{"id":"242e70e7-a7cd-4b2a-8d85-d09ba8be1003","arxiv_id":"2507.07538","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Strong L^p averaging principle with rates for nonautonomous slow-fast SPDEs driven by α-stable processes, extending prior autonomous and Gaussian-noise results.","lead":"This paper proves that the slow part of a two-timescale stochastic system driven by heavy-tailed Lévy noise is well approximated by a simpler averaged equation, with explicit convergence rates. The result covers time-dependent coefficients, an extension of earlier work that only handled time-independent systems with Gaussian noise.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Lemma 3.2 uses 1−e^{−x} ≤ C x^{αθ/2}, which fails when θ > 2/α; since A2 permits θ>2/α, the stated rate in Theorem 2.3 is not established in that regime.","rationale":"The reader's weakest_assumption (L_G<β1) is a standard dissipativity condition and is not the soft spot. The omitted Proposition 4.4 is a genuine expositional gap, but the stated exponential convergence (4.8) is plausible from Lemma 4.1 and can be supplied by coupling; it is not the central obstruction. The false elementary inequality in Lemma 3.2 is different: it is an explicit incorrect step, it is used in every main theorem, and A2 as written allows the offending regime. The reader noted this inequality in the rationale but did not make it the weakest assumption, hence partial agreement on the single most load-bearing concern. The heat-equation example satisfies θ<1/α and is safe, so the appropriate outcome is to keep the conditional acceptance pending a restriction or a corrected rate. No evidence of deliberate overclaim; the issue is technical and localized.","tokens_in":27545,"tokens_out":16256,"duration_ms":177183,"concrete_test":"Take α=3/2, θ=9/5 so that αθ/2=27/20>1, and choose λ_k=2^k, ρ_k^α=2^{-0.45k}/k^2, so Assumption A2 holds. Compute S(δ)=∑_{k≥1} ρ_k^α (1−e^{−λ_k δ})/λ_k for δ=2^{−m}, m=1,...,30. If S(δ)/δ^{θ/2}→∞ (expected: S(δ)∼δ), then the displayed inequality behind (3.9) is false in this admissible regime. Then redo the proof of Theorem 2.3 using S(δ)^{1/α}∼δ^{1/α} in place of δ^{θ/2}; the resulting exponent (p−1)/(α+p−1) is strictly smaller than (2.7) for p∈(1,α), showing the theorem as stated needs a θ≤2/α restriction or a modified rate.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Lemma 3.2, estimate (3.9) bounds the time increment of the stochastic convolution by δ^{θ/2} using the inequality 1−e^{−x} ≤ C x^{αθ/2} for all x>0. This inequality is true only when αθ/2 ≤ 1; for αθ/2 > 1, as x↓0 we have 1−e^{−x} ∼ x, which is much larger than x^{αθ/2}, so no constant works. Assumption A2 does not exclude θ > 2/α: it only asks ∑ ρ_k^α / λ_k^{1−αθ/2} < ∞, and with ρ_k decaying fast this can hold while αθ/2 > 1. When it does, the claimed δ^{θ/2} temporal regularity of the stochastic convolution and hence of X^ε in Lemma 3.2 fails; a correct estimate gives at best order δ^{1/α} (or a ρ-dependent slower rate). This δ^{θ/2} enters the proof of Theorem 2.3 through I_1^ε and I_2^ε, equations (5.8)–(5.9), and through Lemma 3.3 it propagates into Theorems 2.5 and 2.6. Consequently, the explicit rate (2.7) is not proved for the full range of θ permitted by A2. The heat-equation example has θ < 1/α < 2/α and is unaffected, so this is a patchable restriction rather than a collapse of the main idea.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the nonautonomous slow-fast SPDE system (1.1) driven by cylindrical alpha-stable processes with alpha in (1,2). The authors introduce an evolution system of measures for the time-inhomogeneous frozen fast equation and define averaged drift coefficients. The main result, Theorem 2.3, asserts strong L^p convergence of the slow component to the solution of an averaged equation with an explicit rate under Assumptions A1-A3. Under additional periodicity or asymptotic-convergence assumptions, Theorems 2.5 and 2.6 give convergence to averaged equations with epsilon-independent coefficients. The paper closes with a stochastic heat equation example verifying the hypotheses. The proof strategy follows Khasminskii's time discretization combined with exponential contraction estimates for the frozen flow.","tokens_in":27828,"tokens_out":15909,"duration_ms":165393,"significance":"If correct, the results would