{"id":"e666b517-efd2-472f-8b4e-2e3961e4ed35","arxiv_id":"2501.15416","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Under periodic distribution-dependent Lyapunov conditions, the paper constructs T-periodic solutions for McKean-Vlasov SDEs by lifting the dynamics to the product space R^d times P(R^d).","lead":"This paper proves existence of periodic solutions for McKean-Vlasov stochastic differential equations when the Lyapunov function depends on the distribution as well as on position and time. It also claims convergence and continuous dependence of the periodic solutions, with worked examples.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.3's projection step is unjustified: averaging diagonal-start kernels P(s,(x,δ_x),·) over π_s does not reproduce the MVSDE flow from initial law π_s, and the proof never establishes that the periodic lifted measure has the required consistent form.","rationale":"The reader's weakest assumption is exactly the projection/consistency step in Theorem 4.3, and I find that this is the single most load-bearing gap in the paper. The lifted-space construction and the periodic Markov process results in Section 3 are plausible, and the Lyapunov condition (H) is a reasonable framework. However, the final step from a periodic probability on R^d×P(R^d) to a periodic solution of the original MVSDE is not justified in the text. The proof writes an integral involving P(s,(x,δ_x),t,·) and calls the result the MVSDE distribution with initial law π_s; this is incorrect because the kernel with second coordinate δ_x freezes the starting mean-field law at each x, whereas the true MVSDE flow uses a single common law for all particles. The paper's own Remark 4.2 acknowledges that a periodic probability for the coupled system need not have the diagonal form δ_x×μ, but Theorem 4.3 does not supply the missing consistency argument. The flaw is not merely cosmetic: without proving that the Krylov-Bogolioubov limit lies in C={ν(dy)δ_ν(dμ)}, the marginal π_t is not shown to satisfy the nonlinear flow. The concern is concrete and testable, as the linear example in the concrete_test field shows the exact formula in the proof fails. I do not see a different, more fundamental objection; if the consistency-invariance argument is added, the main theorem may be salvageable. Since the reader's verdict of REJECT is supported by this gap, I leave the verdict unchanged.","tokens_in":18669,"tokens_out":17569,"duration_ms":171892,"concrete_test":"Take d=1, b(t,x,μ)=-x+m(μ) with m(μ)=∫z μ(dz), σ=1, T-any, s=0, and π_0 with Var(π_0)>0. For h>0 small, the true MVSDE flow from π_0 has X_h=e^{-h}ξ+(1-e^{-h})E[ξ]+Gaussian(0,(1-e^{-2h})/2), ξ~π_0, so Var(X_h)=e^{-2h}Var(π_0)+(1-e^{-2h})/2. The expression used in the proof of Theorem 4.3, ∫P(0,(x,δ_0),h,·)π_0(dx), has law x+Gaussian(0,(1-e^{-2h})/2) with x~π_0, so its variance is Var(π_0)+(1-e^{-2h})/2. These differ by (1-e^{-2h})Var(π_0)>0 for h>0. Computing this explicit comparison settles that the projection formula in Theorem 4.3 does not give the MVSDE distribution; a correct proof must instead show the periodic lifted measure is in C and use P(s,(x,π_s),·).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central existence claim rests on the sentence in Theorem 4.3: '∫ P(s,(x,δ_x),t,·)π_s(dx) × δ_{π_s}(dδ_x) is the above MVSDE's distribution at time t with initial distribution π_s × δ_{π_s}.' This is false as written. For the coupled MVSDE (4.1), the transition P(s,(x,μ),t,·) propagates the pair (\\bar X_t, L_{X_t}) with \\bar X_s=x and L_{X_s}=μ. The special kernel P(s,(x,δ_x),t,·) is the law of (X_t, L_{X_t}) when X_s=x almost surely, so its first marginal is the MVSDE flow started from the deterministic point x, with the mean-field coefficient L_{X_t} depending on that same starting point. Averaging these kernels over x~π_s does not give the MVSDE flow started from the law π_s, because in the true flow there is a single common mean-field coefficient L_{X_t^{s,π_s}}, while the averaged expression uses a different coefficient L_{X_t^{s,x}} for each x. To