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arxiv: 2606.18815 · v1 · pith:VZKLECRUnew · submitted 2026-06-17 · 🧮 math.NA · cs.NA

Exponential ergodicity of exact and numerical solutions for McKean-Vlasov SDEs driven by L\'evy noise

Yang Sun , Yuhang Zhang , Minghui Song This is my paper

Pith reviewed 2026-06-26 20:18 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords McKean-Vlasov SDELévy noiseexponential ergodicitytamed Euler methodpropagation of chaosinvariant measurenumerical approximation
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The pith

McKean-Vlasov SDEs driven by Lévy noise and their tamed Euler discretizations both exhibit exponential ergodicity, with the numerical invariant measure converging to the exact one.

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

The paper proves exponential ergodicity for the exact solution of McKean-Vlasov SDEs with Lévy noise and for the corresponding tamed Euler numerical method. It then establishes that the invariant measure of the numerical scheme converges to the invariant measure of the continuous equation, and derives an explicit convergence rate by combining uniform-in-time propagation of chaos with uniform strong convergence of the scheme. These results matter because they provide rigorous justification for using the tamed Euler method to approximate long-time statistical behavior in mean-field systems with jumps.

Core claim

Exponential ergodicity holds for both the original McKean-Vlasov SDE and the tamed Euler method. Convergence of the numerical invariant measure to the exact invariant measure follows from propagation of chaos combined with strong convergence of the tamed Euler scheme. An explicit convergence rate is obtained by establishing uniform-in-time propagation of chaos and uniform-in-time strong convergence of the tamed Euler method.

What carries the argument

The tamed Euler scheme applied to the McKean-Vlasov SDE, whose uniform-in-time convergence properties, when paired with propagation of chaos, transfer ergodicity and control the distance between invariant measures.

If this is right

  • The tamed Euler method inherits exponential ergodicity from the underlying McKean-Vlasov SDE.
  • The numerical invariant measure converges to the exact invariant measure.
  • An explicit rate of convergence between the two invariant measures follows from the uniform-in-time propagation of chaos and uniform strong convergence.
  • Numerical experiments can be used to verify the predicted convergence behavior of the invariant measures.

Where Pith is reading between the lines

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

  • The same combination of uniform propagation of chaos and uniform convergence arguments could be tested on other discretization schemes such as Milstein-type methods for jump processes.
  • The rate results supply quantitative error bounds that could be used when approximating stationary distributions in applications involving large systems of interacting particles with jumps.
  • The framework suggests checking whether adaptive time-stepping versions of the tamed Euler scheme retain the uniform-in-time properties needed for invariant-measure convergence.

Load-bearing premise

The coefficients must satisfy the conditions that deliver propagation of chaos, strong convergence of the tamed Euler scheme, and exponential ergodicity for both the continuous and discrete processes.

What would settle it

A concrete McKean-Vlasov SDE with Lévy noise for which either the exact or numerical solution fails to be exponentially ergodic, or for which the numerical invariant measure does not approach the exact one at the rate predicted by the uniform propagation of chaos and uniform convergence results.

Figures

Figures reproduced from arXiv: 2606.18815 by MingHui Song, Yang Sun, Yuhang Zhang.

Figure 1
Figure 1. Figure 1: Approximation of the invariant distribution of Example 7.1 with N = 2000 particles. [PITH_FULL_IMAGE:figures/full_fig_p031_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The empirical distribution functions for Example 7.1 with N = 2000 particles. [PITH_FULL_IMAGE:figures/full_fig_p032_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Approximation of the invariant distribution of Example 7.2 with N = 2000 particles. [PITH_FULL_IMAGE:figures/full_fig_p033_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Convergence rates of invariant measures for Example 7.2 with [PITH_FULL_IMAGE:figures/full_fig_p033_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Approximation of the invariant distribution of Example 7.3 with N = 2000 particles. [PITH_FULL_IMAGE:figures/full_fig_p035_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of empirical density functions for Example 7.3 with N = 3000 particles. [PITH_FULL_IMAGE:figures/full_fig_p035_6.png] view at source ↗
read the original abstract

This paper investigates the exponential ergodicity of the exact solution and the tamed Euler solution for McKean-Vlasov stochastic differential equations driven by L\'evy noise. First, we establish exponential ergodicity for both the original equation and the tamed Euler method. Then we prove the convergence of the numerical invariant measure to the exact invariant measure, which is obtained by combining the propagation of chaos (PoC) result with the strong convergence of the tamed Euler scheme. Furthermore, we derive a convergence rate for the numerical invariant measure by establishing uniform-in-time PoC and uniform-in-time convergence of the tamed Euler method. Finally, numerical experiments are presented to illustrate the theoretical results.

