{"id":"ab59f880-0872-4457-a8d4-bccdd268b346","arxiv_id":"2506.18558","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For time-inhomogeneous two-scale SDEs with partially dissipative fast drift, the slow component converges strongly or weakly to an averaged diffusion whose coefficients are built from an evolution system of measures.","lead":"This paper proves strong and weak averaging principles for multiscale stochastic systems whose fast component is time-dependent and only partially dissipative. It provides a rigorous reduction of such systems to a single averaged stochastic equation, extending classical results beyond uniformly dissipative fast dynamics.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 2.8's proof uses an unjustified conditioning step; the O(δ^{1/2}) bound underpinning both main theorems is not established as written.","rationale":"The strongest-claim assertion is the two averaging theorems. For these to hold, one needs a quantitative replacement of the fast process Y^ε by a locally frozen process \\hat Y^ε with a small weighted-integral error, and that is exactly Lemma 2.8. The proof of Lemma 2.8 is not merely a matter of omitted routine detail; the displayed conditioning equality is unjustified because the weight Z_{t0} lies in F_{t0}, not F_{kδ}. This appears in the derivation of (2.21), which feeds directly into I1 in Lemma 3.2 and into Q_{11}/Q_{13} in Theorem 1.2. The theorems may still be true, and the conditioning issue is plausibly fixable by an interval-splitting argument, but as written the central estimate is missing a required justification. I therefore adjust the verdict to CONDITIONAL rather than ACCEPT: the claims are credible, but the paper should supply a corrected proof of Lemma 2.8. The reader focused on A2 as the weakest structural assumption; our concern is a consequence of using A2 in the key coupling estimate, so agreement is partial. Minor typos and the deferred proofs of \\bar f's properties do not alter this assessment.","tokens_in":33589,"tokens_out":42990,"duration_ms":415209,"concrete_test":"Independently re-derive Lemma 2.8 with the interval containing t0 split off: for kδ≥t0 the weight Z_{t0} is F_{kδ}-measurable so the conditioning step is valid, and the summed contribution is O(δ^{1/2}); the single interval containing t0 is bounded trivially by O(δ). If this re-derivation fails, test a linear model f(t,x,y)=-y+x, b=y, ε=1 numerically or analytically to check whether E∫_{t0}^{t}(Y_s-\\hat Y_s)ds (Y_{t0}-\\hat Y_{t0}) is really O(δ^{1/2}); if not, (3.3) and Theorem 1.1 lack support.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Lemma 2.8 is the key estimate used to control E∫(b(X^ε,Y^ε)-b(X^ε,\\hat Y^ε))Z ds in Lemma 3.2 and in the martingale-projector terms of Theorem 1.2. In its proof, after the piecewise-constant replacement X_s≈X_{kδ}, the authors write E[Z_{t0}(F(Y_s)-F(\\hat Y_s))] = E[Z_{t0} E(F(Y_s)-F(\\hat Y_s)|F_{kδ})] and then estimate the conditional expectation. This equality requires Z_{t0} to be F_{kδ}-measurable. In the actual use, Z_{t0}=b(r/ε,X_r^ε,Y_r^ε)-b(r/ε,X_r^ε,\\hat Y_r^ε) for arbitrary r, which is only F_r-measurable; for r inside (kδ,(k+1)δ) it is not F_{kδ}-measurable. The displayed derivation therefore does not justify the O(δ^{1/2}) bound. The statement is likely repairable by handling the single interval containing t0 with a trivial O(δ) bound and using F_{t0}⊂F_{kδ} on all later intervals, but that argument is absent. Since Lemma 3.2's estimate (3.3) and the weak-limit identification (4.9)-(4.10) both rest on Lemma 2.8, the proofs of Theorems 1.1 and 1.2 are incomplete as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves strong and weak averaging principles for the two-scale SDE system (1.6), where all coefficients are time-inhomogeneous and the fast drift is only partially dissipative in the sense of Assumption A2. Under Assumptions A1, A2, A3, and A4 the authors establish L2 convergence of the slow component X^ε to the averaged diffusion (1.17), with an explicit rate of order ε^{1/3}. Under A1, A2, A3, and A5 they establish weak convergence in C([0,T];R^n) to the solution of (1.20), identified through a martingale problem. The technical core is the construction of an evolution system of measures for the time-inhomogeneous frozen SDE (1.21) with partially dissipative drift, using asymptotic reflection couplings to obtain exponential Wasserstein contraction, followed by a priori moment bounds and a quantitative comparison between the fast component and a piecewise-constant auxiliary process. Theorems 1.1 