{"id":"ccb2b033-499e-4ff8-a9f7-8b511f464b39","arxiv_id":"2608.06011","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Multiscale SDEs with multiplicative alpha-stable noise: the slow component converges in L^p to an averaged equation at rate epsilon^((p-1)/(alpha+p-1)).","lead":"Slow-fast systems with heavy-tailed jumps are shown to obey a strong averaging principle: the slow component stays close, in L^p, to a system whose fast motion is averaged out. The result is new for multiplicative alpha-stable noises and produces quasi-periodic averaged systems when the two timescales have incompatible periods.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Assumption (A4) fixes λ>q(p) for a single p, while q(p) contains (α−p)^{-1} and blows up as p↑α; Theorems 2.2 and 2.5 claim 'any p∈(1,α)', so the exponential ergodicity used throughout is not available for all stated p.","rationale":"The paper's proof is a Khasminskii-discretization argument that is internally coherent for any p for which the dissipativity constant λ actually exceeds the technical constant q(p). The reader's weakest assumption identifies exactly the load-bearing problem: a single λ fixed in Assumption (A4) cannot support the theorems' universal statement 'any p∈(1,α)' because q(p)→∞ as p↑α, and every exponential ergodicity and averaging estimate in the proof depends on λ−q being strictly positive. I considered whether other issues—unproved supporting lemmas, the Section 4 coefficient inconsistencies (660 vs 600, 3796 vs 3739, 2.858 vs 2.853), or the asserted extension to general Hölder exponents in Remark 2.6—are more central. They are not: the unproved lemmas are standard or analogous to proved ones, and the Section 4 inconsistencies are numerical slips that do not affect the abstract theorem. The λ>q(p) issue directly undermines the claimed range of p in the main theorems, but it is correctable by quantifying p honestly, so conditional acceptance remains the right posture.","tokens_in":32844,"tokens_out":6002,"duration_ms":59815,"concrete_test":"Analytical check with α=1.5, C_g=1, and C_{α,d2}S_{d2}>0: choose p0=1.1 and set λ=q(p0)+1, so Assumption (A4) holds for p0. Now re-derive Lemma 3.8 for p=1.49 using this fixed λ. Since (α−p)^{-1} jumps from 2.5 at p=1.1 to about 100 at p=1.49, q(p) exceeds λ, making p(λ−q(p)) negative. The comparison theorem then gives an exponentially growing upper bound instead of the contraction in (3.11), so Theorem 3.9(3.12) and the I31 estimate in (3.29) fail for this p. This settles the universal quantification: either the theorem must be restricted to p with λ>q(p), or the assumptions must be made p-dependent.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Assumption (A4), Eq. (2.4), fixes one constant λ with λ>q, where q=2^{α−1}C_{α,d2}S_{d2}((2−α)^{-1}+(α−1)^{-1}+(α−p)^{-1})C_g^α 'for some 1<p<α'. Because (α−p)^{-1}→∞ as p↑α, no fixed finite λ can dominate q(p) for every p∈(1,α). The proofs of Theorem 3.9 (see (3.12)), Lemma 3.10, Lemma 3.8 (see (3.11)), and the Khasminskii estimate I31 (see (3.29)) all use factors of the form e^{-(λ−q)(t−s)} or e^{-(λ−q)(u−s−k∆)/ε}. For any p with q(p)≥λ these factors no longer decay; the comparison theorem then gives non-decaying or growing bounds and the claimed rate ε^{(p−1)/(α+p−1)} is not obtained. Thus the theorems' quantification 'any p∈(1,α)' is stronger than what the assumptions actually support. This is a statement mismatch, not an error in the estimate chain for one fixed admissible p: the fix is to either fix p in (A4) and state the theorems for that p, or let λ and all constants depend on p and require λ>q(p) for every p appearing in the claimed range.