{"id":"86b6dea8-4a7a-4768-b261-e43079d0b490","arxiv_id":"2412.09850","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For time-inhomogeneous slow-fast SDEs, the slow component converges to an averaged equation at explicit rates governed by a decay function alpha(t), with periodic and convergent coefficients as special cases.","lead":"This paper proves explicit convergence rates for slow-fast stochastic differential equations whose fast component depends on time, using a new tool called nonautonomous Poisson equations. The result quantifies how quickly the slow variable follows an averaged equation, with rates that reduce to classical ones when time dependence disappears.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The weak averaging proof (Theorem 3.9) applies Proposition 2.7 to a parameter-dependent family F^t and asserts bounds on Phi^t and d_r Phi^t without proof; this is the key unverified step supporting the weak-rate claim (3.28).","rationale":"The paper's central contribution is the pair of explicit rates in Theorems 3.5 and 3.9. Theorem 3.5 is proved in detail with the Poisson-equation machinery, and its internal estimates check out. The weak theorem, however, contains the most significant unverified step: it applies the nonautonomous Poisson framework to a family F^t that depends on an extra parameter r and whose r-derivative grows in x. Proposition 2.7, as written, does not cover this situation, and the assertion that its proof can be repeated is not backed by a derivation. This is exactly the kind of omitted regularity estimate that warrants a conditional verdict. It is not an internal inconsistency: the claim is plausible, and the missing argument can likely be supplied. But until it is written out, the weak-rate claim (3.28), and consequently the rates in the convergent and periodic weak theorems (4.19), (5.11), rest on an unverified estimate. I do not see a reason to reject the paper; the strong theorem and examples give independent support, and the gap is local and fixable. Hence UNCHANGED with respect to the reader's CONDITIONAL verdict.","tokens_in":46988,"tokens_out":30075,"duration_ms":315272,"concrete_test":"Independently derive the claimed bound for d_r Phi^t(r,s,x,y) = integral_s^infinity E d_r F^t(r,u,x,Y^{s,x,y}_u) du, tracking every constant. Specifically: (i) show d_r F^t(r,u,x,.) has zero integral against mu_u^x; (ii) bound |d_r F^t(r,u,x,y1) - d_r F^t(r,u,x,y2)| by K(u,x)|y1-y2| with K(u,x) <= C_T(1+|x|); (iii) use E|Y^{s,x,y}_u - eta^x_u| and E|d_x Y - d_x eta| decays from Lemma 2.4 to integrate over u. If the resulting estimate is exactly C_T Lambda_gamma(s)(1+|x|^2+|y|^2) with no extra (1+s) factor, the step is sound; if an extra factor appears, modify (3.27) or the rate in (3.28).","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Theorem 3.9, after defining F^t(r,s,x,y) with parameters r, s, the paper states that Proposition 2.7 and its proof imply a solution Phi^t(r,s,x,y) satisfying sup_{0<=r<=t<=T}(|Phi^t| + |d_r Phi^t| + ||d_x Phi^t|| + ||d_xx Phi^t||) <= C_T Lambda_gamma(s)(1+|x|^2+|y|^2). This is not a direct consequence of Proposition 2.7 as stated: Proposition 2.7 assumes H(s,x,y) with no auxiliary parameter r, and condition (2.25) requires sup_{t,x,y}||d_x^i d_y^j H|| < infinity for j>=1. For F^t, the mixed derivative d_y d_r F^t is only bounded by C_T(1+|x|) (from (3.26), (3.27)), so (2.25) fails uniformly in x. The needed estimate for d_r Phi^t requires an additional argument: differentiating (2.28) under the integral, using the centering of d_r F^t, and controlling E[d_r F^t(r,u,x,Y) - d_r F^t(r,u,x,eta^x_u)] with a Lipschitz constant that may grow linearly in |x|. The paper does not supply that argument, and it is not immediately obvious that no extra factor such as (1+s) or an extra Lambda(s) appears. Since Theorem 3.9 is one of the two central claims, and Theorems 4.5 and 5.5 build on it, this unproved step is load-bearing.