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arxiv: 1907.09256 · v1 · pith:BIKFCOWHnew · submitted 2019-07-22 · 🧮 math.PR

Strong and weak convergence in the averaging principle for SDEs with H\"older coefficients

Michael R\"ockner , Xiaobin Sun , Longjie Xie This is my paper

Pith reviewed 2026-05-24 18:02 UTC · model grok-4.3

classification 🧮 math.PR
keywords averaging principlestochastic differential equationsHölder continuous coefficientsstrong convergenceweak convergenceZvonkin's transformPoisson equationmulti-scale systems
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The pith

The averaging principle holds for SDEs with time-dependent Hölder continuous coefficients at sharp strong and weak rates.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes the averaging principle for stochastic differential equations whose coefficients are only Hölder continuous in both time and space. It applies Zvonkin's transform to reduce the problem to a parameter-dependent Poisson equation whose solutions yield the effective averaged dynamics. The resulting rates are of order (α∧1)/2 for strong convergence and (α/2)∧1 for weak convergence, where α denotes the Hölder exponent. Convergence depends only on the regularity of the slow variable's coefficients and is unaffected by the fast variable's regularity. Readers care because many applied multi-scale models have coefficients too rough for classical Lipschitz theory yet still admit reliable averaging approximations under these weaker assumptions.

Core claim

Using Zvonkin's transform and the Poisson equation in R^d with a parameter, we prove the averaging principle for stochastic differential equations with time-dependent Hölder continuous coefficients. Sharp convergence rates with order (α∧1)/2 in the strong sense and (α/2)∧1 in the weak sense are obtained, considerably extending the existing results in the literature. Moreover, we prove that the convergence of the multi-scale system to the effective equation depends only on the regularity of the coefficients of the equation for the slow variable, and does not depend on the regularity of the coefficients of the equation for the fast component.

What carries the argument

Zvonkin's transform combined with solutions to the parameter-dependent Poisson equation in R^d

If this is right

  • Strong convergence of the slow process to the averaged limit holds at rate (α∧1)/2.
  • Weak convergence holds at rate (α/2)∧1.
  • The rates and convergence are independent of the regularity of the fast-component coefficients.
  • The result applies directly to time-dependent coefficients without additional smoothing assumptions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Numerical schemes for multi-scale SDEs can safely ignore high regularity on fast variables and focus computational effort on the slow equation.
  • The same transform-plus-Poisson strategy may extend to other singularly perturbed SDE systems that lack Lipschitz coefficients.
  • Applications in finance or fluid models with rough coefficients can now invoke averaging with explicit error bounds rather than formal limits.

Load-bearing premise

The Poisson equation in R^d with a parameter admits sufficiently regular solutions under the stated Hölder assumptions on the coefficients.

What would settle it

A concrete counterexample SDE with Hölder coefficients where strong convergence to the averaged equation occurs at a rate strictly slower than (α∧1)/2 would falsify the claim.

read the original abstract

Using Zvonkin's transform and the Poisson equation in $R^d$ with a parameter, we prove the averaging principle for stochastic differential equations with time-dependent H\"older continuous coefficients. Sharp convergence rates with order $(\alpha\wedge1)/2$ in the strong sense and $(\alpha/2)\wedge1$ in the weak sense are obtained, considerably extending the existing results in the literature. Moreover, we prove that the convergence of the multi-scale system to the effective equation depends only on the regularity of the coefficients of the equation for the slow variable, and does not depend on the regularity of the coefficients of the equation for the fast component.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript proves the averaging principle for multi-scale SDEs with time-dependent Hölder-α coefficients by combining Zvonkin's transform with estimates derived from solutions of a parameterized Poisson equation in R^d. It obtains sharp strong convergence rates of order (α ∧ 1)/2 and weak rates of (α/2) ∧ 1, and asserts that these rates and the convergence itself depend only on the Hölder regularity of the slow-component coefficients, independent of the fast-component regularity.