evolve the lifted initial distribution corresponding to law π_s one must use ∫ P(s,(x,π_s),t,·)π_s(dx), not P(s,(x,δ_x),·). Moreover, the paper never proves that the T-periodic measure P0 obtained from Lemma 4.1 lies in the consistent set C={ν(dy)δ_ν(dμ)} — the set of lifted measures whose second coordinate is the first marginal. Starting the Krylov-Bogolioubov construction from a diagonal point (x0,δ_{x0}) would make each Cesàro average lie in C, and C is closed under weak limits, so the proof is likely repairable; but that argument is absent. As written, the projection step in Section 4.1 does not establish that π_t solves the MVSDE, so Theorem 4.3 is unproved. This is exactly the gap identified by the reader.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies T-periodic solutions of McKean–Vlasov SDEs under periodic, distribution-dependent Lyapunov conditions. The strategy is to lift the original nonlinear problem to a Markov process on R^d × P(R^d), prove general criteria for periodic Markov processes in Section 3, and then apply them to coupled MVSDEs in Section 4. The main existence result is Theorem 4.3, with convergence and parameter-continuity results in Theorems 4.7 and 4.8, and illustrative examples in Section 5.","tokens_in":19120,"tokens_out":18233,"duration_ms":156564,"significance":"The high-level idea of linearizing the MVSDE semigroup by adding the law as a second coordinate is appropriate and gives a clean framework for Krylov–Bogoliubov arguments; the paper also develops useful equivalent conditions for periodic Markov processes on the product space. If Theorem 4.3 were proved, the result would be a meaningful extension of Khasminskii's periodic-Lyapunov method to distribution-dependent coefficients. However, the central projection step is not justified as written, and the regularity and uniformity assumptions are not fully specified. These are repairable in principle, so the manuscript has potential, but the main theorem is not established in the current form.","major_comments":[{"comment":"The proof asserts that ∫ P(s,(x,δ_x),t,·)π_s(dx) × δ_{π_s}(dδ_x) is the MVSDE distribution at time t with initial distribution π_s. This is false. For each fixed x, P(s,(x,δ_x),t,·) is the law of (X_t^{s,x}, L_{X_t^{s,x}}), where L_{X_t^{s,x}} is the marginal of the point-start MVSDE, not the common mean-field law μ_t^{s,π_s} of the flow started from π_s. Averaging these kernels over x ∼ π_s gives a mixture of point-start flows, whereas the MVSDE flow from initial law π_s is generated by a single common law μ_t. The correct kernel would be P(s,(x,π_s),t,·), with the second coordinate fixed to π_s. Moreover, the paper never proves that the periodic lifted measure P0 is supported on the consistent set {ν(dy)δ_ν(dμ)}: Lemma 4.1 starts from an arbitrary (x0,μ0), and a weak limit of Cesàro averages need not be diagonal. Without consistency, the marginal π_t does not evolve under the nonlinear semigroup. This gap is load-bearing for the paper's central claim; it is likely repairable, for example by starting the Krylov–Bogoliubov construction from (x0,δ_{x0}) and observing that the diagonal is closed under the lift, but that argument is absent.","section":"Theorem 4.3, §4.1"},{"comment":"The verification of condition (3.4) of Theorem 3.10 is incomplete. Inequality (4.10) gives, for each initial (x,μ), P(s,(x,μ),t,U_R^c) ≤ (V(s,x,μ)+λ(t-s))/V_R. To obtain (3.4), one needs the supremum over (x,μ) ∈ U_{β(R)} of this ratio to vanish as R → ∞. Condition (H) only states V_R → ∞; it gives no control on sup_{U_{β(R)}} V(s,x,μ) relative to V_R, and P_2 balls are not compact in W_2, so the required uniformity is not automatic. As a result, Lemma 4.1's application of Theorem 3.10 is not justified.","section":"Lemma 4.1, eq. (4.10)"},{"comment":"The paper makes 'regular solution' an assumption without stating sufficient conditions. The text says, 'for simplicity we do not introduce the conditions in [19], and directly define that the solutions ... is called regular.' Since Theorems 4.3, 4.7, and 4.8 all inherit this