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a multiscale time-inhomogeneous SDE system (1.1) whose fast component is scaled by 1/ε and whose noise is multiplicative and rotationally invariant α-stable. Under Lipschitz/linear-growth assumptions and a dissipativity condition on f, the authors construct a τ2-periodic measure for the frozen fast equation via Wasserstein contraction and exponential convergence (Theorem 3.9). They then prove L^p strong averaging principles: Theorem 2.2 compares the slow component X^ε with the ε-dependent averaged equation (2.7) at rate ε^{(p-1)/(α+p-1)}, and Theorem 2.5 gives the same rate against the ε-independent averaged equation (2.10). A climate-weather example is presented in Section 4.","tokens_in":33072,"tokens_out":26026,"duration_ms":233344,"significance":"The paper addresses a genuinely open direction: strong averaging for multiscale SDEs with heavy-tailed multiplicative noise and fully coupled drifts. The Khasminskii discretization is implemented carefully, with an explicit error decomposition into freezing, averaging, and ergodicity terms, and the periodic-measure construction for the time-inhomogeneous fast equation is a useful contribution. If the quantification of p is corrected, the result would be a solid advance. As stated, however, the advertised scope 'any p∈(1,α)' is not supported by Assumption (A4); this is a correctness issue in the theorem statements, while the underlying argument appears sound for a fixed admissible p.","major_comments":[{"comment":"Assumption (A4), Eq. (2.4), fixes a single constant λ with λ>q(p0), where q(p0)=2^{α−1}C_{α,d2}S_{d2}((2−α)^{-1}+(α−1)^{-1}+(α−p0)^{-1})C_g^α for a particular p0∈(1,α). Theorems 2.2 and 2.5 then assert the estimates (2.6) and (2.9) for every p∈(1,α). This quantification is not implied by the assumptions: because q(p) contains (α−p)^{-1}, q(p)→∞ as p↑α, so for any finite λ there exist p<α with q(p)≥λ. The proofs of Lemma 3.8 (Eq. (3.11)), Lemma 3.10, and the I31 estimate (Eq. (3.29)) rely on exponential factors of the form e^{−(λ−q)(t−s)} or e^{−(λ−q)(u−s−kΔ)/ε}; when q(p)≥λ these factors no longer decay, and the comparison steps give non-decaying or even growing bounds. The same issue appears in the statements 'for all p∈[1,α)' in Lemmas 3.5, 3.7 and 3.8, whose proofs compare the q' built with the lemma's p against the q of Assumption (A4); that comparison is valid only when the p's agree. The main results should be restated for the p fixed in (A4), or for all p∈(1,α) with q(p)<λ, with constants allowed to degenerate as q(p)↑λ; Section 4's 'for any p∈(1,α)' inherits the same problem.","section":"§2, Assumption (A4); Theorems 2.2 and 2.5"}],"minor_comments":[{"comment":"The auxiliary process is defined with Δ=τ2/N for some integer N, but the proof of Theorem 2.2 later sets Δ=ε^{α/(α+p−1)}, which is not generally of the form τ2/N; choose N=N(ε) so that Δ_N=τ2/N has the same order, or remove the divisibility requirement.","section":"§3.2"},{"comment":"The coefficient 3796 appearing in (4.1) is written as 3739 in the displayed definition of f, and the phase 2.858 becomes 2.853 in the frozen equation; please reconcile these numbers.","section":"§4"},{"comment":"The final line writes 'sup_{s∈[t,T]}' where the supremum should be over t∈[s,T]; as printed, the inequality is not well-formed.","section":"Lemma 3.2 proof"},{"comment":"The term 'random quasi-periodicity' is used for the averaged system, but the argument only shows that the averaged drift \\bar B^ε(t,x) is quasi-periodic in t; the solution-level random quasi-periodic property is not defined or proved, so please clarify the terminology.","section":"Remark 2.4 and abstract"},{"comment":"The letter p is used both for the exponent fixed in the definition of q and for the arbitrary exponent in the theorems; renaming the former p0 would avoid ambiguity.","section":"Assumption (A4)"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main thing you should know: this paper proves a strong L^p averaging principle for multiscale time-inhomogeneous SDEs with multiplicative α-stable noise, extending the additive-noise results of Sun-Xie-Xie and Li-Sun-Wang-Xie to the case where the slow drift depends on the fast process and the fast drift depends on the slow process. The proof uses Khasminskii's discretization and constructs periodic measures for the frozen fast equation; the estimates are detailed and mostly coherent. If the p-range issue is fixed, this is a useful and honest contribution.