Significance. If the uniform-in-parameter regularity of the Poisson solutions can be established under the stated assumptions, the result would extend the literature on stochastic averaging to the time-dependent Hölder setting while delivering sharp rates and a decoupling of slow/fast regularities; this is a technically substantive contribution to the analysis of multi-scale diffusions with limited smoothness.

major comments (2)
  1. [Proof of main theorem (via Zvonkin's transform and Poisson equation)] The central argument invokes Zvonkin's transform to reduce the averaging problem to remainder estimates controlled by derivatives of solutions to the parameterized Poisson equation -L^y u(x,y) = f(x,y). The manuscript does not supply (or cite) the uniform-in-x C^{2,α} estimates and Hölder moduli for these solutions when the coefficients are time-dependent and only Hölder-α; this uniformity is load-bearing for both the claimed rates and the independence from fast-component regularity.
  2. [Section on Poisson equation regularity with parameter] The asserted sharp rates ((α ∧ 1)/2 strong, (α/2) ∧ 1 weak) and the claim that convergence depends only on slow-variable regularity both presuppose that the Poisson solutions furnish bounds sufficient to absorb the time-dependent terms after the transform. No explicit verification or reference is given showing that the Hölder-α assumption on the fast generator yields the required uniform control when the slow coefficients are also time-dependent.
minor comments (1)
  1. [Notation and setup] The precise statement of the Poisson equation (including the form of L^y and the right-hand side f) should be written explicitly with all coefficient assumptions listed, to make the regularity invocation self-contained.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need for greater explicitness in the Poisson-equation estimates that underpin the main argument. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Proof of main theorem (via Zvonkin's transform and Poisson equation)] The central argument invokes Zvonkin's transform to reduce the averaging problem to remainder estimates controlled by derivatives of solutions to the parameterized Poisson equation -L^y u(x,y) = f(x,y). The manuscript does not supply (or cite) the uniform-in-x C^{2,α} estimates and Hölder moduli for these solutions when the coefficients are time-dependent and only Hölder-α; this uniformity is load-bearing for both the claimed rates and the independence from fast-component regularity.

    Authors: We agree that the uniform-in-x C^{2,α} estimates and Hölder moduli for the solutions of the parameterized Poisson equation are essential and were not stated with sufficient detail or citation. In the revision we will add an explicit subsection (or appendix) deriving these bounds from standard parabolic Schauder theory for time-dependent Hölder-α coefficients, verifying uniformity in the slow-variable parameter x and confirming that the estimates depend only on the Hölder regularity of the slow coefficients. This will also make transparent why the fast-component regularity beyond α is not required. revision: yes

  2. Referee: [Section on Poisson equation regularity with parameter] The asserted sharp rates ((α ∧ 1)/2 strong, (α/2) ∧ 1 weak) and the claim that convergence depends only on slow-variable regularity both presuppose that the Poisson solutions furnish bounds sufficient to absorb the time-dependent terms after the transform. No explicit verification or reference is given showing that the Hölder-α assumption on the fast generator yields the required uniform control when the slow coefficients are also time-dependent.

    Authors: We concur that the sharp rates and the claimed decoupling rest on these uniform bounds being sufficient to control the time-dependent remainder terms. The revision will supply the missing verification, showing how the Hölder-α assumption on the fast generator produces the necessary C^{2,α} control that absorbs the time-dependent slow coefficients after Zvonkin's transform, together with appropriate references to the underlying Schauder estimates. revision: yes

Circularity Check

0 steps flagged

No circularity: direct analytic proof via Zvonkin transform and Poisson estimates

full rationale

The paper derives strong and weak convergence rates for the averaging principle under time-dependent Hölder coefficients by applying Zvonkin's transform to reduce the problem to estimates on solutions of a parameterized Poisson equation. This is a standard analytic strategy with no data fitting, no self-definitional loops (e.g., no quantity defined in terms of the claimed rate), and no load-bearing self-citations that substitute for independent verification. The claimed rates follow from uniform ellipticity and Hölder regularity assumptions on the coefficients, which are external to the derivation itself. The result is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof rests on existence of solutions to a parameter-dependent Poisson equation and applicability of Zvonkin's transform under Hölder regularity; these are standard but domain-specific assumptions in stochastic PDE theory.

axioms (1)
  • domain assumption The Poisson equation in R^d with a parameter admits solutions with sufficient regularity when coefficients are Hölder continuous.
    Invoked as the key analytic tool alongside Zvonkin's transform.