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

0 major / 1 minor

Summary. The paper claims to establish exponential ergodicity for both the exact solution and the tamed Euler discretization of McKean-Vlasov SDEs driven by Lévy noise. It then proves convergence of the numerical invariant measure to the exact invariant measure by combining propagation of chaos with strong convergence of the scheme, derives a convergence rate via uniform-in-time propagation of chaos and uniform-in-time strong convergence, and includes numerical experiments to illustrate the results.

Significance. If the proofs hold under appropriate conditions on the coefficients, the results would provide a coherent framework for long-time behavior and invariant-measure approximation in numerical schemes for mean-field SDEs with jumps. The combination of ergodicity, propagation of chaos, and uniform-in-time estimates is technically non-trivial and relevant for applications involving Lévy-driven interacting systems.

minor comments (1)
  1. The abstract refers to coefficient conditions needed for propagation of chaos, strong convergence, and ergodicity, but does not state them; the full manuscript should make these assumptions explicit in the statements of the main theorems to allow verification of the scope.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful summary of our manuscript and for acknowledging the technical non-triviality of combining exponential ergodicity, propagation of chaos, and uniform-in-time estimates for Lévy-driven McKean-Vlasov SDEs. We are pleased that the potential relevance to applications is recognized, provided the proofs hold under the stated conditions on the coefficients.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation follows a standard sequential structure: exponential ergodicity is established first for the McKean-Vlasov SDE and its tamed Euler discretization, followed by convergence of invariant measures via propagation of chaos combined with strong convergence, and rates derived from uniform-in-time versions of those results. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations appear in the abstract or high-level strategy. The argument relies on external standard tools (PoC, strong convergence under dissipativity conditions) that are independent of the target claims, rendering the chain self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, invented entities, or ad-hoc axioms; the work rests on standard domain assumptions for existence, uniqueness, and ergodicity of McKean-Vlasov Lévy SDEs that are not enumerated here.

axioms (1)
  • domain assumption Coefficients satisfy conditions sufficient for propagation of chaos, strong convergence of tamed Euler, and exponential ergodicity
    Invoked implicitly to obtain all stated results; concrete form not given in abstract

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Reference graph

Works this paper leans on

39 extracted references · 6 canonical work pages · 1 internal anchor

  1. [1]

    H. P. McKean, Propagation of chaos for a class of non-linear parabolic equations, Proceedings of the National Academy of Sciences 56 (1967) 1907–1911

    1967

  2. [2]

    Benazzoli, L

    C. Benazzoli, L. Campi, L. Di Persio, Mean field games with controlled jump- diffusion dynamics: Existence results and an illiquid interbank market model, Stochastic Processes and their Applications 130 (2020) 6927–6964

    2020

  3. [3]

    Kolokolnikov, J

    T. Kolokolnikov, J. A. Carrillo, A. Bertozzi, R. Fetecau, M. Lewis, Emergent be- haviour in multi-particle systems with non-local interactions, Physica D: Nonlinear Phenomena 260 (2013) 1–4

    2013

  4. [4]

    Sabbar, K

    Y. Sabbar, K. S. Nisar, A selective review of modern stochastic modeling: SDE/SPDE numerics, data-driven identification, and generative methods with ap- plications in biomathematics, Transactions on Computational Modeling and Intel- ligent Systems 2 (2026) 10028

    2026

  5. [5]

    Neufeld, T

    A. Neufeld, T. A. Nguyen, Rectified deep neural networks overcome the curse of dimensionality when approximating solutions of McKean–Vlasov stochastic differ- ential equations, Journal of Mathematical Analysis and Applications 541 (2025) 128661

    2025

  6. [6]

    Baladron, D

    J. Baladron, D. Fasoli, O. Faugeras, J. Touboul, Mean-field description and prop- agation of chaos in networks of Hodgkin–Huxley and FitzHugh–Nagumo neurons, Journal of Mathematical Neuroscience 2 (2012) 10

    2012

  7. [7]

    mean-field descriptionandpropagationofchaosinnetworksofHodgkin–HuxleyandFitzHugh– Nagumo neurons