and 1.2 are the central load-bearing assertions.","tokens_in":33890,"tokens_out":13870,"duration_ms":130576,"significance":"If the results are correct, they substantially extend the available averaging principles for time-inhomogeneous slow-fast SDEs beyond the uniformly dissipative and periodic settings; the partial dissipativity condition (1.9) is a genuine structural novelty in this context. The use of asymptotic reflection coupling to obtain contraction (2.3) and to construct the evolution system of measures is elegant and appears to be new for averaging problems. The paper is largely self-contained, gives explicit convergence rates, and the main estimates (Lemmas 2.1, 2.6, 2.7, 3.2, 3.3) are written in detail. The results are falsifiable and quantitatively precise, and I see no circularity or hidden free parameters. However, as detailed below, one key estimate (Lemma 2.8) has a gap in its proof as written, and one step in the weak-convergence identification ((4.10)) is only sketched.","major_comments":[{"comment":"The conditioning step is not justified for the interval containing t0. In the displayed equality the authors write E[Z_{t0}(F(Y_s)-F(\\hat Y_s))] = E[Z_{t0} E(F(Y_s)-F(\\hat Y_s)|F_{kδ})] for s in [kδ,(k+1)δ]. This is valid only if Z_{t0} is F_{kδ}-measurable. For the unique interval with kδ < t0 ≤ s, the hypothesis gives only Z_{t0} ∈ F_{t0}, which is generally not contained in F_{kδ}. Since Lemma 2.8 is used in Lemma 3.2 (estimate (3.5) for I1) and in the proof of Theorem 1.2 (the bound on Q^{ε_k}_{11}), the proofs of Theorems 1.1 and 1.2 are incomplete as written. The gap is repairable: the contribution from the single interval containing t0 can be bounded directly by Cauchy-Schwarz and the moment bounds of Lemmas 2.6-2.7, giving O(δ) (which is stronger than the claimed O(δ^{1/2})), while for all later intervals kδ ≥ t0 one has F_{t0} ⊂ F_{kδ} and the displayed conditioning is valid. I request that this argument be added and the proof be rewritten accordingly.","section":"Lemma 2.8, proof (second displayed equality in the block after (2.18))"},{"comment":"The verification of (4.10) is dismissed with 'one can follow the proof of (4.9)', but the function C(t,x,y) = Tr[((σσ*)(t,x,y) - \\bar Σ(x))∇²U(x)] is not uniformly Lipschitz in y; its y-Lipschitz constant grows as 1+|x|+|y|. The argument for (4.9) critically uses the uniform y-Lipschitz property of B, so the reduction is not immediate and requires weighted estimates exploiting the moment bounds from Lemma 2.6 and Lemma 2.7. Since (4.10) is needed to identify the diffusion coefficient of the limiting process, please provide the detailed estimates (or a precise reference to a stated weighted contraction estimate) rather than an omission.","section":"Section 4, proof of Theorem 1.2, treatment of (4.10)"}],"minor_comments":[{"comment":"The notation '|y 2|' appears in several places and should read '|y|²' (e.g., in the statement of Lemma 3.2 and in the displays (3.3), (3.4), and (3.7)).","section":"Lemma 3.2 and equations (3.3)-(3.7)"},{"comment":"The symbol s(δ) = [s/δ]δ is used before it is defined; please define it at the start of the proof.","section":"Lemma 2.8, proof"},{"comment":"The symbol ϕ_5 appears without definition; based on the analogous estimate (3.21), it should presumably be the function \\tilde ϕ_2 defined in (3.20). Please correct the notation.","section":"Proof of Theorem 1.2, bound for Q^{ε_k}_2 (4.13)"},{"comment":"The ergodicity properties (i)-(iii) of the limit frozen SDE (1.14) are asserted by saying that one can follow the proofs of Lemmas 2.1 and 2.3. Since the partially dissipative setting is a novelty of the paper, it would be helpful to include at least a brief indication of the argument, rather than a direct omission.","section":"Section 3, paragraph after (3.8)"}],"recommendation":"major_revision","confidential_remarks":"The gap in Lemma 2.8 is real but clearly localized and repairable within the scope of the paper; I do not see grounds for rejection. The omission in the proof of (4.10) is more substantial because the function involved is not uniformly Lipschitz, but it is still within the reach of the paper's techniques. The paper is a good fit for the journal once these points are addressed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a real contribution to the averaging-principle literature—time-inhomogeneous fast drift with only partial dissipativity, handled via asymptotic reflection coupling and evolution systems of measures. The main strong and weak averaging theorems are new, and the proof strategy is sound in outline. But the written proof of Lemma 2.8, which both theorems lean on, has a genuine gap that needs fixing before the results are fully established.