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Works this paper leans on

43 extracted references · 43 canonical work pages · 2 internal anchors

  1. [1]

    and Rubin J

    Bertram R. and Rubin J. E.: Multi-timescale systems and fast-slow analysis. Math. Biosci. 287 (2017), 105–121

    work page 2017

  2. [2]

    I., Shaposhnikov S

    Bogachev V. I., Shaposhnikov S. V. and Veretennikov A. Yu.: Diffe rentiability of solutions of sta- tionary Fokker-Pllanck-Kolmogorov equations with respect to a pa rameter. Discrete and Continuous Dynamical Systems - Series B , 36 (2016), 3519–3543

    work page 2016

  3. [3]

    E.: Strong and weak orders in averaging for SPDEs

    Br´ ehier C. E.: Strong and weak orders in averaging for SPDEs. Stoch. Process. Appl. , 122 (2012), 2553–2593

    work page 2012

  4. [4]

    E.: Analysis of an HMM time-discretization scheme for a system of stochastic PDEs

    Br´ ehier C. E.: Analysis of an HMM time-discretization scheme for a system of stochastic PDEs. SIAM J. Numer. Anal. , 51 (2013), 1185–1210

    work page 2013

  5. [5]

    Orders of convergence in the averaging principle for SPDEs: the case of a stochastically forced slow component

    Br´ ehier C. E.: Orders of convergence in the averaging principle for SPDEs: the case of a stochastically forced slow component. https://arxiv.org/abs/1810.06448

    work page internal anchor Pith review Pith/arXiv arXiv

  6. [6]

    Cerrai S.: A Khasminskii type averaging principle for stochastic r eaction-diffusion equations. Ann. Appl. Probab. 19 (2009), 899–948

    work page 2009

  7. [7]

    and Freidli M.: Averaging principle for stochastic react ion-diffusion equations

    Cerrai S. and Freidli M.: Averaging principle for stochastic react ion-diffusion equations. Probab. Theory Related Fields , 144 (2009), 137–177

    work page 2009

  8. [8]

    and Zhang X.: Heat kernels for non-symmetric diffusions operators with jumps

    Chen Z., Hu E., Xie L. and Zhang X.: Heat kernels for non-symmetric diffusions operators with jumps. J. Diff. Equations , 263 (2017), 6576–6634

    work page 2017

  9. [9]

    and Zhai J.: Averaging principle for one dimen sional stochastic Burgers equation

    Dong Z., Sun X., Xiao H. and Zhai J.: Averaging principle for one dimen sional stochastic Burgers equation. J. Diff. Equations , 265 (2018), 4749–4797

    work page 2018

  10. [10]

    and Vanden-Eijnden E.: Analysis of multiscale methods f or stochastic differential equations

    E W., Liu D. and Vanden-Eijnden E.: Analysis of multiscale methods f or stochastic differential equations. Comm. Pure Appl. Math. , 58 (2005), 1544–1585

    work page 2005

  11. [11]

    Press, Prince- ton, N.J., 1985

    Friedlin M.: Functional integration and partial differential equat ions, Princeton Univ. Press, Prince- ton, N.J., 1985

    work page 1985

  12. [12]

    and Wentzell A.: Random perturbations of dynamical s ystems, Springer Science & Business Media, Berlin, Heidelberg, 2012

    Freidlin M. and Wentzell A.: Random perturbations of dynamical s ystems, Springer Science & Business Media, Berlin, Heidelberg, 2012

    work page 2012

  13. [13]

    and Duan J.: An averaging principle for two-scale stochast ic partial differential equations

    Fu H. and Duan J.: An averaging principle for two-scale stochast ic partial differential equations. Stoch. Dyn. , 11 (2011), 353–367. 28

    work page 2011

  14. [14]

    Multiscale Model

    Givon D.: Strong convergence rate for two-time-scale jump-d iffusion stochastic differential systems. Multiscale Model. Simul. 6 (2007), 577–594

    work page 2007

  15. [15]

    Givon D., Kevrekidis I. G. and Kupferman R.: Strong convergenc e of projective integeration schemes for singularly perturbed stochastic differential systems . Comm. Math. Sci. , 4 (2006), 707– 729

    work page 2006

  16. [16]

    and Sneyd J.: Multiple timesca les, mixed mode oscillations and canards in models of intracellular calcium dynamics

    Harvey E., Kirk V., Wechselberger M. and Sneyd J.: Multiple timesca les, mixed mode oscillations and canards in models of intracellular calcium dynamics. J. Nonlinear Sci. , 21 (2011), 639–683

    work page 2011

  17. [17]

    Z.: A limit theorem for the solutions of differential e quations with random right- hand sides

    Khasminskii R. Z.: A limit theorem for the solutions of differential e quations with random right- hand sides. Theory Probab. Appl. , 11 (1966), 390–406

    work page 1966

  18. [18]

    Z.: On an averging principle for Itˆ o stochastic diff erential equations

    Khasminskii R. Z.: On an averging principle for Itˆ o stochastic diff erential equations. Kibernetica, 4 (1968), 260–279

    work page 1968

  19. [19]

    Khasminskii R. Z. and Yin G.: On averaging principles: an asymptot ic expansion approach. SIAM J. Math. Anal. , 35 (2004), 1534–1560

    work page 2004

  20. [20]

    Z and Yin G.: Limit behavior of two-time-scale diffusio ns revised

    Khasminskii R. Z and Yin G.: Limit behavior of two-time-scale diffusio ns revised. J.Diff. Equ. , 212 (2005), 85–113

    work page 2005

  21. [21]

    Kifer Y.: Diffusion approximation for slow motion in fully coupled aver aging. Proba. Theor. Relat. Fields, 129 (2004) 157–181

    work page 2004

  22. [22]

    Discrete Contin

    Kifer Y.: Another proof of the averaging principle for fully couple d dynamical systems with hyper- bolic fast motions. Discrete Contin. Dyn. Syst. , 13 (2005), 1187–1201

    work page 2005

  23. [23]

    V.: Controlled diffusion processes

    Krylov N. V.: Controlled diffusion processes. Translated from the Russian by A.B. Aries. Applica- tions of Mathematics, 14. Springer-Verlag, New York-Berlin, 1980

    work page 1980

  24. [24]

    Krylov N. V. and Bogolyubov N.: Les proprietes ergodiques des s uites des probabilites en chaine. C. R. Acad. Sci. Paris , 204 (1937), 1454–1456

    work page 1937

  25. [25]

    Krylov N. V. and R¨ ockner M.: Strong solutions of stochastic equations with singular time dependent drift. Probab. Theory Related Fields , 131 (2005), 154–196

    work page 2005

  26. [26]

    Springer, Cham, 2015

    Kuehn C.: Multiple time scale dynamics, volume 191 of Applied Mathem atical Sciences. Springer, Cham, 2015

    work page 2015

  27. [27]

    and Ural’ceva N.N.: Linear and Quasi-linear Equations of Parabolic Type

    Lady˘ zenskaja O.A., Solonnikov V.A. and Ural’ceva N.N.: Linear and Quasi-linear Equations of Parabolic Type. Translated from Russian by S.Smmith. Amercian Math ematical Society, 1968

    work page 1968

  28. [28]

    M.: An averaging principle for a completely integrable stocha stic Hamiltonian system

    Li X. M.: An averaging principle for a completely integrable stocha stic Hamiltonian system. Non- linearity, 21 (2008), 803–822

    work page 2008

  29. [29]

    Liu D.: Strong convergence of principle of averaging for multisca le stochastic dynamical systems. Commun. Math. Sci. , 8 (2010), 999–1020

    work page 2010

  30. [30]

    and Veretennikov A

    Pardoux E. and Veretennikov A. Yu.: On the Poisson equation an d diffusion approximation. I. Ann. Prob., 29 (2001), 1061–1085

    work page 2001

  31. [31]

    and Veretennikov A

    Pardoux E. and Veretennikov A. Yu.: On the Poisson equation an d diffusion approximation 2. Ann. Prob., 31 (2003), 1166–1192

    work page 2003

  32. [32]

    Pavliotis G. A. and Stuart A. M.: Multiscale methods: averaging an d homogenization, volume 53 of Texts in Applied Mathematics. Springer, New York, 2008

    work page 2008

  33. [33]

    Commun Math Sci

    Vanden-Eijnden E: Numerical techniques for multi-scale dynam ical systems with stochastic effects. Commun Math Sci. , 1 (2003), 377–384

    work page 2003

  34. [34]

    Yu.: On the strong solutions of stochastic diffe rential equations

    Veretennikov A. Yu.: On the strong solutions of stochastic diffe rential equations. Theory Probab. Appl., 24 (1979), 354–366

    work page 1979

  35. [35]

    Yu.: On the averaging principle for systems of s tochastic differential equations

    Veretennikov A. Yu.: On the averaging principle for systems of s tochastic differential equations. Math. USSR Sborn. , 69 (1991), 271–284

    work page 1991

  36. [36]

    Yu.: On polynomial mixing bounds for stochastic differential equations

    Veretennikov A. Yu.: On polynomial mixing bounds for stochastic differential equations. Stoch. Processes Appl., 70 (1997), 115–127

    work page 1997

  37. [37]

    Yu.: On Sobolev solutions of poisson equations in Rd with a parameter

    Veretennikov A. Yu.: On Sobolev solutions of poisson equations in Rd with a parameter. J. Math. Sci., 179 (2011), 1–32

    work page 2011

  38. [38]

    and Roberts A

    Wang W. and Roberts A. J.: Average and deviation for slow-fast stochastic partial differential equations. J. Differential Equations , 253 (2012), 1265–1286. 29

    work page 2012

  39. [39]

    and Zhang X.: Sobolev differentiable flows of SDEs with local S obolev and super-linear growth coefficients

    Xie L. and Zhang X.: Sobolev differentiable flows of SDEs with local S obolev and super-linear growth coefficients. Ann. Prob., 44 (2016), 3661–3687

    work page 2016

  40. [40]

    Ergodicity of stochastic differential equations with jumps and singular coefficients

    Xie L. and Zhang X.: Ergodicity of stochastic differential equatio ns with jumps and singular coefficients. Accepted by Ann. Inst Henri Poincare-Pr. , https://arxiv.org/pdf/1705.07402.pdf

    work page internal anchor Pith review Pith/arXiv arXiv

  41. [41]

    and Liu J.: Weak order in averaging principle f or stochastic differential equations with jumps

    Zhang B., Fu H., Wan L. and Liu J.: Weak order in averaging principle f or stochastic differential equations with jumps. Adv. Difference Equ. , 2018, Paper No. 197, 20 pp

    work page 2018

  42. [42]

    Electron

    Zhang X.: Stochastic homemomorphism flows of SDEs with singular drifts and Sobolev diffusion coefficients. Electron. J. Probab., 16 (2011), 1096–1116

    work page 2011

  43. [43]

    K.: A transformation of the phase space of a diffusion process that removes the drift

    Zvonkin A. K.: A transformation of the phase space of a diffusion process that removes the drift. Mat. Sb. , 135 (1974), 129–149. Michael R ¨ockner: F akult¨at f ¨ur Mathematik, Universit ¨at Bielefeld, D-33501 Biele- feld, Germany, and Academy of Mathematics and Systems Scien ce, Chinese Academy of Sciences (CAS), Beijing, 100190, P.R.China, Email: roeckn...

    work page 1974