\n\nWhat is new is genuine: prior strong averaging results for α-stable-driven systems handle additive noise [38, 29], and the closest multiplicative-noise paper [12] assumes the fast process is not an SDE and is independent of the slow variable. The quasi-periodicity remark (Remark 2.4) is a nice byproduct, and the second averaged equation independent of ε via the ergodic theorem is a reasonable addition.\n\nThe soft spot is load-bearing but correctable. Assumption (A4) fixes a single λ > q(p) for one p, where q(p) contains (α-p)^{-1} and blows up as p↑α. Theorems 2.2 and 2.5 claim the rate for 'any p ∈ (1,α)', but the exponential decay e^{-(λ-q)(t-s)} used in Lemmas 3.7–3.10 and in the key estimate I31 (3.29) requires λ > q(p) for the p in question. As written, the theorems are not implied by the assumptions. The fix is straightforward: either fix p in (A4) and state the theorems for that p (or for all p below the implied threshold), or let λ and the constants depend on p with λ > q(p) for every p in the claimed range.\n\nThere are also minor issues: Section 4's climate example has inconsistent coefficients (660 vs 600, 3739 vs 3796, phase 2.858 vs 2.853), and Lemmas 3.4, 3.12, and 3.13 are asserted without proof; the first is analogous to Lemma 3.2 and the others are standard, so these are minor. The claim of 'fully coupled' is slightly overstated because the fast diffusion g does not depend on the slow variable, but the drift coupling is what matters and the comparison with prior work is accurate.\n\nOverall: this deserves a serious referee. The core proof is coherent for a fixed admissible p, the novelty is real, and the main flaw is a statement mismatch, not a broken estimate chain. I would engage with it after the authors tighten the p-quantification.","headline":"A solid extension of strong averaging to multiplicative α-stable noise, but the 'any p∈(1,α)' claim outruns Assumption (A4); fix the p-quantification and it's a good paper.","tokens_in":33718,"tokens_out":4942,"would_cite":true,"duration_ms":42925,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H10","37A50","60G52","34C29"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves a strong averaging principle for multiscale time-inhomogeneous SDEs driven by multiplicative $\\alpha$-stable noise, with $L^p$ convergence to the averaged system at rate $\\varepsilon^{(p-1)/(\\alpha+p-1)}$.","keywords":["strong averaging principle","multiscale SDEs","α-stable Lévy noise","multiplicative noise","periodic measure","Khasminskii discretization","time-inhomogeneous SDEs","quasi-periodicity"],"falsifier":"Take the frozen fast equation with $f(t,x,y)=-\\lambda y$, $g(t,y)=\\mathrm{Id}$, and a rotationally invariant $\\alpha$-stable driver, then compute $q(p)$ from Assumption (A4); for a fixed $\\lambda$, choose $p<\\alpha$ with $q(p)>\\lambda$. If that system still satisfies the theorem's hypotheses, the inequality in Theorem 2.2 cannot hold with a finite constant, so the theorem's 'any $p$' claim would be refuted. A direct check of whether $\\lambda>q(p)$ holds for all $p\\in(1,\\alpha)$ settles the issue.","tokens_in":32528,"feed_emoji":"⚡","tokens_out":12312,"duration_ms":100384,"temperature":0.7,"pith_summary":"The paper establishes a strong averaging principle for multiscale stochastic differential equations whose slow and fast components are driven by multiplicative $\\alpha$-stable L\\'evy noise with stability index $\\alpha\\in(1,2)$. The central result is an $L^p$ error bound: the slow process $X^\\varepsilon$ converges to an averaged process at rate $\\varepsilon^{(p-1)/(\\alpha+p-1)}$ on finite time intervals, uniformly in $\\varepsilon$, for any $p\\in(1,\\alpha)$. The averaged system is built from a periodic measure of the frozen fast equation, exploiting time-periodicity of the coefficients. A second, $\\varepsilon$-independent averaged system is obtained by further averaging over the fast period, and the paper shows the same convergence rate holds for it. This is the first strong averaging result for fully coupled multiscale systems with multiplicative $\\alpha$-stable noise, including in the time-homogeneous case.","feed_headline":"Averaging principle proved for multiscale SDEs with α-stable noise","feed_subtitle":"Slow component converges in p-th mean at rate ε^((p-1)/(α+p-1)), even with multiplicative heavy tails.","key_machinery":"The carrying mechanism is Khasminskii's time-discretization: an auxiliary fast process $\\hat Y^\\varepsilon$ with the slow variable frozen on each subinterval. The frozen time-inhomogeneous fast SDE admits a unique $\\tau_2$-periodic measure $\\rho^x_t$ with exponential ergodicity in the 1-Wasserstein distance at rate $e^{-(\\lambda-q)t}$, and a second-order expansion estimate (Lemma 2.7) controls the heavy-tailed jump terms. Balancing the two error terms in the splitting gives the rate $\\varepsilon^{(p-1)/(\\alpha+p-1)}$ and the optimal choice $\\Delta=\\varepsilon^{\\alpha/(\\alpha+p-1)}$.","core_discovery":"Under Assumptions (A) and (B), the paper proves Theorems 2.2 and 2.5: for any $p\\in(1,\\alpha)$, $\\sup_{t\\in[s,T]}\\mathbb{E}|X^\\varepsilon_t-\\bar X^\\varepsilon_t|^p \\le C_{\\kappa,p,s,T}(1+|x|^p+|y|^p)\\varepsilon^{(p-1)/(\\alpha+p-1)}$, where $\\bar X^\\varepsilon$ solves the averaged equation whose drift is $b$ averaged against the unique periodic measure $\\rho^x_t$ of the frozen fast equation, and the same bound holds with $\\bar X$, the further-period-averaged system independent of $\\varepsilon$. The proof uses Khasminskii's discretization: freeze the slow variable on intervals of length $\\Delta$, estimate the error by the fast dynamics' exponential relaxation with rate $\\lambda-q$, and optimize $\\Delta$ to balance the two error terms. A corollary is that when the two periods are rationally linearly independent, the $\\varepsilon$-dependent averaged system is random quasi-periodic.","pith_inferences":["The constant $q$ in Assumption (A4) depends on $p$ through $(\\alpha-p)^{-1}$, so the theorem's claim of 'any $p\\in(1,\\alpha)$' implicitly requires either $\\lambda$ to grow with $p$ or the admissible range of $p$ to shrink; this is a gap between the stated assumption and the stated theorem.","The rate exponent $(p-1)/(\\alpha+p-1)$ tends to $0$ as $p\\uparrow\\alpha$, so strong convergence is slowest for moments close to the stability index, consistent with heavier tails giving weaker integrability.","One could test the quasi-periodicity claim numerically by simulating the Section 4 averaged equation and checking whether sample paths are quasi-periodic with periods $1$ and $\\varepsilon$ in the sense of Definition 2.3.","Remark 2.6(ii) indicates the same discretization extends to $\\gamma$-Hölder drift in time with rate $\\varepsilon^{\\beta/(1+\\beta)}$, $\\beta=\\min\\{(p-1)/\\alpha,\\gamma\\}$; a natural next step is to push the method to SPDEs with multiplicative $\\alpha$-stable noise."],"forward_implications":["The slow component of systems like (1.1) can be replaced by the one-dimensional averaged SDE, with a computable $L^p$ error of order $\\varepsilon^{(p-1)/(\\alpha+p-1)}$ on finite time intervals.","When the slow and fast periods are rationally independent, the averaged system is random quasi-periodic, so the long-time behaviour is captured by a two-parameter periodic drift.","The result applies to the climate-weather model in Section 4, where $\\varepsilon\\approx 10^{-5}$, yielding a quasi-periodic averaged climate system.","Because multiplicative noise is allowed, the diffusion coefficient of the slow equation need not be independent of the fast component, removing a restriction that appears in earlier averaging results.","The same argument yields a new strong averaging theorem in the time-homogeneous fully coupled case, which the paper notes was previously open for multiplicative $\\alpha$-stable noise."],"supporting_citations":[{"why":"Supplies the time-discretization method on which the whole proof is built.","marker":"[26]"},{"why":"Provides the periodic-measure ergodic theory and law of large numbers used to define the averaged drift.","marker":"[21]"},{"why":"Recent strong averaging result for time-inhomogeneous SPDEs with α-stable noise that the present work extends from additive to multiplicative settings.","marker":"[29]"},{"why":"Establishes the additive-α-stable averaging rates that this paper generalizes to multiplicative noise.","marker":"[38]"},{"why":"Supplies the Lévy–Itô decomposition and Itô formula used in the a priori moment estimates.","marker":"[1]"},{"why":"Source of the random quasi-periodicity concept applied in Definition 2.3 and Remark 2.4.","marker":"[20]"},{"why":"Previous averaging result for multiplicative α-stable noise under a different fast-process structure, against which the paper positions its novelty.","marker":"[12]"}],"fun_headline_variants":["Strong averaging for multiscale SDEs with α-stable noise","Multiscale SDEs with α-stable noise: strong averaging","Averaging principle for multiscale SDEs with multiplicative α-stable noise","Time-inhomogeneous multiscale SDEs: strong averaging with α-stable noise","α-stable noise: strong averaging for multiscale SDEs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the fast drift is strongly dissipative with a rate $\\lambda$ larger than a constant $q$, and $q$ grows without bound as the moment order $p$ approaches the stability index $\\alpha$; the theorems state the convergence for every $p\\in(1,\\alpha)$ under a single $\\lambda$, which requires the assumption and the theorem's quantifiers to be reconciled.","fun_headline_variants_meta":{"raw":{"variants":["Strong averaging for multiscale SDEs with α-stable noise","Multiscale SDEs with α-stable noise: strong averaging","Averaging principle for multiscale SDEs with multiplicative α-stable noise","Time-inhomogeneous multiscale SDEs: strong averaging with α-stable noise","α-stable noise: strong averaging for multiscale SDEs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000981,"raw_usage":{"total_tokens":4180,"prompt_tokens":975,"completion_tokens":3205,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":591,"completion_tokens_details":{"reasoning_tokens":3112}},"tokens_in":591,"tokens_out":3205,"duration_ms":20469,"temperature":1.0,"reasoning_tokens":3112,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T18:21:57.972924+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the frozen fast equation with $f(t,x,y)=-\\lambda y$, $g(t,y)=\\mathrm{Id}$, and a rotationally invariant $\\alpha$-stable driver, then compute $q(p)$ from Assumption (A4); for a fixed $\\lambda$, choose $p<\\alpha$ with $q(p)>\\lambda$. If that system still satisfies the theorem's hypotheses, the inequality in Theorem 2.2 cannot hold with a finite constant, so the theorem's 'any $p$' claim would be refuted. A direct check of whether $\\lambda>q(p)$ holds for all $p\\in(1,\\alpha)$ settles the issue.","supporting_citations":[{"cited_title":"Applebaum.L´ evy processes and stochastic calculus","cited_arxiv_id":null,"evidence_quote":"Supplies the Lévy–Itô decomposition and Itô formula used in the a priori moment estimates."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Source of the random quasi-periodicity concept applied in Definition 2.3 and Remark 2.4."},{"cited_title":"Khasminskii","cited_arxiv_id":null,"evidence_quote":"Supplies the time-discretization method on which the whole proof is built."},{"cited_title":"Feng and H","cited_arxiv_id":null,"evidence_quote":"Provides the periodic-measure ergodic theory and law of large numbers used to define the averaged drift."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the additive-α-stable averaging rates that this paper generalizes to multiplicative noise."},{"cited_title":"Cheng, Z","cited_arxiv_id":null,"evidence_quote":"Previous averaging result for multiplicative α-stable noise under a different fast-process structure, against which the paper positions its novelty."}],"review_version":1}