\n\nWhat's new: the relaxation from uniformly dissipative to partially dissipative fast drift in a time-inhomogeneous setting appears absent from prior work, including the authors' own [29]. The use of asymptotic reflection coupling to get exponential Wasserstein contraction for the frozen SDE, and then to build an evolution system of measures, is a coherent and nonstandard way to get the averaged coefficients. The paper also handles the strong case (sigma independent of y) and the weak case (sigma depending on y) with the right tools: direct L2 estimates and martingale problem identification.\n\nThe soft spot: in Lemma 2.8, the step\n\nE[Z_{t0} (F(x',Y)-F(x',\\hat Y))|_{x'=X_{k\\delta},y'=Y_{k\\delta}}]\n\nrequires Z_{t0} to be F_{k\\delta}-measurable. That's true when kδ ≥ t0, but not for the single interval containing t0. For that interval the integrand is non-negligible, and the proof as written doesn't justify the O(δ^{1/2}) bound. The fix is straightforward—bound that interval by O(δ) and run the argument on the later intervals—but the gap is real and affects both Theorem 1.1 and Theorem 1.2 through Lemma 3.2 and the weak-limit identification.\n\nOther, smaller issues: the ergodicity claims for the limit frozen SDE (3.8)–(3.12) are just stated as 'we omit the details', and (4.10) is waved off with 'one can follow'. Those are probably fine, but they'd be better with a few lines.\n\nThe citation pattern is clean, and the two self-citations are to standard technical facts, not to a hidden main result.\n\nBottom line: the paper deserves a serious referee. With a corrected proof of Lemma 2.8 and a few added details, it's an accept. I'd bring it to my reading group.","headline":"Genuine extension of averaging to partially dissipative time-inhomogeneous fast drift, but Lemma 2.8 has a repairable gap in the conditioning step that both main theorems rely on.","tokens_in":34438,"tokens_out":6490,"would_cite":true,"duration_ms":54989,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["34D08","34D25","60H20"],"pacs":[],"model":"deepseek-v4-flash","headline":"For time-inhomogeneous slow-fast SDEs, a fast drift with only a dissipative tail is enough for strong and weak averaging.","keywords":["averaging principle","time-inhomogeneous SDE","multi-scale SDE","partially dissipative drift","evolution system of measures","asymptotic reflection coupling","L1-Wasserstein contraction","martingale problem"],"falsifier":"A direct numerical check would settle the claim: take the one-dimensional fast drift $f(y)=\\sin(y)+\\alpha y$ with small $\\alpha>0$, which satisfies Assumption A2 with $K=\\alpha$ and $C=1$; the theorem predicts $\\sup_{t\\in[0,T]}\\mathbb{E}|X^\\varepsilon_t-\\bar X_t|^2\\to0$ as $\\varepsilon\\to0$. If the empirical distance does not vanish, the assumptions are not sufficient; if it vanishes but at a rate much worse than the bound (3.22), the stated rate is not sharp.","tokens_in":33366,"feed_emoji":"🎯","tokens_out":13442,"duration_ms":126616,"temperature":0.7,"pith_summary":"This paper establishes averaging principles for two-scale SDEs with time-dependent coefficients, where the fast component's drift is only partially dissipative: it may fail to contract nearby trajectories but must push them together at a uniform rate when they are far apart. Under these assumptions the slow component is shown to converge to a single averaged SDE, strongly in mean square when the slow diffusion coefficient is independent of the fast variable, and weakly in path space when the slow diffusion depends on the fast variable. The averaged coefficients are explicit integrals of the time-averaged coefficients against the invariant measure of a limiting autonomous frozen equation. The proof builds a unique evolution system of measures for the time-inhomogeneous frozen equation via asymptotic reflection coupling, and that construction is the new ingredient that makes averaging work without uniform dissipation.","feed_headline":"Partially dissipative fast drift still averages","feed_subtitle":"The slow component still averages even when the fast drift is time-dependent and only partially dissipative.","key_machinery":"The machine that carries the argument is the asymptotic reflection coupling for the time-inhomogeneous frozen fast equation, defined by building two copies of the diffusion that share noise when they are far apart and reflect one copy's noise off the