is Theorem 2.1 together with its applications: if the decoupled family $dX_t^\\mu = b(X_t^\\mu,\\mu)dt + \\sigma(X_t^\\mu,\\mu)dZ_t$ has $W_1$-contraction to a unique invariant measure $\\Gamma(\\mu)$ with rate uniform in $\\mu$, and the map $\\mu\\mapsto\\Gamma(\\mu)$ is contractive after solving a fixed-point condition involving $G(t)/(1-c_0e^{-\\lambda_0 t})$, then the original distribution-dependent SDE has a unique invariant measure $\\mu_*$ and exponential $W_1$-ergodicity $W_1(P_t^*\\eta,\\mu_*)\\le ce^{-\\lambda t}W_1(\\eta,\\mu_*)$. The applications verify the two ingredients for four noise regimes, each time with the coupling-strength constant $\\kappa$ below an explicit threshold $\\delta_0$; the noise coefficients may depend on the distribution, and the drifts need only be dissipative at long distances or dissipative in a partial set of components.","pith_inferences":["The uniform contraction rate of the decoupled family is likely the sharp bottleneck: if that rate fails for some frozen measure, the current perturbation argument collapses even if the original SDE is still ergodic.","The threshold $\\delta_0$ is explicit but probably conservative; computing the infima in $G(t)$ and $H(t)$ numerically for concrete models could give substantially larger ergodicity regions.","The same freeze-then-perturb scheme may extend to other Wasserstein distances $W_p$ or to relative entropy, provided a suitable coupling and Lyapunov structure replace reflection coupling.","For $\\alpha$-stable noise, the time-change lemma could also yield explicit propagation-of-chaos rates for the associated mean-field particle system, since the same decoupled contraction controls particle laws."],"forward_implications":["When $\\kappa<\\delta_0$, the solution semigroup has a unique invariant probability measure and every initial law in $P_1$ converges to it exponentially in $W_1$.","The noise coefficient may depend on the law, not only the drift, improving on prior results that required distribution-free $\\sigma$.","Partially dissipative drifts suffice: kinetic Langevin systems can be exponentially ergodic in $W_1$ without uniform dissipation in every coordinate.","The $\\alpha$-stable cases extend exponential $W_1$-ergodicity to heavy-tailed noise with distribution-dependent diffusion coefficients, in both non-degenerate and kinetic forms.","The smallness condition on $\\kappa$ is not purely technical: Example 3.3 shows that for large distribution-coupling strength the stationary measure can cease to exist, so ergodicity can fail."],"supporting_citations":[{"why":"Supplies the quantitative Harris and reflection-coupling framework for diffusions that the Brownian decoupled contraction builds on.","marker":"[8]"},{"why":"Provides the reflection-coupling and concave-function argument for non-dissipative McKean–Vlasov SDEs adapted here for the Brownian decoupled family.","marker":"[25]"},{"why":"Yields the uniform $W_1$-contraction for the kinetic (second-order) decoupled Langevin family used in Theorem 3.6.","marker":"[23]"},{"why":"Supplies the $W_1$-contraction for multiplicative L\\'evy noise used for the non-degenerate $\\alpha$-stable decoupled family in Theorem 4.1.","marker":"[18]"},{"why":"Provides the $W_1$-contraction for kinetic Langevin dynamics with L\\'evy noise used in Theorem 4.2.","marker":"[14]"},{"why":"Justifies well-posedness in $P_1$ for the distribution-dependent SDEs considered.","marker":"[22]"},{"why":"Provides the explicit $\\psi$ satisfying $\\psi''\\le 0$ with linear growth constants used in Corollary 3.4.","marker":"[12]"},{"why":"Supplies the time-change regularization argument behind Lemma 5.1 for $\\alpha$-stable noise.","marker":"[27]"}],"fun_headline_variants":["Exponential ergodicity holds for law-dependent noise and partial dissipation","Law-dependent noise doesn't block exponential convergence to equilibrium","Exponential mixing survives distribution-dependent noise in SDEs","McKean-Vlasov SDEs: exponential ergodicity despite law-dependent coefficients","Partially dissipative drift still yields exponential ergodicity with law-dependent noise"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument rests on the assumption that the decoupled SDE family, with a frozen measure $\\mu$, contracts in $W_1$ to its invariant measure at a rate $c_0e^{-\\lambda_0 t}$ that is uniform over all $\\mu$; if this uniform rate fails, the perturbation and fixed-point step collapses even if the original SDE is still ergodic.","fun_headline_variants_meta":{"raw":{"variants":["Exponential ergodicity holds for law-dependent noise and partial dissipation","Law-dependent noise doesn't block exponential convergence to equilibrium","Exponential mixing survives distribution-dependent noise in SDEs","McKean-Vlasov SDEs: exponential ergodicity despite law-dependent coefficients","Partially dissipative drift still yields exponential ergodicity with law-dependent noise"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00103,"raw_usage":{"total_tokens":4304,"prompt_tokens":875,"completion_tokens":3429,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":491,"completion_tokens_details":{"reasoning_tokens":3336}},"tokens_in":491,"tokens_out":3429,"duration_ms":23934,"temperature":1.0,"reasoning_tokens":3336,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:34:42.619163+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the stationary equations for a one-dimensional model of the form of Example 3.3 with the distribution-coupling strength below the theorem's $\\delta_0$; if two distinct stationary densities solve the equilibrium Fokker–Planck equation, the uniqueness claim of Theorem 3.1 is false. Alternatively, for the kinetic case, exhibit a drift satisfying the partial dissipativity condition but violating the uniform decoupled contraction estimate, and check whether the original distribution-dependent SDE nevertheless has two invariant measures.","supporting_citations":[{"cited_title":"Eberle, A","cited_arxiv_id":null,"evidence_quote":"Supplies the quantitative Harris and reflection-coupling framework for diffusions that the Brownian decoupled contraction builds on."},{"cited_title":"Wang, Exponential Ergodicity for Non-Dissipative McKean –Vlasov SDEs, Bernoulli, 29 (2023), 1035–1062","cited_arxiv_id":null,"evidence_quote":"Provides the reflection-coupling and concave-function argument for non-dissipative McKean–Vlasov SDEs adapted here for the Brownian decoupled family."},{"cited_title":"Schuh, Global contractivity for Langevin dynamics with distr ibution-dependent forces and uniform in time propagation of chaos, Ann","cited_arxiv_id":null,"evidence_quote":"Yields the uniform $W_1$-contraction for the kinetic (second-order) decoupled Langevin family used in Theorem 3.6."},{"cited_title":"Liang, J","cited_arxiv_id":null,"evidence_quote":"Supplies the $W_1$-contraction for multiplicative L\\'evy noise used for the non-degenerate $\\alpha$-stable decoupled family in Theorem 4.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the $W_1$-contraction for kinetic Langevin dynamics with L\\'evy noise used in Theorem 4.2."},{"cited_title":"Ren, F.-Y","cited_arxiv_id":null,"evidence_quote":"Justifies well-posedness in $P_1$ for the distribution-dependent SDEs considered."},{"cited_title":"Long Time $\\W_0$-$\\widetilde{\\W}_1$ type Propagation of Chaos for Mean Field Interacting Particle System","cited_arxiv_id":"2404.01795","evidence_quote":"Provides the explicit $\\psi$ satisfying $\\psi''\\le 0$ with linear growth constants used in Corollary 3.4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the time-change regularization argument behind Lemma 5.1 for $\\alpha$-stable noise."}],"review_version":1}