{"id":"ad9a5f6b-aadb-40e3-8366-6c21346ef20d","arxiv_id":"2501.09176","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Lyapunov-type conditions are shown to guarantee theta-periodic and stationary solutions for McKean-Vlasov SDEs.","lead":"This paper proves existence criteria for periodic and stationary solutions of distribution-dependent stochastic differential equations. The criteria combine weak convergence and Lyapunov function methods, with worked examples including a Landau-type equation.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Corollary 1.5, the stationary-solution half of the central claim, is stated without proof; the 'proof is very similar' assertion leaves a nontrivial invariance-for-all-times argument unverified.","rationale":"The reader's verdict of CONDITIONAL is appropriate. The proof of Theorem 1.3 is internally consistent: the Cesaro averages are tight, the shift-by-θ argument gives weak convergence of P_θ^*ν_n to ν, and H2 converts this to W_θ-convergence so the fixed-point conclusion follows. H2 is a real assumption and is the natural point where nonlinearity is controlled; it is imported from Wang's theorem in the examples, and I do not find a concrete failure of that transfer. The strongest concrete weakness is the omitted proof of Corollary 1.5, which the reader also noted in the rationale but did not select as the weakest assumption. The stationary case requires invariance under every time shift, not just one period, so the omission is not merely cosmetic. My recommendation is to keep the CONDITIONAL verdict until the proof of Corollary 1.5 is supplied; no adjustment to the verdict is needed because the conditional status already reflects this gap.","tokens_in":16428,"tokens_out":21067,"duration_ms":211085,"concrete_test":"Write a complete proof of Corollary 1.5 using the Krylov-Bogoliubov scheme: for H3a'/H3b', define ν_T = (1/T)∫_0^T P_t^*δ_0 dt, prove tightness from the Lyapunov estimate, extract a W_θ-limit ν, and for any fixed h > 0 show W_θ(P_h^*ν_{T_n}, ν_{T_n}) → 0 by the boundary-term shift, then use H2' to conclude P_h^*ν = ν for all h. For H3c', adapt the Schauder argument or use the same Cesaro route. If this proof requires any condition beyond (H1')–(H3c'), the statement of Corollary 1.5 must be amended.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 2 ends the proof of Theorem 1.3 with: 'The proof of Corollary 1.5 is very similar, which we will omit.' This is a load-bearing gap because Corollary 1.5 is not a special case of Theorem 1.3 in a one-line sense. Theorem 1.3 produces a measure ν satisfying P_θ^*ν = ν for the fixed period θ, while Corollary 1.5 requires a measure ν with P_t^*ν = ν for every t > 0. The natural adaptation (Cesaro averages over [0,T] sent to infinity, then shifting the average by h and using H2' to pass to the limit) must be written out, and for a single measure to be invariant for all h simultaneously one needs a tightness/diagonalization argument. The manuscript supplies none of this. No contradiction with the periodic theorem is apparent, and H2/H2' are legitimate standard assumptions; the weak point is the missing derivation of the stationary conclusion, not a false step in the periodic proof. Because the abstract and introduction advertise both periodic and stationary solutions as main results, the stationary half is under-supported as it stands.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript establishes criteria for the existence of θ-periodic and stationary solutions for distribution-dependent SDEs (McKean-Vlasov equations). Under structural assumptions (H0)-(H2), Theorem 1.3 asserts that any of three Lyapunov-type conditions (H3a), (H3b), or (H3c) yields a θ-periodic solution, and Corollary 1.5 states the time-homogeneous analogue. Section 2 proves the H3a/H3b cases by a Krylov-Bogoliubov argument with Cesàro averages and the H3c case by a Schauder fixed-point argument on a compact convex set of measures. Section 3 gives applications, including a time-dependent Landau-type example. The Schauder part and the reduction of periodic solutions to periodic measures in Lemma 2.2 are coherent, but the Cesàro part is invalid because it uses linearity of the nonlinear semigroup.","tokens_in":16651,"tokens_out":23786,"duration_ms":239359,"significance":"If the results were correct, the paper would provide flexible Lyapunov criteria for periodic solutions of time-inhomogeneous McKean-Vlasov SDEs, a topic with few existing tools, and the Landau-type example would be a useful application. The manuscript is also careful to allow non-unique stationary and periodic measures. However, the H3a/H3b half of Theorem 1.3 and the corresponding half of Corollary 1.5 rest on an unjustified linearity step, so the main advertised result is not established. The H3c/Schauder argument and the examples built on it, such as Example 3.7, are not affected by this defect.","major_comments":[{"comment":"The displayed identity (P^*_θ ν_n)(G) = (1/T_n)∫_0^{T_n}(P^*_{0,θ}P^*_{0,s}δ_0)(G)ds treats P^*_{0,θ} as if it commuted with convex combinations of initial laws. This is exactly the linearity that the authors show fails for DDSDEs in Section 1: in the example dX_t = Var(X_t)dt + dB_t, they compute P^*_t(ν0) ≠ ∫P^*_t δ_x ν0(dx). Starting the SDE from the mixture ν_n does not produce the mixture of the laws obtained from the components P^*_{0,s}δ_0, because the coefficients at time u depend on the global law of the solution at time u. Consequently the weak-convergence step (2.7) and the conclusion P^*_θ ν = ν are not justified for the H3a/H3b cases; the Krylov-Bogoliubov argument does not close. The H3c/Schauder proof is independent and appears sound.","section":"Section 2, after Eq. (2.6)"},{"comment":"Corollary 1.5 is advertised as a main result, but its proof is omitted with the sentence 'The proof of Corollary 1.5 is very similar, which we will omit.' The conclusion requires a measure ν with P^*_t ν = ν for every t > 0, whereas Theorem 1.3 only produces P^*_θ ν = ν for one fixed θ. The natural adaptation of the Cesàro argument must be written out and, in particular, one must show that a single limit measure works for all h simultaneously; the manuscript supplies no such argument. Remark 2.4 also asserts the existence of an invariant measure under the assumptions of Corollary 1.5 without proof. Even setting aside the linearity issue above, this is a load-bearing gap in a central advertised claim.","section":"Section 2, end of proof of Theorem 1.3; Corollary 1.5"}],"minor_comments":[{"comment":"The line 'Thus, condition (H3a) holds' is too quick: the displayed upper bound contains a positive K_{σ,1}|x|^2 term, and one must invoke Young's inequality with r > 2 to absorb it into -K_{b,4}|x|^r before the bound has the form required by (H3a).","section":"Example 3.2, (A4) case"},{"comment":"In the definition of σ(t,x,μ), the integral is written over R^d, although the example is set in R^3; the domain should be R^3.","section":"Example 3.5"},{"comment":"The notation P^*_{s+θ}δ_0 is used in the displayed computation for P^*_θ ν_n where elsewhere the notation means P^*_{0,s+θ}δ_0; please define this explicitly to avoid ambiguity.","section":"Section 2, Cesàro computation"},{"comment":"There is a line-break typo in the first sentence ('dis tribution'); this should be corrected.","section":"Abstract"}],"recommendation":"reject","confidential_remarks":"The fundamental issue is not presentation: the identity at the center of the H3a/H3b proof is refuted by the authors' own nonlinearity example in Section 1. I therefore do not think a major revision can repair Theorem 1.3 without changing the assumptions or the proof strategy. The H3c/Schauder part, however, appears sound and could be developed into a revised manuscript with different Lyapunov conditions or with a genuinely nonlinear fixed-point argument for the H3a/H3b cases."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The periodic half of this paper is a real, mostly correct result; the stationary half is advertised but not proved. Sun and Wong prove existence of theta-periodic solutions for time-inhomogeneous McKean-Vlasov SDEs under Lyapunov conditions, using Krylov-Bogoliubov weak convergence plus a Schauder fixed point argument for the supermartingale-type condition H3c. The proof is careful and, as far as I checked, the steps hold: the moment bounds from Ito's formula are clean, the Cesaro averages have uniform r-th moments and hence are tight, convergence in W_theta follows from Villani's criteria, and H2 does the job of passing P^*_theta through the limit. The nonlinearity of the semigroup is handled honestly, and the Schauder setup with the Kantorovich-Rubinstein norm is legitimate. The examples are concrete and show the criteria are checkable; the Landau equation example is a nice test.\n\nThe soft spot is exactly what the stress-test says. Corollary 1.5 is a stated main result, but Section 2 ends with 'The proof of Corollary 1.5 is very similar, which we will omit.' It is not a one-line special case of Theorem 1.3. The periodic theorem only gives a measure invariant under P^*_theta for one fixed period; stationarity needs P^*_t invariance for every t > 0. That requires the usual shift/diagonalization argument to get one measure that works for all t simultaneously. That argument may be standard, but it is not included, and Remark 2.4 asserts the conclusion without proof. A referee should require a written proof before the paper is accepted. I don't see a false step in the periodic proof, so this is a completeness problem, not a correctness problem.\n\nThe assumptions H2/H2' are imported from Wang's theorem; they are not automatic for nonlinear semigroups, but the paper cites the right result and checks it in the examples. The distinction from Bao-Reis-Wu's random periodic solutions is clearly drawn, and the citation pattern looks fair.