other. It produces the exponential $L^1$-Wasserstein contraction $W_1(\\delta_{y_1}P^{x_1}_{s,t},\\delta_{y_2}P^{x_2}_{s,t})\\le C e^{-\\beta(t-s)}|y_1-y_2|+C|x_1-x_2|$, which then yields a unique evolution system of measures $\\{\\mu^x_t\\}$ and the convergence of $\\mu^x_t$ to the invariant measure $\\mu^x$ of the limiting autonomous frozen equation. This contraction is also what controls the difference between the true fast process and the auxiliary process $\\hat Y^\\varepsilon$ with the slow variable frozen on short intervals, giving the strong averaging error, and it underpins the tightness and martingale-problem identification used in the weak result.","core_discovery":"The paper's central claim is that a time-inhomogeneous fast drift that is only partially dissipative is still sufficient for averaging: under Assumptions A1, A2, A3 and A4, the slow component $X^\\varepsilon$ converges in mean square, uniformly on $[0,T]$, to the unique solution $\\bar X$ of the averaged SDE $d\\bar X_t=\\bar b(\\bar X_t)dt+\\bar\\sigma(\\bar X_t)dW^1_t$, with $\\bar b(x)=\\int \\hat b(x,y)\\mu^x(dy)$; under A1, A2, A3 and A5 it converges weakly in $C([0,T];\\mathbb{R}^n)$ to the solution of the same-drift averaged SDE with diffusion $\\Theta(x)=(\\int\\Sigma(x,y)\\mu^x(dy))^{1/2}$. The measure $\\mu^x$ is the unique invariant measure of the limiting autonomous frozen equation $d\\bar Y_t=\\bar f(x,\\bar Y_t)dt+dW^2_t$, and the time-inhomogeneous frozen equation is shown to have a unique evolution system of measures that converges to $\\mu^x$ as the frozen time goes to infinity. This gives the first averaging-principle treatment in which the fast drift is only partially dissipative rather than uniformly dissipative.","pith_inferences":["Editorial inference: because the argument only needs the Wasserstein contraction plus moment bounds, the averaging limit should also hold when the fast noise is multiplicative or the fast dynamics are hypoelliptic, as long as an analogous exponential contraction is available; the paper notes the multiplicative case but leaves the degenerate-noise details open.","Editorial inference: the $\\varepsilon^{1/3}$ strong rate reflects the choice $\\delta=\\varepsilon^{2/3}$ and is not optimal; sharper time-average estimates should push the strong rate toward $\\varepsilon^{1/2}$, matching classical averaging rates.","Editorial inference: the weak limit is identified via the martingale problem, which leaves room for a subsequent central limit theorem or large-deviation analysis of the fluctuations $X^\\varepsilon-\\bar X$ when $\\sigma$ depends on the fast variable.","Editorial inference: the evolution system of measures gives an interpretation of the averaged coefficients as pullback averages of the nonautonomous frozen dynamics; this suggests the same averaging limits should hold under almost periodic or other recurrent time dependence, not only the Cesàro conditions in Assumption A3."],"forward_implications":["When the slow noise coefficient does not depend on the fast variable, the slow path converges in mean square on any finite time horizon to the averaged SDE, with error of order $\\varepsilon^{1/3}$ up to the rates at which the time-average errors $\\phi_1,\\phi_2,\\phi_3$ vanish.","When the slow noise coefficient depends on the fast variable, the path laws converge weakly in $C([0,T];\\mathbb{R}^n)$ to the unique solution of the averaged SDE with diffusion matrix $\\Theta(x)=(\\bar\\Sigma(x))^{1/2}$.","The averaged drift and diffusion coefficients are explicit: integrate the time-averaged coefficient against the invariant measure of the limiting autonomous frozen equation.","The time-inhomogeneous frozen equation admits one and only one evolution system of measures with finite first moment, so the averaging limit is well defined and unambiguous.","The exponential $L^1$-Wasserstein contraction of the frozen fast semigroup replaces the uniform dissipativity used in earlier averaging results, so the same style of proof covers drifts that are only weakly contracting near the origin."],"supporting_citations":[{"why":"Establishes the classical averaging principle whose time-average conditions (1.1)-(1.2) motivate Assumptions A3 and A5.","marker":"[17]"},{"why":"Provides the asymptotic reflection coupling contraction for diffusions with non-uniformly dissipative drift that yields the key estimate (2.3).","marker":"[11]"},{"why":"Supplies the elementary reflection-coupling argument adapted in Lemma 2.1 to two different drifts.","marker":"[9]"},{"why":"Gives the generator formula for the asymptotic coupling used to compute the contraction in Lemma 2.1.","marker":"[25]"},{"why":"Defines evolution systems of measures and supplies the criterion used to construct the measures in Proposition 2.5.","marker":"[7]"},{"why":"Provides the uniqueness result for evolution systems of measures with finite first moment used to identify the averaged coefficients.","marker":"[8]"},{"why":"Shows the strong averaging setting and the role of a slow diffusion independent of the fast variable, formalized as Assumption A4.","marker":"[18]"},{"why":"Supplies the weak averaging principle template and the invariant-measure identification of averaged coefficients.","marker":"[32]"},{"why":"Is the authors' previous treatment of time-inhomogeneous multi-scale SDEs under uniform dissipation, which the present paper extends to partial dissipation.","marker":"[29]"}],"fun_headline_variants":["Time-inhomogeneous fast drift need not be fully dissipative","Partial dissipation suffices for SDE averaging","Averaging holds for non-uniform fast drift","Weak and strong averaging with partial dissipation","Time-dependent fast noise still averages out"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole proof rests on the fast drift having a dissipative tail: two fast trajectories that are far apart must be pushed toward each other at a uniform rate, and this exponential contraction is what allows every later averaging step to go through.","fun_headline_variants_meta":{"raw":{"variants":["Time-inhomogeneous fast drift need not be fully dissipative","Partial dissipation suffices for SDE averaging","Averaging holds for non-uniform fast drift","Weak and strong averaging with partial dissipation","Time-dependent fast noise still averages out"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000571,"raw_usage":{"total_tokens":2731,"prompt_tokens":1004,"completion_tokens":1727,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":620,"completion_tokens_details":{"reasoning_tokens":1658}},"tokens_in":620,"tokens_out":1727,"duration_ms":13128,"temperature":1.0,"reasoning_tokens":1658,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:47:58.950916+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct numerical check would settle the claim: take the one-dimensional fast drift $f(y)=\\sin(y)+\\alpha y$ with small $\\alpha>0$, which satisfies Assumption A2 with $K=\\alpha$ and $C=1$; the theorem predicts $\\sup_{t\\in[0,T]}\\mathbb{E}|X^\\varepsilon_t-\\bar X_t|^2\\to0$ as $\\varepsilon\\to0$. If the empirical distance does not vanish, the assumptions are not sufficient; if it vanishes but at a rate much worse than the bound (3.22), the stated rate is not sharp.","supporting_citations":[{"cited_title":"Khasminskii: On an averging principle for Itô stochastic differential equations,Kibernetica, 4 (1968), 260– 279","cited_arxiv_id":null,"evidence_quote":"Establishes the classical averaging principle whose time-average conditions (1.1)-(1.2) motivate Assumptions A3 and A5."},{"cited_title":"Eberle: Reflection couplings and contraction rates for diffusions,Probab","cited_arxiv_id":null,"evidence_quote":"Provides the asymptotic reflection coupling contraction for diffusions with non-uniformly dissipative drift that yields the key estimate (2.3)."},{"cited_title":"Durmus, A","cited_arxiv_id":null,"evidence_quote":"Supplies the elementary reflection-coupling argument adapted in Lemma 2.1 to two different drifts."},{"cited_title":"Priola and F.-Y","cited_arxiv_id":null,"evidence_quote":"Gives the generator formula for the asymptotic coupling used to compute the contraction in Lemma 2.1."},{"cited_title":"Da Prato and M","cited_arxiv_id":null,"evidence_quote":"Defines evolution systems of measures and supplies the criterion used to construct the measures in Proposition 2.5."},{"cited_title":"Da Prato and M","cited_arxiv_id":null,"evidence_quote":"Provides the uniqueness result for evolution systems of measures with finite first moment used to identify the averaged coefficients."},{"cited_title":"Liu: Strong convergence of principle of averaging for multiscale stochastic dynamical systems,Commun","cited_arxiv_id":null,"evidence_quote":"Shows the strong averaging setting and the role of a slow diffusion independent of the fast variable, formalized as Assumption A4."},{"cited_title":"Veretennikov: On the averaging principle for systems of stochastic differential equations,Math","cited_arxiv_id":null,"evidence_quote":"Supplies the weak averaging principle template and the invariant-measure identification of averaged coefficients."}],"review_version":2}