provide the first strong averaging principle for nonautonomous slow-fast SPDEs driven by alpha-stable noise, and they would resolve the open problem mentioned in [1, Remark 3.3] by removing the uniform boundedness condition on F. The use of evolution systems of measures is a natural extension of the Brownian framework to heavy-tailed noise, and the explicit convergence rates are a useful quantitative feature. The paper is clearly organized, and the main estimates are largely worked out. However, the current proofs contain a few correctable but load-bearing gaps, in particular in the temporal regularity estimate for the stochastic convolution and in the rigorous justification of the Itô-type chain rule for mild solutions.","major_comments":[{"comment":"The inequality 1 - e^{-x} <= C x^{alpha theta / 2} used in (3.9) is not valid for all x > 0 when alpha theta / 2 > 1; in that regime 1 - e^{-x} ~ x as x decreases to 0. Since Assumption A2 does not exclude alpha theta / 2 > 1, the claimed bound (3.4) is not established for the full range of theta. A corrected argument gives at best delta^{1/alpha} in place of delta^{theta/2} for the stochastic convolution term, which changes the balancing of epsilon/delta and delta^{theta(p-1)/2} in the proof of Theorem 2.3 and propagates through Lemma 3.3 to Theorems 2.5 and 2.6. The authors should either add the restriction theta < 2/alpha to A2 or reformulate the rates with min(theta/2, 1/alpha).","section":"§3, Lemma 3.2, Eq. (3.9)"},{"comment":"The argument below Eq. (5.4) applies the chain rule to U_rho(Z^epsilon_t), where Z^epsilon is a difference of two mild solutions and is not known to be a strong solution. Although the noise terms cancel in the equation for Z^epsilon, the mild solution need not be differentiable, so the displayed identity for E U_rho(Z^epsilon_t) requires a justification via Yosida or Galerkin approximations. This is a standard but necessary step, and the main L^p estimate depends on it.","section":"§5.1, proof of Theorem 2.3, Step 1"},{"comment":"Proposition 4.4 is a central ingredient: it establishes that {mu^x_t} is an evolution system of measures and provides the exponential convergence (4.8) used in Lemma 5.1, Theorem 2.3, and Theorem 2.6. The proof is omitted with a reference to [11], which treats Brownian noise. Because the present setting has alpha-stable noise and only p < alpha moments, the adaptation is not immediate; the authors should give a proof or a detailed sketch of the proposition.","section":"§4, Proposition 4.4"}],"minor_comments":[{"comment":"There are several typographical errors, e.g., \"This paper focus\" in the introduction and \"Lipshcitz\" in Section 4; the text should be proofread carefully.","section":"§1 and §4"},{"comment":"The proof of Theorem 2.5 uses the same symbol delta for the discretization step as in Theorem 2.3 but chooses a different value delta = epsilon^{2/(theta+2)}; the two choices should be distinguished to avoid confusion.","section":"§5.1 and §5.2"},{"comment":"The H_theta norm of the initial condition term is bounded by C t^{-theta/2}|x|, which diverges as t to 0; this is acceptable because the subsequent integrals start at delta > 0, but this should be stated explicitly.","section":"§3, Lemma 3.2, Eq. (3.6)"},{"comment":"The exponent p theta/(2+theta) in (5.20) is obtained by balancing (epsilon/delta)^p and delta^{p theta/2}; adding one line showing this balance would improve readability.","section":"§5.2, Eq. (5.20)"},{"comment":"In the asymptotic convergence case of the example, the verification that Assumption A5 holds for the Nemytskii operators is only sketched; a direct statement of the operator-level estimates would make the example self-contained.","section":"§6, Example"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the journal's scope, and the referee believes the core approach is sound. The major comments are all fixable; the most serious issue is the overclaimed rate in Lemma 3.2 for theta > 2/alpha, which can be repaired either by restricting Assumption A2 or by adjusting the stated rates. The dependence on the authors' own unpublished preprints [28] and [29] for elementary lemmas is a minor concern that the editor may wish to flag, as the referees cannot readily check those results if the preprints are not available."