\n\nWho is this for: people working on long-time behavior of McKean-Vlasov equations. It deserves a serious referee. Send it out, but the referee report should insist on the proof of Corollary 1.5, or a revision that downgrades it to a conjecture or remark.","headline":"Solid new existence criterion for theta-periodic McKean-Vlasov solutions, but the advertised stationary corollary is stated without its proof.","tokens_in":17180,"tokens_out":2455,"would_cite":true,"duration_ms":23037,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H10","37A50"],"pacs":[],"model":"deepseek-v4-flash","headline":"Distribution-dependent SDEs with periodic coefficients admit periodic-in-law solutions whenever a Lyapunov function controls the drift, and stationary solutions in the time-homogeneous case.","keywords":["distribution-dependent SDE","McKean-Vlasov SDE","periodic solution","stationary solution","Lyapunov function","weak convergence","Wasserstein metric"],"falsifier":"Find coefficients $b,\\sigma$ satisfying (H0) and (H1) and one of (H3a)–(H3c), but for which $P_\\theta^*$ is discontinuous at some $\\mu \\in P_\\vartheta(\\mathbb{R}^d)$ in the $W_\\vartheta$ metric; then Theorem 1.3 would predict a periodic solution but the proof's fixed-point step breaks down. Concretely, compute two initial laws $\\mu_n \\to \\mu$ in $W_\\vartheta$ and check whether $W_\\vartheta(P_\\theta^*\\mu_n, P_\\theta^*\\mu)$ fails to go to zero.","tokens_in":16214,"feed_emoji":"🔄","tokens_out":5252,"duration_ms":43941,"temperature":0.7,"pith_summary":"The paper proves that a distribution-dependent stochastic differential equation with periodic coefficients has a periodic-in-law solution whenever the coefficients admit a Lyapunov function with a drift bound, provided the one-step semigroup is well behaved on a Wasserstein space. This matters because most existing long-time results for McKean–Vlasov equations are time-homogeneous, and the nonlinearity of the distribution-dependent semigroup blocks the standard fixed-point routes to periodic solutions. The authors give three distinct Lyapunov conditions — a polynomial drift bound, a stronger interacting bound, and a pure supermartingale condition — and verify them on concrete models including a homogeneous Landau equation.","feed_headline":"Three criteria yield periodic solutions for McKean-Vlasov SDEs","feed_subtitle":"Weak convergence plus Lyapunov functions overcome the nonlinearity of the distribution-dependent semigroup.","key_machinery":"The central object is the time-θ semigroup $P_\\theta^*$ acting on the Polish space $(P_\\vartheta(\\mathbb{R}^d), W_\\vartheta)$ of probability measures with finite ϑ-th moment, equipped with the Wasserstein-ϑ metric. For distribution-dependent equations this semigroup is generally nonlinear, so the proof cannot appeal to linear Markov semigroup ergodic theory; instead the Lyapunov function V and its generator $\\mathcal{L}V$ supply the drift inequalities that make the Cesàro averages of $\\{P_s^*\\delta_0\\}$ tight, and condition (H2) — continuity of $P_\\theta^*$ in $W_\\vartheta$ — is exactly what turns the weak convergence of those averages into the fixed-point equation $P_\\theta^*\\nu = \\nu$. For condition (H3c), the same semigroup is shown to map a compact convex set $K = \\{\\mu : \\int V(0,x)\\,\\mu(dx) \\le |V(0,0)|\\}$ into itself continuously, so Schauder's fixed point theorem applies.","core_discovery":"The central claim is Theorem 1.3: under periodicity of coefficients (H0), strong existence and uniqueness (H1), and Wasserstein-θ continuity of the time-θ semigroup (H2), each of the Lyapunov conditions (H3a), (H3b), or (H3c) implies the existence of a θ-periodic measure ν in $P_\\vartheta(\\mathbb{R}^d)$ with $P_\\theta^*\\nu = \\nu$, and hence a θ-periodic solution whose finite-dimensional distributions are invariant under time shifts by θ. The proof splits: for (H3a) and (H3b) it runs a Krylov–Bogoliubov argument on the Cesàro averages of the law of the solution started at 0, extracting a weak limit and using (H2) to show the limit is periodic; for (H3c) it constructs a compact convex set of laws on which $P_\\theta^*$ acts continuously and applies the Schauder fixed point theorem. Corollary 1.5 gives the time-homogeneous analogue: a stationary solution exists if the coefficients do not depend on t and the same Lyapunov conditions hold with t omitted.","pith_inferences":["The proof's reliance on (H2) suggests that the periodic measure may fail to exist if the semigroup is only continuous in a weaker topology; a natural test is to look for DDSDEs where Cesàro averages converge weakly but $P_\\theta^*$ is discontinuous at the limit.","Because uniqueness is not established, the framework could be extended to phase-transition phenomena where multiple periodic measures coexist, analogous to known non-uniqueness results for stationary measures of McKean–Vlasov SDEs.","The Lyapunov conditions are stated for the solution from a deterministic