assumption, the results are conditional on an unverified hypothesis. In particular, Examples 5.1–5.3 check only the Lyapunov condition; they do not verify existence of unique regular solutions for the exhibited non-Lipschitz coefficients. The revision should either state the regularity conditions explicitly or cite them in a way that makes the hypotheses of the main theorems checkable.","section":"Paragraph after (4.6), Lemma 4.1, Theorem 4.3"},{"comment":"The result stated in Theorem 4.8 is only tightness of the family {L_{X_{k,t}}}, not the continuous dependence on parameters claimed in the abstract and introduction. Remark 4.9 explicitly leaves the convergence ν_t = L_{X_t} to a condition deferred to reference [21]. Thus the section does not prove the announced continuous-dependence result. Either the statement should be weakened to a tightness result, or the missing growth conditions from [21] should be stated and verified.","section":"Theorem 4.8 and Remark 4.9, §4.3"}],"minor_comments":[{"comment":"Lemma 4.6 refers to 'conditions of Theorem 4.1', but no Theorem 4.1 exists in the paper; it should refer to Lemma 4.1.","section":"Lemma 4.6"},{"comment":"In the proof of Theorem 3.13 the text says 'We next prove equality (4.11)', but equation (4.11) is introduced later in Section 4.2; the reference should be to (3.5).","section":"Theorem 3.13 proof"},{"comment":"The set U_R is used in Theorem 3.10 and Corollary 3.11, but only U_R^c is defined, in Theorem 3.9; the notation should be defined explicitly where it first appears.","section":"Section 3, U_R notation"},{"comment":"In the first part of the proof of Theorem 3.10, the sentence 'by limits (3.2) and (3.3)' appears to be a typo; the argument uses limits (3.3) and (3.4) to derive (3.2).","section":"Theorem 3.10 proof"},{"comment":"There is a typo in 'coupled McKean-Vlaosv SDEs'; it should be 'McKean-Vlasov'.","section":"Section 4.1, before (4.1)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript relies heavily on the author's own companion works [19] and [21] for regularity and continuous-dependence inputs; the revision should state precisely which results are imported and make the hypotheses of the main theorems self-contained. The gap in Theorem 4.3 appears repairable, so I do not recommend rejection, but the current version should not be accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe genuinely new thing here is a periodic distribution-dependent Lyapunov condition for McKean-Vlasov SDEs, obtained by working on the lifted space R^d × P(R^d). Section 3 gives a coherent Khasminskii-style criterion for existence, convergence, and parameter dependence of T-periodic Markov processes on that product space. The operator semigroup is linear on the product space, so the Krylov-Bogolioubov argument is legitimate there. Credit is due for seeing that the Lyapunov function should live on the joint space when the coefficients do.\n\nThe problem is Theorem 4.3, the central existence claim for the original MVSDE. The proof defines π_t(·) = P0(t, · × P(R^d)) from a T-periodic measure P0 on the lifted space. But the transition kernel P(s,(x,δ_x),t,·) describes the pair (X_t,L_{X_t}) when the first coordinate starts deterministically at x. Averaging this kernel over x ~ π_s does not evolve the MVSDE from initial law π_s: in the true MVSDE there is a single common mean-field coefficient L_{X_t^{s,π_s}}, while the averaged expression uses a different coefficient L_{X_t^{s,x}} for each starting point. One would need something like ∫ P(s,(x,π_s),t,·)π_s(dx), and then one would need to prove the periodic lifted measure has the consistent form ν(dy)δ_ν(dμ). The paper asserts this consistency rather than proving it. This is not cosmetic; it is the step that connects the lifted periodic Markov process to a solution of the nonlinear MVSDE. The stress-test note is right.