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper develops an averaging theory for the time-inhomogeneous slow-fast SDE (3.1), in which the fast component has coefficients f(t/ε,x,y), g(t/ε,x,y). Under a dissipativity condition (Assumption A1) with time-dependent rate α(·), the authors construct an evolution system of measures (Proposition 2.6) and solve nonautonomous Poisson equations of the form (2.26) with representation (2.28) and regularity estimates (2.29). These tools are applied to prove: (i) a strong mean-square error bound (Theorem 3.5) for σ(x,y)=σ(x); (ii) a weak error bound (Theorem 3.9) for general σ; (iii) averaged equations and rates when the fast coefficients converge as t→∞ (Theorems 4.4 and 4.5) or are τ-periodic (Theorems 5.3 and 5.5). The paper closes with two explicit one-dimensional examples; Example 6.1 gives an exact computation of the strong error for an Ornstein-Uhlenbeck fast component and shows the rate can be ε^{(1+β)/2}, ε(log 1/ε)^{1/2}, or ε depending on the growth of α, matching the bounds of Theorem 3.5.","tokens_in":47384,"tokens_out":14254,"duration_ms":138946,"significance":"The paper's main contribution is the derivation of explicit, parameter-free strong and weak averaging rates for time-inhomogeneous multi-scale SDEs, a regime where previous works (e.g., [8,40,42]) did not provide rates. The nonautonomous Poisson framework is natural and the proof itinerary is largely standard: Itô expansion, Poisson solution, martingale estimates, and Gronwall. A particular strength is that the strong rate in Theorem 3.5 is not a fitted constant bound: it is compared with an exact Gaussian computation in Example 6.1, which confirms the sharp order. The extensions to convergent and periodic fast coefficients are useful, and the dependence on the convergence profile φ(·) in Theorems 4.4 and 4.5 is new. However, the weak-rate proof in Theorem 3.9 rests on an unverified regularity statement for the parameter-dependent Poisson solution, and Lemma 2.3 omits the proof of the x-differentiability estimates on which Proposition 2.7 relies. These are load-bearing and need to be completed before the results can be accepted as established.","major_comments":[{"comment":"The statement that Proposition 2.7 'and its proof' yield a solution Φ^t of (3.29) with the displayed bound, including the term |∂_r Φ^t|, is not justified. Proposition 2.7 is stated for a single function H(s,x,y), and its proof uses condition (2.25), which requires uniform control of ∂_x^i ∂_y^j H for j≥1; it contains no statement about differentiating the solution with respect to an auxiliary parameter r. For the family F^t(r,s,x,y), the mixed derivative ∂_y ∂_r F^t is only bounded by C_T(1+|x|) by (3.26)–(3.27), and such a bound is not enough to apply Proposition 2.7 directly to obtain |∂_r Φ^t| with the factor Λ_γ(s). A separate argument differentiating (2.28) in r, using the centering of ∂_r F^t and controlling E[∂_r F^t(r,u,x,Y) - ∂_r F^t(r,u,x,η)] with a Lipschitz constant that may grow linearly in |x|, is required but absent. Since (3.28) is one of the two central claims and Theorems 4.5 and 5.5 build on it, this is a load-bearing gap.","section":"§3.2, Theorem 3.9 (Eqs. (3.28)–(3.29))"},{"comment":"The assertions that Y^{s,x,y}_t is twice mean-square differentiable in x and that sup_{t≥s} E||∂_x Y^{s,x,y}_t||^4 and sup_{t≥s} E||∂_x^2 Y^{s,x,y}_t||^2 are bounded are stated with the proof 'omitted here.' These bounds enter directly into Proposition 2.7 through the estimates for ∂_x Φ and ∂_x^2 Φ in (2.29), and hence into Theorem 3.5 and the later averaging theorems. The x-derivative equation contains additional terms involving ∂_x f, ∂_x g, ∂_y f, and ∂_y g that are not present in the y-derivative argument, so the reduction to the y-argument is not immediate. A detailed proof, or a precise statement of the analogous argument, is needed.","section":"§2.1, Lemma 2.3 (Eq. (2.9))"}],"minor_comments":[{"comment":"The final display 'sup_{t∈[0,T]} E|X^ε_t - \\bar X^ε_t| ≍ ...' is inconsistent with the preceding computation, which yields E|X^ε_t - \\bar X_t|^2 ≍ ε^{1+β} for β<1; the square and the halved exponent are missing, and the averaged solution is denoted \\bar X_t (independent of ε). Since this example is used to assert optimality, the display should be corrected.","section":"Example 6.1, final display"},{"comment":"The display (3.27) is missing a supremum over x; as written, the left-hand side still depends on x, so the displayed bound with C_T(1+|x|) is not a uniform estimate in x.","section":"§3.2, Eq. (3.27)"},{"comment":"The sentence claiming that Theorems 3.5 and 3.9 imply sup |EX^ε_t - E\\bar X^ε_t| ≤ Cε is not a direct consequence of Theorem 3.9, because φ(x)=x is not in C_b^4(R^n). The exact computation in the example already proves this estimate, so the reference to Theorem 3.9 should be removed or justified by an approximation argument.","section":"Example 6.2"},{"comment":"There are numerous typographical errors: 'Grownall' for Gronwall, 't/greaterorequalslants' for t≥s, '0 ≤ i ≤ 4 and 0 ≤ i ≤ 5' in Remark 2.5 (presumably the second index is j), and the notation in (2.4) for improper integrals over (-∞,0). These should be corrected.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"To the editor: The paper is within the scope of the journal and the underlying idea is sound. The main issue is the unproved parameter-dependent Poisson regularity in §3.2; this is not a matter of exposition but a gap in the proof of the central weak-rate theorem. I recommend major revision. The self-citations to the authors' earlier Poisson and ergodic works are frequent but not inappropriate; they are used as background tools. The paper would benefit from a short derivation of the ∂_r Φ^t estimate in Theorem 3.9, either by differentiating the representation (2.28) and supplying the required centering and Lipschitz bounds, or by an alternative argument."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a serious paper, worth refereeing, but the weak averaging theorem has a load-bearing gap. The strong averaging result (Theorem 3.5) looks solid: with sigma independent of y, the Poisson-equation decomposition and the estimates do lead to the epsilon^2-bound, and Example 6.1's closed-form lower bound matching the upper rate is a strong sign the rate is right. The nonautonomous Poisson equation setup is new and should be reusable. The paper also does well to state assumptions precisely and to give two explicit examples that check the claims.\n\nThe soft spot is Theorem 3.9. Proposition 2.7 only covers H with no auxiliary parameter r, and condition (2.25) requires sup_{t,x,y}||d_x^i d_y^j H|| finite for j>=1. For F^t, the mixed derivative d_y d_r F^t is only bounded by C_T(1+|x|), so the proposition does not apply as stated, and the asserted bound on d_r Phi^t does not follow. The stress-test note is accurate. You need an extra argument differentiating the integral representation of Phi^t, using centering of d_r F^t and controlling the difference involving the frozen flow; that argument may hold but it is not in the paper. Since Theorems 4.5 and 5.5 rely on Theorem 3.9, this is not a cosmetic issue.\n\nOther concerns are minor. Lemma 2.3 asserts x-differentiability with details omitted, but the pattern is standard and the omitted proof is likely routine. The symmetric extension to negative times in Remark 3.1 is natural but should be checked carefully. The two-sided divergence condition in A1 is restrictive, but the paper is explicit about it. The citation pattern is fine: self-citations are used as background tools, not to force the claim, and there is no circularity.\n\nWho this is for: stochastic analysts working on averaging and homogenization, and anyone who needs explicit convergence rates for time-inhomogeneous slow-fast systems. The strong rate with the sharp example is the core takeaway; the weak rate is plausible but not proven as written.\n\nRecommendation: send to a serious referee. Ask specifically for the missing argument in Theorem 3.9 before trusting (3.28) or the later theorems built on it.","headline":"New explicit rates for nonautonomous multiscale averaging, but the weak-rate proof has a gap that needs a real fix before the main claim can be trusted.","tokens_in":47885,"tokens_out":1883,"would_cite":true,"duration_ms":21356,"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":"The paper proves explicit strong and weak averaging rates for time-inhomogeneous multi-scale SDEs, with rates that depend on the time-dependent dissipation of the fast process.","keywords":["averaging principle","time-inhomogeneous stochastic differential equations","multi-scale SDE","nonautonomous Poisson equation","evolution system of measures","explicit convergence rates","slow-fast systems","periodic coefficients"],"falsifier":"The sharpness claim in Example 6.1 is directly checkable: with $\\alpha(t)=c_0(1+t)^{\\beta}$, the paper's formulas say the mean-square error is of order $\\varepsilon^{1+\\beta}$ for $-1<\\beta<1$, $\\varepsilon^2\\log(1/\\varepsilon)$ for $\\beta=1$, and $\\varepsilon^2$ for $\\beta>1$; computing the exact covariance of the linear system would reveal whether these exponents hold, and any mismatch would invalidate the rate theorems.","tokens_in":46821,"feed_emoji":"🎲","tokens_out":14639,"duration_ms":128701,"temperature":0.7,"pith_summary":"The paper proves that the slow component of a two-scale stochastic system can be approximated by an averaged equation even when the fast dynamics depends on time through the scaled clock $t/\\varepsilon$, and it gives explicit rates for the error in mean square and in law. The rates are controlled by a single function $\\alpha(t)$ that measures how fast the frozen fast process forgets its past; depending on $\\alpha$, the strong error can be smaller or larger than the classical $\\varepsilon^{1/2}$, and explicit formulas are obtained when $\\alpha(t)=c_0(1+t)^{\\beta}$. To reach these rates, the paper replaces the classical invariant measure with an evolution system of measures and replaces the autonomous Poisson equation with a nonautonomous one, whose solution is a conditional integral of the centred fluctuation. When the fast coefficients converge or are periodic in time, the averaged equation can be chosen independent of $\\varepsilon$, and the paper tracks how the convergence modulus or the period enters the rate. Two solvable one-dimensional examples show that the rates are sharp and that earlier periodic averaging statements missed the optimal strong order.","feed_headline":"Time-varying slow-fast SDEs now have explicit averaging rates","feed_subtitle":"The error rate is set by how fast the fast variable forgets its past, so it can beat or lag the classic square-root-of-epsilon order.","key_machinery":"The load-bearing object is the nonautonomous Poisson equation $\\partial_s\\Phi(s,x,y)+\\mathcal{L}_x(s)\\Phi(s,x,\\cdot)(y)=-H(s,x,y)$, where $\\mathcal{L}_x(s)$ is the generator of the frozen fast SDE; its solution is represented as $\\Phi(s,x,y)=\\int_s^{\\infty}\\mathbb{E}H(r,x,Y_r^{s,x,y})\\,dr$, which is the time-inhomogeneous analogue of the classical Poisson equation used for optimal rates in autonomous slow-fast systems. The companion object is the evolution system of measures $\\{\\mu_t^x\\}$, the family of laws reached from the infinite past, which replaces the invariant measure and defines the averaged coefficients $\\bar b(t,x)=\\int b(x,y)\\,\\mu_t^x(dy)$ and $\\overline{\\sigma\\sigma^*}(t,x)=\\int(\\sigma\\sigma^*)(x,y)\\,\\mu_t^x(dy)$. The derivative estimates (2.29) on $\\Phi$, together with Itô's formula applied along the slow-fast trajectories, convert the error into $\\varepsilon$ times integrals of $\\alpha$ and the kernels $\\Lambda$ and $\\Lambda_\\gamma$.","core_discovery":"The central discovery is that the Poisson-equation route to optimal averaging rates works in the time-inhomogeneous setting, provided one uses the evolution system of measures $\\{\\mu_t^x\\}_{t\\in\\mathbb{R}}$ of the frozen fast SDE $dY_t=f(t,x,Y_t)\\,dt+g(t,x,Y_t)\\,dW_t^2$ and solves the nonautonomous Poisson equation $\\partial_s\\Phi(s,x,y)+\\mathcal{L}_x(s)\\Phi(s,x,\\cdot)(y)=-H(s,x,y)$ by $\\Phi(s,x,y)=\\int_s^{\\infty}\\mathbb{E}H(r,x,Y_r^{s,x,y})\\,dr$ for centred $H$. With the derivative bounds (2.29) this yields the strong error bound $\\sup_{0\\le t\\le T}\\mathbb{E}|X_t^\\varepsilon-\\bar X_t^\\varepsilon|^2\\le C_{T,x,y}\\,\\varepsilon^2[\\sup_{0\\le t\\le T}|\\Lambda_\\gamma(t/\\varepsilon)|^2+\\int_0^{T/\\varepsilon}\\alpha(s)\\Lambda^2(s)\\,ds]$ when $\\sigma\\equiv\\sigma(x)$ (Theorem 3.5), and the weak error bound $\\sup_{0\\le t\\le T}|\\mathbb{E}\\phi(X_t^\\varepsilon)-\\mathbb{E}\\phi(\\bar X_t^\\varepsilon)|\\le C_{\\phi,T,x,y}\\,\\varepsilon\\sup_{t\\in[0,T]}\\Lambda_\\gamma(t/\\varepsilon)$ for general $\\sigma$ (Theorem 3.9). When the fast coefficients converge to a time-homogeneous limit or are $\\tau$-periodic, the averaged coefficients can be taken independent of $\\varepsilon$, and the same Poisson machinery produces rates involving the convergence modulus $\\varphi$ or the period $\\tau$ (Theorems 4.4, 4.5, 5.3 and 5.5).","pith_inferences":["The same nonautonomous Poisson representation, with its derivative bounds, is exactly the input a central limit theorem and a diffusion approximation for time-inhomogeneous slow-fast systems would need; those limit theorems are a natural next step even though the paper does not state them.","The $\\varepsilon$-dependence of the averaged equation in the general case is a modelling choice, not a flaw: a numerical method that wants an $\\varepsilon$-free model must first know whether the fast coefficients converge or are periodic, because the two regimes lead to different error structures.","The rate formulas give a ready diagnostic for multiscale integrators: from the frozen fast coefficients one can compute $\\alpha$, $\\Lambda$, and $\\int_0^{T/\\varepsilon}\\alpha(s)\\Lambda^2(s)\\,ds$, then use that closed-form quantity as an a priori estimate of the averaging bias before running a simulation."],"forward_implications":["For $\\alpha(t)=c_0(1+t)^{\\beta}$, the strong error scales as $\\varepsilon^{(1+\\beta)/2}$ for $-1<\\beta<1$, $\\varepsilon\\sqrt{\\log(1/\\varepsilon)}$ at $\\beta=1$, and $\\varepsilon$ for $\\beta>1$, so time-dependent dissipation can improve or worsen the classical $\\varepsilon^{1/2}$ strong order (Remark 3.7).","The weak error is of order $\\varepsilon^{1+\\beta}$ for $-1<\\beta<0$ and of order $\\varepsilon$ for $\\beta\\ge 0$, recovering order $1$ in the stationary case and slowing down when the dissipation decays (Remark 3.10).","In the convergent-coefficients case the error contains the convolution term $\\int_0^{T/\\varepsilon}(\\int_0^s e^{-2\\beta\\alpha(s-r)}\\varphi^2(r)\\,dr)^{1/2}\\,ds$, so the transient $\\varphi$ of the fast coefficients directly sets the effective convergence rate (Theorems 4.4 and 4.5).","For $\\tau$-periodic fast coefficients, the averaged equation is the period average of the $\\varepsilon$-dependent averaged coefficients, with general strong order $\\varepsilon^{2/3}$ and weak order $\\varepsilon^{1/3}$; the strong order improves to $\\varepsilon$ when $\\alpha$ is constant and $\\sigma\\equiv 0$ with the supremum outside the expectation (Theorems 5.3 and 5.5).","Example 6.1 shows the rates are sharp: for a linear test system the mean-square error is comparable to the expression predicted by Theorem 3.5, confirming that the $\\alpha$-dependent formulas are the correct leading behaviour."],"supporting_citations":[{"why":"Supplies the concept and existence theory of evolution systems of measures for dissipative time-dependent SDEs, the basis for the averaged coefficients.","marker":"[12]"},{"why":"Gives the definition and basic properties of evolution systems of measures that Section 2 uses to construct the time-dependent averages.","marker":"[13]"},{"why":"Establishes the Poisson-equation method for optimal strong rates in autonomous diffusion settings, which the paper adapts to the nonautonomous case.","marker":"[30]"},{"why":"Provides the regularity theory for autonomous Poisson equations that motivates the derivative estimates needed in Theorem 3.5.","marker":"[31]"},{"why":"Gives the classical optimal strong order $1/2$ and the counterexample showing $\\sigma$ cannot depend on $y$ for strong averaging, motivating the weak theorem.","marker":"[28]"},{"why":"Explains why explicit averaging rates feed numerical error analysis and supplies the autonomous weak order $1$ baseline.","marker":"[6]"},{"why":"Treats time-inhomogeneous slow-fast SPDEs with almost-periodic coefficients without convergence rates, the prior setting this paper makes quantitative.","marker":"[8]"},{"why":"Establishes a double-averaging statement for periodically forced slow-fast systems; Remark 5.4 compares and corrects its strong order claim.","marker":"[42]"},{"why":"Used to prove that a periodic fast semigroup has a periodic evolution system of measures, which underlies the epsilon-free periodic averaged equation.","marker":"[14]"}],"fun_headline_variants":["Explicit averaging rates for time-inhomogeneous multiscale SDEs","Nonautonomous Poisson equations power new averaging bounds","Time-varying slow-fast SDEs: explicit rates for averaged limit","Averaging with rates when coefficients drift in time","Optimal averaging rates for time-dependent multiscale systems"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on the frozen fast equation being dissipative at a time-dependent rate $\\alpha(t)$ whose two-sided integral diverges, so that an evolution system of measures exists and the nonautonomous Poisson solution can be written as an integral from $s$ to infinity.","fun_headline_variants_meta":{"raw":{"variants":["Explicit averaging rates for time-inhomogeneous multiscale SDEs","Nonautonomous Poisson equations power new averaging bounds","Time-varying slow-fast SDEs: explicit rates for averaged limit","Averaging with rates when coefficients drift in time","Optimal averaging rates for time-dependent multiscale systems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000753,"raw_usage":{"total_tokens":3431,"prompt_tokens":1105,"completion_tokens":2326,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":721,"completion_tokens_details":{"reasoning_tokens":2242}},"tokens_in":721,"tokens_out":2326,"duration_ms":17430,"temperature":1.0,"reasoning_tokens":2242,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T16:39:59.963088+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The sharpness claim in Example 6.1 is directly checkable: with $\\alpha(t)=c_0(1+t)^{\\beta}$, the paper's formulas say the mean-square error is of order $\\varepsilon^{1+\\beta}$ for $-1<\\beta<1$, $\\varepsilon^2\\log(1/\\varepsilon)$ for $\\beta=1$, and $\\varepsilon^2$ for $\\beta>1$; computing the exact covariance of the linear system would reveal whether these exponents hold, and any mismatch would invalidate the rate theorems.","supporting_citations":[{"cited_title":"Da Prato and M","cited_arxiv_id":null,"evidence_quote":"Supplies the concept and existence theory of evolution systems of measures for dissipative time-dependent SDEs, the basis for the averaged coefficients."},{"cited_title":"Da Prato and M","cited_arxiv_id":null,"evidence_quote":"Gives the definition and basic properties of evolution systems of measures that Section 2 uses to construct the time-dependent averages."},{"cited_title":"Pardoux and A.Y","cited_arxiv_id":null,"evidence_quote":"Establishes the Poisson-equation method for optimal strong rates in autonomous diffusion settings, which the paper adapts to the nonautonomous case."},{"cited_title":"Pardoux and A.Y","cited_arxiv_id":null,"evidence_quote":"Provides the regularity theory for autonomous Poisson equations that motivates the derivative estimates needed in Theorem 3.5."},{"cited_title":"Liu: Strong convergence of principle of averaging fo r multiscale stochastic dynamical systems, Commun","cited_arxiv_id":null,"evidence_quote":"Gives the classical optimal strong order $1/2$ and the counterexample showing $\\sigma$ cannot depend on $y$ for strong averaging, motivating the weak theorem."},{"cited_title":"Bréhier: Orders of convergence in the averaging pri nciple for SPDEs: the case of a stochastically forced slow component, Stochastic Process","cited_arxiv_id":null,"evidence_quote":"Explains why explicit averaging rates feed numerical error analysis and supplies the autonomous weak order $1$ baseline."},{"cited_title":"Cerrai and A","cited_arxiv_id":null,"evidence_quote":"Treats time-inhomogeneous slow-fast SPDEs with almost-periodic coefficients without convergence rates, the prior setting this paper makes quantitative."},{"cited_title":"Wainrib: Double averaging principle for periodical ly forced slow-fast stochastic systems, Electron","cited_arxiv_id":null,"evidence_quote":"Establishes a double-averaging statement for periodically forced slow-fast systems; Remark 5.4 compares and corrects its strong order claim."},{"cited_title":"Da Prato and C","cited_arxiv_id":null,"evidence_quote":"Used to prove that a periodic fast semigroup has a periodic evolution system of measures, which underlies the epsilon-free periodic averaged equation."}],"review_version":1}