start; one could try weakening them to require only a bound on the Cesàro averages of the law."],"forward_implications":["If Theorem 1.3 is correct, every DDSDE with θ-periodic coefficients satisfying (H0)–(H2) and one of the three Lyapunov inequalities has at least one θ-periodic solution; uniqueness is not claimed.","Corollary 1.5 gives existence of a stationary solution for time-homogeneous DDSDEs under the same type of conditions, without requiring the coefficient maps to be contractions.","The results cover coefficients that are not necessarily monotone in the distribution variable, as long as the Lyapunov inequality holds.","The examples include a concrete nonlinear SDE (Example 3.2) and a time-dependent homogeneous Landau equation (Example 3.5), showing the criteria apply beyond additive noise."],"supporting_citations":[{"why":"Supplies the continuity, monotonicity and growth conditions under which (H1) and (H2) hold, used to verify the examples.","marker":"[36]"},{"why":"Theorem 3.3.1 is the weak-convergence criterion used to pass from Cesàro averages to a limiting periodic measure.","marker":"[11]"},{"why":"Villani's characterization of $W_\\vartheta$ convergence and compactness is used to show the extracted limit lies in $P_\\vartheta$ and to convert weak convergence into $W_\\vartheta$ convergence.","marker":"[35]"},{"why":"Provides the Schauder fixed point theorem used in the (H3c) route.","marker":"[31]"},{"why":"Supplies the Lyapunov-function method for stochastic stability that inspires the drift inequalities (H3a)–(H3c).","marker":"[19]"},{"why":"Used for the Kantorovich–Rubinstein norm on the space of signed measures that underlies the compact-set argument.","marker":"[6]"}],"fun_headline_variants":["Lyapunov and weak convergence criteria yield periodic and stationary solutions for SDEs","Periodic and stationary solutions for distribution-dependent SDEs via Lyapunov functions","New existence criteria for periodic and stationary solutions of McKean-Vlasov SDEs","Weak convergence plus Lyapunov functions give periodic solutions for mean-field SDEs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is condition (H2): the one-step semigroup $P_\\theta^*$ must map $P_\\vartheta$ into itself continuously with respect to the Wasserstein-ϑ metric; for distribution-dependent equations this continuity is not automatic, and if it fails the Krylov–Bogoliubov or Schauder argument does not close.","fun_headline_variants_meta":{"raw":{"variants":["Lyapunov and weak convergence criteria yield periodic and stationary solutions for SDEs","Periodic and stationary solutions for distribution-dependent SDEs via Lyapunov functions","New existence criteria for periodic and stationary solutions of McKean-Vlasov SDEs","Weak convergence plus Lyapunov functions give periodic solutions for mean-field SDEs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0007,"raw_usage":{"total_tokens":3106,"prompt_tokens":833,"completion_tokens":2273,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":449,"completion_tokens_details":{"reasoning_tokens":2184}},"tokens_in":449,"tokens_out":2273,"duration_ms":17194,"temperature":1.0,"reasoning_tokens":2184,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T20:11:03.743472+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find coefficients $b,\\sigma$ satisfying (H0) and (H1) and one of (H3a)–(H3c), but for which $P_\\theta^*$ is discontinuous at some $\\mu \\in P_\\vartheta(\\mathbb{R}^d)$ in the $W_\\vartheta$ metric; then Theorem 1.3 would predict a periodic solution but the proof's fixed-point step breaks down. Concretely, compute two initial laws $\\mu_n \\to \\mu$ in $W_\\vartheta$ and check whether $W_\\vartheta(P_\\theta^*\\mu_n, P_\\theta^*\\mu)$ fails to go to zero.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the continuity, monotonicity and growth conditions under which (H1) and (H2) hold, used to verify the examples."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Theorem 3.3.1 is the weak-convergence criterion used to pass from Cesàro averages to a limiting periodic measure."},{"cited_title":"Villani, Optimal Transport: Old and New , Springer, 2016","cited_arxiv_id":null,"evidence_quote":"Villani's characterization of $W_\\vartheta$ convergence and compactness is used to show the extracted limit lies in $P_\\vartheta$ and to convert weak convergence into $W_\\vartheta$ convergence."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Schauder fixed point theorem used in the (H3c) route."},{"cited_title":"Khasminskii, Stochastic Stability of Differential Equations , Springer, 2011","cited_arxiv_id":null,"evidence_quote":"Supplies the Lyapunov-function method for stochastic stability that inspires the drift inequalities (H3a)–(H3c)."},{"cited_title":"Carmona and F","cited_arxiv_id":null,"evidence_quote":"Used for the Kantorovich–Rubinstein norm on the space of signed measures that underlies the compact-set argument."}],"review_version":1}