\n\nThere is also an unstated regularity burden. The paper imports the notion of 'regular solutions' from the author's own works [19,21] without stating sufficient conditions for the convergence Y^n → X and for applying Itô's formula with the Lions derivative. A referee would have to verify those hypotheses in the examples. Theorem 4.8 is weaker than its title: it only proves tightness of periodic solutions, and Remark 4.9 concedes that the actual continuous-dependence conclusion needs growth conditions from [21].\n\nBottom line: the lifted Lyapunov framework is a sensible approach and Section 3 may be publishable on its own. But Theorem 4.3 as written is unproved. The gap is likely repairable—starting the Krylov-Bogolioubov construction from a diagonal point preserves consistency, and the consistent set is closed under weak limits—but the repair is real work, not a typo. I would send this to a referee with a clear request to examine the projection consistency and the regularity assumptions, not desk-reject it. Current form should not be accepted.","headline":"A useful lifted-space Lyapunov framework, but Theorem 4.3's projection step does not prove a periodic solution for the original MVSDE as written.","tokens_in":19618,"tokens_out":1907,"would_cite":false,"duration_ms":18133,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["34C25","60H10"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves existence, convergence, and parameter continuity of T-periodic measure-valued solutions for McKean-Vlasov SDEs under periodic distribution-dependent Lyapunov conditions.","keywords":["periodic solutions","McKean-Vlasov stochastic differential equations","distribution-dependent Lyapunov conditions","periodic Markov processes","Krylov-Bogoliubov theorem","convergence","continuous dependence"],"falsifier":"For any coefficient pair satisfying (H), compute the marginal $\\pi_t(\\cdot)=P_0(t,\\cdot\\times\\mathcal P(\\mathbb{R}^d))$ of the lifted $T$-periodic measure and compare it with the law at time $t$ of the original McKean-Vlasov equation started from $\\pi_s$; a discrepancy at any $t>s$ would falsify Theorem 4.3. Alternatively, exhibiting a point $(x,\\mu)$ in the support of $P_0$ with $\\mu$ different from the law of the first coordinate would show that the projection step is not justified.","tokens_in":18464,"feed_emoji":"🔄","tokens_out":11099,"duration_ms":85657,"temperature":0.7,"pith_summary":"The paper asks whether a McKean-Vlasov stochastic differential equation, whose drift and diffusion may depend on the law of the solution itself, has a solution whose distribution repeats after a fixed period $T$. It proves that the answer is yes under a Lyapunov condition in which the Lyapunov function is periodic in time and depends on time, space, and the distribution variable. The proof lifts the dynamics to a Markov process on the product space $\\mathbb{R}^d \\times \\mathcal P(\\mathbb{R}^d)$, where the associated semigroup becomes linear and the classical Krylov-Bogolioubov compactness argument yields a $T$-periodic probability; projecting onto the space variable gives the desired $T$-periodic law for the original equation. The paper also proves weak convergence of solutions to the periodic regime along a subsequence, full convergence when the periodic law is unique, and continuous dependence of periodic solutions on parameters.","feed_headline":"Mean-field SDEs admit periodic solutions under periodic Lyapunov bounds","feed_subtitle":"Existence, convergence, parameter continuity for T-periodic measure-valued solutions via lifted Markov process","key_machinery":"The central object is the coupled McKean-Vlasov system (4.1), a pair of SDEs in which the second component has coefficients depending on the law of the first, so its transition probabilities define a Markov process on the lifted state space $\\mathbb{R}^d \\times \\mathcal P(\\mathbb{R}^d)$. Because the measure appears as an explicit coordinate, the associated semigroup (3.1) is linear, which is what makes a Krylov-Bogolioubov compactness argument available for a nonlinear mean-field problem. The periodic distribution-dependent Lyapunov condition (H) supplies tightness: the generator $\\mathcal L V$ tends to $-\\infty$ outside large sets, giving the averaged