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this is a real result in the right setting, and the main machinery is sound. The one thing to fix before relying on the stated rates: Lemma 3.2's inequality (3.9) is only valid for θ ≤ 2/α, and Assumption A2 allows θ up to 2. So the advertised rate in Theorem 2.3 is not proven in the full range.\n\nWhat's genuinely new: the first averaging principle for nonautonomous slow-fast SPDEs with α-stable noise. The previous literature had autonomous α-stable (Bao-Yin-Yuan and Sun-Xie) and nonautonomous Gaussian (Cerrai-Lunardi). The paper also removes the boundedness condition on F flagged in [1, Remark 3.3] by working in L^p with p<α, avoiding second moments. That is a legitimate technical advance. The proof strategy is the expected one: evolution system of measures for the frozen equation, Khasminskii time discretization, then a Gronwall argument. The heat equation example is worked out cleanly.\n\nThe main problem is (3.9). The bound 1−e^{−x} ≤ C x^{αθ/2} holds for all x>0 only when αθ/2 ≤ 1. For larger θ, the left side is ~x for small x and cannot be bounded by a higher power. Since A2 does not exclude θ > 2/α, the temporal regularity estimate in Lemma 3.2, and with it the explicit rate, is not established for that range. A correct treatment would either restrict θ to (0, 2/α] or use a different modulus (e.g. δ^{1/α}) giving a slower rate. The heat equation example has θ < 1/α, so it is unaffected. This looks patchable, not fatal.\n\nTwo smaller issues: the chain rule is applied to a mild solution in Theorem 2.3 without approximation argument; standard but should be justified. And Proposition 4.4, which underpins the exponential convergence (4.8) used everywhere, is cited without proof. I'd want the details or a precise reference.\n\nFor a reader in averaging principles for SPDEs, this is worth the time. It deserves a serious referee: the core idea is right and the gap is narrow. I'd send it out, with the request that the authors either fix the θ range or weaken the statement accordingly. I would not cite the current version for the full θ range, but would cite a corrected version.","headline":"Genuinely new averaging principle for nonautonomous slow-fast SPDEs with α-stable noise, but the stated θ range in A2 is too broad because Lemma 3.2 uses an inequality that only holds for θ ≤ 2/α.","tokens_in":28432,"tokens_out":5663,"would_cite":false,"duration_ms":56638,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H35"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that in nonautonomous slow-fast SPDEs driven by $\\alpha$-stable processes, the slow component converges strongly in $L^p$ to the solution of an averaged equation with an explicit rate, even when the drift is unbounded.","keywords":["averaging principle","slow-fast SPDE","stochastic partial differential equations","evolution system of measures","nonautonomous","α-stable process","time periodic","strong convergence"],"falsifier":"Run the stochastic heat equation example of Section 6 with $p=1.5$, $\\alpha=1.7$, $\\theta=0.5$, compute the empirical $L^p$ error between $X^{\\varepsilon}$ and the averaged solution for $\\varepsilon = 10^{-1}, 10^{-2}, 10^{-3}$, and plot the error against $\\varepsilon$ on a log-log scale: the slope should equal $\\theta(p-1)/(\\theta(p-1)+2)$. A slope clearly below that value, or a non-vanishing error, would falsify Theorem 2.3.","tokens_in":27302,"feed_emoji":"⚡","tokens_out":10375,"duration_ms":91535,"temperature":0.7,"pith_summary":"Slow-fast systems couple a quickly changing variable to a slowly changing one; over long times the slow variable often behaves like an averaged, simpler system. This paper proves such an averaging principle for stochastic partial differential equations driven by heavy-tailed $\\alpha$-stable noise, where the coefficients depend on time and the drift is allowed to grow linearly rather than be bounded. The central result is a strong $L^p$ convergence (for $p$ strictly below the stability index $\\alpha$) of the slow component to the solution of an averaged equation, with an explicit polynomial rate in the scale parameter $\\varepsilon$. The paper also shows that when the coefficients are time-periodic, or converge asymptotically, the averaged equation can be chosen independent of $\\varepsilon$. A concrete stochastic heat equation shows the assumptions are