    M. Bossy, O. Faugeras, D. Talay, Clarification and complement to “mean-field descriptionandpropagationofchaosinnetworksofHodgkin–HuxleyandFitzHugh– Nagumo neurons”, Journal of Mathematical Neuroscience 5 (2014) 31

    2014

  8. [8]

    Ullner, A

    E. Ullner, A. Politi, A. Torcini, Ubiquity of collective irregular dynamics in balanced networks of spiking neurons, Chaos 28 (2018) 081106

    2018

  9. [9]

    Bensoussan, J

    A. Bensoussan, J. Frehse, P. Yam, Mean Field Games and Mean Field Type Control Theory, Springer, New York, 2013

    2013

  10. [10]

    Carmona, F

    R. Carmona, F. Delarue, Probabilistic Theory of Mean Field Games with Applica- tions I: Mean Field FBSDEs, Control, and Games, volume 83 ofProbability Theory and Stochastic Modelling, Springer, Cham, 2018

    2018

  11. [11]

    Carmona, F

    R. Carmona, F. Delarue, Probabilistic Theory of Mean Field Games with Applica- tions II: Mean Field Games with Common Noise and Master Equations, volume 84 ofProbability Theory and Stochastic Modelling, Springer, Cham, 2018. 36

    2018

  12. [12]

    Y. Li, X. Mao, Q. Song, F. Wu, G. Yin, Strong convergence of Euler–Maruyama schemes for McKean–Vlasov stochastic differential equations under local lipschitz conditions of state variables, IMA Journal of Numerical Analysis 43 (2022) 1001– 1035

    2022

  13. [13]

    J. Bao, C. Reisinger, P. Ren, W. Stockinger, First-order convergence of Milstein schemes for McKean–Vlasov equations and interacting particle systems, Proceed- ings of the Royal Society A 477 (2021) 20200258

    2021

  14. [14]

    Reisinger, W

    C. Reisinger, W. Stockinger, An adaptive Euler–Maruyama scheme for McKean–Vlasov SDEs with super-linear growth and application to the mean-field FitzHugh–Nagumo model, Journal of Computational and Applied Mathematics 400 (2022) 113725

    2022

  15. [15]

    X. Ding, H. Qiao, Euler–Maruyama approximations for stochastic McKean–Vlasov equations with non-lipschitz coefficients, Journal of Theoretical Probability 34 (2021) 1408–1425

    2021

  16. [16]

    C. Kumar, Neelima, On explicit Milstein-type scheme for McKean–Vlasov stochas- tic differential equations with super-linear drift coefficient, Electronic Journal of Probability 26 (2021) 1–32

    2021

  17. [17]

    H.Amini, Z.Cao, A.Sulem, Stochasticgraphongameswithjumpsandapproximate Nash equilibria, arXiv preprint arXiv:2304.04112 (2023)

    work page arXiv 2023

  18. [18]

    Andreis, P

    L. Andreis, P. Dai Pra, M. Fischer, McKean–Vlasov limit for interacting systems with simultaneous jumps, Stochastic Analysis and Applications 36 (2018) 960–995

    2018

  19. [19]

    Mehri, M

    S. Mehri, M. Scheutzow, W. Stannat, B. Z. Zangeneh, Propagation of chaos for stochastic spatially structured neuronal networks with delay driven by jump diffu- sions, The Annals of Applied Probability 30 (2020) 175–207

    2020

  20. [20]

    Agarwal, S

    A. Agarwal, S. Pagliarani, A Fourier-based Picard-iteration approach for a class of McKean–Vlasov SDEs with Lévy jumps, Stochastics 93 (2021) 592–624

    2021

  21. [21]

    Biswas, C

    Neelima, S. Biswas, C. Kumar, G. dos Reis, C. Reisinger, Well-posedness and tamed Euler schemes for McKean–Vlasov equations driven by Lévy noise, arXiv preprint arXiv:2010.08585 (2020)

    work page arXiv 2010

  22. [22]

    Biswas, C

    S. Biswas, C. Kumar, C. Reisinger, V. Schwarz, Milstein-type schemes for McKean– Vlasov SDEs driven by Brownian motion and Poisson random measure (with super- linear coefficients), arXiv preprint arXiv:2411.11759 (2025)

    work page arXiv 2025

  23. [23]

    N. K. Tran, T.-T. Kieu, D.-T. Luong, H.-L. