arXiv:2608.06011v1

Strong averaging principle for multiscale time-inhomogeneous SDEs with multiplicative $α$ -stable noises

Jiaquan Lu, Huaizhong Zhao

math.PR60H1037A5060G5234C29

Paper metadata

Original submission: August 6, 2026
Claimed MSC: 60H10, 37A50, 60G52, 34C29
arXiv categories: math.PR
Paper license: arXiv non-exclusive distribution license ↗

Abstract

In this paper, we study the strong averaging principle for multiscale time-inhomogeneous stochastic systems driven by multiplicative $α$ -stable processes with $α\in(1,2)$ . Based on Khasminskii's discretization approach, we first establish that the fast component processes with a frozen slow variable admits a periodic measure. We then prove the strong convergence of the slow subsystem to an averaged system that depends on the time scale $\varepsilon$ . For any fixed $\varepsilon$ , if the reciprocals of the two periods $τ_1$ and $\varepsilon τ_2$ are rationally linearly independent, an important consequence is that the averaged system has random quasi-periodicity. Furthermore, by applying the ergodic theorem, we prove the strong convergence of the slow subsystem to another averaged system, a time-inhomogeneous SDEs independent of the time scale $\varepsilon$ . Our result is also novel even in the time-homogeneous case for a fully coupled multiscale system with multiplicative $α$ -stable noises. Finally, we apply the result to a climate-weather system.

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Audit summary

Audited against arXiv v1

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

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Detailed mathematical audit

Generated August 18, 2026

01Statements3 reported findingsContains unsupported statements

The two strong averaging theorems and the all-moment clause of the periodic-measure theorem are not verified for every $p\in(1,\alpha)$ under the printed assumptions. Assumption (A4) gives the required dissipation inequality for only one unspecified exponent, while the proof needs it at the exponent selected in each conclusion. The results are verified for each $p$ that separately satisfies $\lambda>q(p)$ .

Theorem 2.2Not able to verify

Strong averaging to the time-dependent averaged equation

Pages 6 and 21–25 · Assumption (A4), Theorem 2.2, and its proof · arXiv:2608.06011v1

The theorem claims the rate $\sup_{t\in[s,T]}\mathbb E|X_t^\varepsilon-\overline X_t^\varepsilon|^p\leq C\varepsilon^{(p-1)/(\alpha+p-1)}$ for every $p\in(1,\alpha)$ . Assumption (A4), however, requires $\lambda>q(p)$ only for some $p\in(1,\alpha)$ , where $q(p)$ contains $(\alpha-p)^{-1}$ and therefore tends to infinity as $p\uparrow\alpha$ . A finite $\lambda$ can satisfy (A4) at one exponent while failing $\lambda>q(p)$ at another. Lemmas 3.5, 3.7, 3.8, and 3.10 and the block estimate (3.29) use positive exponential contraction $\lambda-q(p)>0$ at the same exponent as the theorem. Hence the printed assumptions do not establish the full all- $p$ conclusion. The statement is verified after restricting it to each $p$ for which $\lambda>q(p)$ ; no counterexample to the broader conclusion is supplied.

Full paper, version 1 ↗

Theorem 2.5Not able to verify

Strong averaging to the epsilon-free averaged equation

Pages 7 and 25–28 · Theorem 2.5 and its proof · arXiv:2608.06011v1

This theorem makes the same assertion for every $p\in(1,\alpha)$ with rate $\varepsilon^{(p-1)/(\alpha+p-1)}$ . Its proof uses the $p$ -moment estimates and the averaged coefficients derived from Theorem 3.9, which in turn depend on contraction at that same $p$ . Assumption (A4) supplies this only at an existentially chosen exponent. The Khasminskii argument does establish the displayed conclusion for each $p$ satisfying $\lambda>q(p)$ , but the paper does not provide a separate argument covering exponents for which that inequality fails.

Full paper, version 1 ↗

Theorem 3.9Not able to verify

Periodic measure and its moment bounds

Pages 17–20 · Theorem 3.9 · arXiv:2608.06011v1

The pullback contraction at an exponent admitted by (A4), followed by Jensen's inequality, correctly gives the stated $W_1$ convergence, existence, uniqueness, periodicity, and continuity. The additional claim that $\int|w|^p\rho_t^x(dw)\leq C_{\kappa,p}(1+|x|^p)$ for every $p\in[1,\alpha)$ is not established under the existential exponent in (A4): Lemma 3.7 derives its uniform $p$ -moment bound using $\lambda>q(p)$ . Thus the periodic-measure conclusion is verified, while its full all- $p$ moment clause is not.

02Proofs5 reported findingsContains incorrect or incomplete proofs

The main proof chain is correct for each exponent satisfying $\lambda>q(p)$ , but the paper applies that contraction to every $p\in(1,\alpha)$ although (A4) assumes it for only one exponent. The periodic-measure proof also equates unbounded moments after only $W_1$ convergence; lower semicontinuity gives a verified repair. One domain mismatch and one omitted constant are harmless local corrections.

Use of Assumption (A4)Incorrect as written

A contraction hypothesis at one exponent is applied at every exponent

Pages 6, 13–17, and 21–28 · (A4), Lemmas 3.5, 3.7–3.10, and proofs of Theorems 2.2 and 2.5 · arXiv:2608.06011v1

Writing the dependence explicitly, $q(p)=2^{\alpha-1}C_{\alpha,d_2}S_{d_2}\left((2-\alpha)^{-1}+(\alpha-1)^{-1}+(\alpha-p)^{-1}\right)C_g^\alpha.$ Assumption (A4) states $\lambda>q(p)$ for some $p\in(1,\alpha)$ , whereas the cited lemmas and both main proofs substitute an arbitrary selected $p$ and use $e^{-(\lambda-q(p))(t-s)}$ as a decaying factor. Because $q(p)\to\infty$ as $p\uparrow\alpha$ , the required sign does not follow for all selected exponents. Downstream dependency: the periodic-measure $p$ -moment estimates, the averaged-drift estimates, (3.29), and both main rates. Repair classification: Verified repair. State (A4) for the exponent used in a conclusion and replace 'for any $p\in(1,\alpha)$ ' by 'for every $p\in(1,\alpha)$ such that $\lambda>q(p)$ .' Every contraction and balancing estimate in the paper then has the needed sign and yields the stated rate.

Proof of Theorem 3.9, Step 4Incorrect as written

First-Wasserstein convergence does not imply convergence of higher moments

Page 20 · first display in Step 4 of Theorem 3.9 · arXiv:2608.06011v1

The proof writes $\int_{\mathbb R^m}|w|^p\rho_t^x(dw)=\lim_{n\to\infty}\int_{\mathbb R^m}|w|^pP^x(s-n\tau_2,y,t,dw)$ after establishing only $W_1$ convergence. For $p>1$ , the integrand $|w|^p$ is unbounded and this equality does not follow from $W_1$ convergence plus a uniform $p$ -moment bound. Repair classification: Verified repair. Weak convergence and the nonnegative lower-semicontinuous function $|w|^p$ give the Portmanteau inequality with $\liminf$ in the needed direction; combining it with Lemma 3.7 yields the displayed upper bound (3.13). No equality of moments is required.

Full paper, version 1 ↗

Restricted averaging proof chainCorrect and complete

Khasminskii discretization and rate optimization

Pages 21–28 · Sections 3.4–3.5 · arXiv:2608.06011v1

For a fixed exponent with $\lambda>q(p)$ , the auxiliary frozen process has the required moment and contraction estimates. The proof controls the discretization, frozen-fast-variable, and averaging-block terms by $\Delta^{(p-1)/\alpha}$ and $\varepsilon\Delta^{-1}$ , then chooses $\Delta=\varepsilon^{\alpha/(\alpha+p-1)}$ . The analogous periodic averaging estimate for the second system has the same balance. After the exponent restriction stated above, these steps yield both displayed strong rates.

Lemma 3.10Minor formal correction

The fast process requires the terminal time to follow the initial time

Pages 14 and 20 · definition of $Y_t^{s,y,x}$ and statement of Lemma 3.10 · arXiv:2608.06011v1

The lemma quantifies over arbitrary $s,t_1,t_2\in\mathbb R$ , but $Y_{t_2}^{s,y,x}$ and the two-parameter transition operator were defined only for $t_2\geq s$ . Add the restriction $s\leq t_2$ . Every application in the averaging proofs satisfies this order, and the proof then applies without change.

Equation (3.34)Typo

The time-Holder estimate omits its coefficient

Page 25 · Equation (3.34) · arXiv:2608.06011v1

Printed: $|\overline b(t,x)-\overline b(s,x)|\leq|t-s|^{1/\alpha}$ . Assumption (A1) has an arbitrary Holder constant, so the mechanically determined correction is $|\overline b(t,x)-\overline b(s,x)|\leq C_\kappa|t-s|^{1/\alpha}$ . All later estimates already absorb such coefficients into generic constants, so no argument or rate changes.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2608.06011v1
Authors listed: Jiaquan Lu, Huaizhong Zhao
Audit date: August 18, 2026

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Strong averaging principle for multiscale time-inhomogeneous SDEs with multiplicative ααα-stable noises

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Strong averaging to the time-dependent averaged equation

Strong averaging to the epsilon-free averaged equation

Periodic measure and its moment bounds

A contraction hypothesis at one exponent is applied at every exponent

First-Wasserstein convergence does not imply convergence of higher moments

Khasminskii discretization and rate optimization

The fast process requires the terminal time to follow the initial time

The time-Holder estimate omits its coefficient

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Strong averaging principle for multiscale time-inhomogeneous SDEs with multiplicative $α$ -stable noises