satisfiable.","feed_headline":"Slow SPDE components converge to averaged equations at explicit rates","feed_subtitle":"Nonautonomous slow-fast SPDEs driven by α-stable noise obey an averaging principle even with unbounded drifts.","key_machinery":"The machinery is the evolution system of measures $\\{\\mu^x_t\\}_{t\\in\\mathbb{R}}$ for the time-inhomogeneous frozen fast equation $dY_t = [BY_t + G(t,x,Y_t)]\\,dt + dZ_t$, which plays the role that an invariant measure plays in autonomous averaging. The load-bearing property is the exponential contraction of the frozen flow, $|Y^{s,x_1,y_1}_t - Y^{s,x_2,y_2}_t| \\le e^{-(\\beta_1-L_G)(t-s)/2}|y_1-y_2| + C|x_1-x_2|$, together with the consequent exponential convergence of the transition semigroup to the evolution system of measures (4.8). The slow-fast proof then uses Khasminskii's time-discretization: on intervals of length $\\delta$ the fast process is replaced by an auxiliary process with the slow component frozen, and the three error terms---time-regularity of the slow path, difference between true and auxiliary fast process, and the mixing error controlled by (4.8)---are balanced by choosing $\\delta = \\varepsilon^{2/(\\theta(p-1)+2)}$. The function $U_\\rho(x)=(|x|^2+\\rho)^{p/2}$ is used to handle $L^p$ estimates for non-integer $p$ without assuming boundedness of $F$.","core_discovery":"Under Assumptions A1--A3, for every $p\\in(1,\\alpha)$ and $T>0$ the paper establishes the estimate $\\sup_{t\\in[0,T]} E|X^{\\varepsilon}_t - \\bar{X}^{\\varepsilon}_t|^p \\le C_{p,T}(1+|x|^p+|y|^p)\\,\\varepsilon^{\\theta(p-1)/(\\theta(p-1)+2)}$, where $\\bar{X}^{\\varepsilon}$ solves the averaged equation $d\\bar{X}^{\\varepsilon}_t = [A\\bar{X}^{\\varepsilon}_t + \\bar{F}(t/\\varepsilon,\\bar{X}^{\\varepsilon}_t)]\\,dt + dL_t$ with $\\bar{F}(t,x)=\\int_H F(t,x,y)\\,\\mu^x_t(dy)$, the average of the slow drift against the evolution system of measures of the frozen fast equation. Theorems 2.5 and 2.6 give the same convergence toward an $\\varepsilon$-independent averaged equation under time-periodicity (A4) or asymptotic convergence (A5) of the coefficients, with the rate in the asymptotic case also depending on how quickly the coefficients approach their limits. The proof removes the uniform boundedness of $F$ that earlier $\\alpha$-stable averaging results required, solving the problem raised in [1, Remark 3.3].","pith_inferences":["The balancing choice $\\delta = \\varepsilon^{2/(\\theta(p-1)+2)}$ suggests the error is dominated jointly by slow-path regularity and fast mixing; a numerical test could probe whether the exponent is sharp or whether a Poisson-equation approach yields the closer-to-optimal rate the authors mention in Remark 2.4.","The strict inequality $L_G < \\beta_1$ is used for exponential contraction; a natural testable extension is whether the averaging principle survives with only partial dissipation in the fast component, as has been explored for time-inhomogeneous SDEs with Poisson techniques.","The same evolution-system-of-measures framework should extend to multiplicative $\\alpha$-stable noise or other pure-jump L\\'evy processes with comparable scaling, as long as the frozen flow contracts; the key bottleneck would be the analogue of estimate (4.8)."],"forward_implications":["The slow component $X^{\\varepsilon}$ can be replaced, in $L^p$ for any $p<\\alpha$, by the solution of the one-equation averaged system with error of order $\\varepsilon^{\\theta(p-1)/(\\theta(p-1)+2)}$, making the approximation quantitative for simulation and control.","When the coefficients are time-periodic, the averaged limit is an autonomous SPDE, so the long-time behaviour of the original nonautonomous system is governed by an autonomous equation with constant coefficients.","Under the asymptotic-convergence assumption, the rate is degraded by the rate at which the coefficients forget their initial time, as measured by $\\phi_1$ and the convolution-weighted $\\tilde{\\phi}_2$.","The result answers the open question from the earlier $\\alpha$-stable averaging literature by removing the uniform boundedness assumption on the drift $F$, at the price of only mild regularity and a strict contraction condition on the fast equation."],"supporting_citations":[{"why":"Provides the prior strong averaging principle for two-time-scale SPDEs driven by $\\alpha$-stable noise and the open problem (Remark 3.3) about removing the uniform boundedness of the nonlinearity, which this paper solves.","marker":"[1]"},{"why":"The previous nonautonomous slow-fast SPDE averaging result, but for Brownian noise; this paper extends it to $\\alpha$-stable driving noise.","marker":"[7]"},{"why":"Supplies the definition and basic properties of evolution systems of measures for time-dependent semigroups, which replace the invariant measure in the nonautonomous setting.","marker":"[11]"},{"why":"The Khasminskii time-discretization idea of freezing the slow component on small intervals, which structures the proof of the main convergence estimates.","marker":"[21]"},{"why":"Gives well-posedness and structural estimates for semilinear SPDEs driven by cylindrical stable processes, used to justify existence and a priori bounds for the fast equation.","marker":"[25]"},{"why":"Provides the $L^p$ moment bounds for stochastic convolutions of stable processes used in the a priori estimates and in Assumption A2.","marker":"[26]"}],"fun_headline_variants":["Strong convergence rates for alpha-stable slow-fast SPDE averaging","Averaging without finite moments in alpha-stable SPDEs","Explicit averaging rates for slow-fast SPDEs with heavy tails","New strong averaging result for alpha-stable SPDEs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on the fast component's self-interaction being strictly weaker than its linear damping ($L_G < \\beta_1$), which forces the frozen fast flow to contract exponentially; if that inequality fails, the averaged coefficient may not even be defined and the rate estimates collapse.","fun_headline_variants_meta":{"raw":{"variants":["Strong convergence rates for alpha-stable slow-fast SPDE averaging","Averaging without finite moments in alpha-stable SPDEs","Explicit averaging rates for slow-fast SPDEs with heavy tails","New strong averaging result for alpha-stable SPDEs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00037,"raw_usage":{"total_tokens":2003,"prompt_tokens":988,"completion_tokens":1015,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":604,"completion_tokens_details":{"reasoning_tokens":943}},"tokens_in":604,"tokens_out":1015,"duration_ms":10275,"temperature":1.0,"reasoning_tokens":943,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T18:39:11.153058+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the stochastic heat equation example of Section 6 with $p=1.5$, $\\alpha=1.7$, $\\theta=0.5$, compute the empirical $L^p$ error between $X^{\\varepsilon}$ and the averaged solution for $\\varepsilon = 10^{-1}, 10^{-2}, 10^{-3}$, and plot the error against $\\varepsilon$ on a log-log scale: the slope should equal $\\theta(p-1)/(\\theta(p-1)+2)$. A slope clearly below that value, or a non-vanishing error, would falsify Theorem 2.3.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the prior strong averaging principle for two-time-scale SPDEs driven by $\\alpha$-stable noise and the open problem (Remark 3.3) about removing the uniform boundedness of the nonlinearity, which this paper solves."},{"cited_title":"Cerrai, A","cited_arxiv_id":null,"evidence_quote":"The previous nonautonomous slow-fast SPDE averaging result, but for Brownian noise; this paper extends it to $\\alpha$-stable driving noise."},{"cited_title":"Da Prato, M","cited_arxiv_id":null,"evidence_quote":"Supplies the definition and basic properties of evolution systems of measures for time-dependent semigroups, which replace the invariant measure in the nonautonomous setting."},{"cited_title":"Khasminskii, On the principle of averaging the Itô stochastic differential equations.Kibernetica4 (1968), 260-279","cited_arxiv_id":null,"evidence_quote":"The Khasminskii time-discretization idea of freezing the slow component on small intervals, which structures the proof of the main convergence estimates."},{"cited_title":"Priola, J","cited_arxiv_id":null,"evidence_quote":"Gives well-posedness and structural estimates for semilinear SPDEs driven by cylindrical stable processes, used to justify existence and a priori bounds for the fast equation."},{"cited_title":"Priola, A","cited_arxiv_id":null,"evidence_quote":"Provides the $L^p$ moment bounds for stochastic convolutions of stable processes used in the a priori estimates and in Assumption A2."}],"review_version":1}