escape-time bound (3.3) and the uniform control (3.4) needed to apply Theorem 3.10 and obtain a $T$-periodic probability $P_0$ on the lifted space. The final step in Theorem 4.3 takes the first-coordinate marginal of $P_0$ as the periodic law of the original MVSDE.","core_discovery":"The central claim is Theorem 4.3: if the coupled McKean-Vlasov system built from $b$ and $\\sigma$ has unique regular solutions and the periodic distribution-dependent Lyapunov condition (H) holds, then the McKean-Vlasov SDE $dX_t = b(t,X_t,\\mathcal L_{X_t})dt + \\sigma(t,X_t,\\mathcal L_{X_t})dW_t$ has a $T$-periodic solution in law. The periodic distribution is constructed as the projection $\\pi_t(\\cdot)=P_0(t,\\cdot\\times\\mathcal P(\\mathbb{R}^d))$ of a $T$-periodic probability $P_0$ on the lifted space $\\mathbb{R}^d\\times\\mathcal P(\\mathbb{R}^d)$ obtained for the coupled system. The paper further claims convergence of arbitrary solutions to this periodic law in the Cesaro sense along a subsequence, full-sequence convergence under uniqueness, and continuity of the periodic laws as the coefficients converge pointwise.","pith_inferences":["A point the paper leaves open is whether the lifted $T$-periodic measure $P_0$ is supported on consistent pairs $(x,\\mu)$ with $\\mu$ equal to the law of $x$; without that, the projection step in Theorem 4.3 is the part of the argument most worth checking.","The same lifting trick is likely to work for almost-periodic, recurrent, or asymptotically autonomous distribution-dependent dynamics, because the linearization by adding the measure as a coordinate does not rely on exact periodicity.","A natural testable strengthening would be to convert the Cesaro convergence into an explicit rate under stronger coercivity, analogous to exponential ergodicity results for time-periodic mean-field equations."],"forward_implications":["Any McKean-Vlasov SDE with $T$-periodic coefficients, a periodic distribution-dependent Lyapunov function, and unique regular solutions has a $T$-periodic measure-valued solution.","The periodic law is obtained as the first-coordinate projection of a $T$-periodic probability on $\\mathbb{R}^d\\times\\mathcal P(\\mathbb{R}^d)$, so the lifted Markov-process framework transfers Krylov-Bogolioubov compactness to nonlinear Fokker-Planck equations.","Weak convergence of solutions to the periodic law holds along a subsequence, and along the full sequence whenever the $T$-periodic law is unique.","Families of periodic solutions are tight and their weak limits are periodic solutions as the coefficients converge pointwise, giving continuous dependence on parameters.","For time-homogeneous coefficients the same theorem yields a stationary solution as a direct corollary."],"supporting_citations":[{"why":"Supplies the periodic Markov process and periodic Lyapunov framework that Section 3 and Theorem 4.3 are built on.","marker":"[15]"},{"why":"Provides the truncation, existence, uniqueness, and Krylov-Bogolioubov ergodicity for distribution-dependent Lyapunov conditions that the coupled-system construction relies on.","marker":"[19]"},{"why":"Is the prior existence result for distribution-dependent SDEs under time-space Lyapunov functions that this paper extends to distribution-dependent Lyapunov functions.","marker":"[27]"},{"why":"Obtains periodic solutions for time-periodic McKean-Vlasov SDEs by exponential ergodicity under monotone conditions, giving the closest prior comparison.","marker":"[25]"},{"why":"Gives the existence and uniqueness theorem for distribution-dependent SDEs used to solve the truncated equations (4.2).","marker":"[30]"},{"why":"Justifies that a transition probability function together with an initial law defines a Markov process on the lifted space, used in Proposition 3.6.","marker":"[26]"},{"why":"Provides the compactness and weak-convergence facts used to extract the $T$-periodic probability from tight averaged laws.","marker":"[23]"}],"fun_headline_variants":["Periodic solutions exist for McKean-Vlasov SDEs under periodic Lyapunov conditions","McKean-Vlasov SDEs get periodic solutions via Lyapunov conditions","Mean-field SDEs with periodic Lyapunov bounds have T-periodic solutions","Periodic Lyapunov conditions ensure periodic solutions for McKean-Vlasov SDEs","T-periodic solutions proven for McKean-Vlasov SDEs under Lyapunov conditions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the $T$-periodic probability $P_0$ on the lifted space $\\mathbb{R}^d\\times\\mathcal P(\\mathbb{R}^d)$ projects to a genuine solution of the original McKean-Vlasov equation, even though the proof does not show that $P_0$ is supported on consistent pairs $(x,\\mu)$ with $\\mu$ equal to the law of $x$.","fun_headline_variants_meta":{"raw":{"variants":["Periodic solutions exist for McKean-Vlasov SDEs under periodic Lyapunov conditions","McKean-Vlasov SDEs get periodic solutions via Lyapunov conditions","Mean-field SDEs with periodic Lyapunov bounds have T-periodic solutions","Periodic Lyapunov conditions ensure periodic solutions for McKean-Vlasov SDEs","T-periodic solutions proven for McKean-Vlasov SDEs under Lyapunov conditions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001564,"raw_usage":{"total_tokens":6199,"prompt_tokens":849,"completion_tokens":5350,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":465,"completion_tokens_details":{"reasoning_tokens":5232}},"tokens_in":465,"tokens_out":5350,"duration_ms":32738,"temperature":1.0,"reasoning_tokens":5232,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T14:19:39.754195+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For any coefficient pair satisfying (H), compute the marginal $\\pi_t(\\cdot)=P_0(t,\\cdot\\times\\mathcal P(\\mathbb{R}^d))$ of the lifted $T$-periodic measure and compare it with the law at time $t$ of the original McKean-Vlasov equation started from $\\pi_s$; a discrepancy at any $t>s$ would falsify Theorem 4.3. Alternatively, exhibiting a point $(x,\\mu)$ in the support of $P_0$ with $\\mu$ different from the law of the first coordinate would show that the projection step is not justified.","supporting_citations":[{"cited_title":"Khasminskii, Stochastic Stability of Differential Equations , Springer Berlin, Heidelberg, 2012, xvi+344 pp","cited_arxiv_id":null,"evidence_quote":"Supplies the periodic Markov process and periodic Lyapunov framework that Section 3 and Theorem 4.3 are built on."},{"cited_title":"Existence, uniqueness and ergodicity for McKean-Vlasov SDEs under distribution-dependent Lyapunov conditions","cited_arxiv_id":"2309.05411","evidence_quote":"Provides the truncation, existence, uniqueness, and Krylov-Bogolioubov ergodicity for distribution-dependent Lyapunov conditions that the coupled-system construction relies on."},{"cited_title":"Existence of Periodic and Stationary Solutions to Distribution-Dependent SDEs","cited_arxiv_id":"2501.09176","evidence_quote":"Is the prior existence result for distribution-dependent SDEs under time-space Lyapunov functions that this paper extends to distribution-dependent Lyapunov functions."},{"cited_title":"Exponential Ergodicity for Time-Periodic McKean-Vlasov SDEs","cited_arxiv_id":"2110.06473","evidence_quote":"Obtains periodic solutions for time-periodic McKean-Vlasov SDEs by exponential ergodicity under monotone conditions, giving the closest prior comparison."},{"cited_title":"Wang, Distribution dependent SDEs for Landau type equations, Stochastic Process","cited_arxiv_id":null,"evidence_quote":"Gives the existence and uniqueness theorem for distribution-dependent SDEs used to solve the truncated equations (4.2)."},{"cited_title":"Revuz and M","cited_arxiv_id":null,"evidence_quote":"Justifies that a transition probability function together with an initial law defines a Markov process on the lifted space, used in Proposition 3.6."},{"cited_title":"Prokhorov, Convergence of random processes and l imit theorems in probability theory, Theory Probab","cited_arxiv_id":null,"evidence_quote":"Provides the compactness and weak-convergence facts used to extract the $T$-periodic probability from tight averaged laws."}],"review_version":1}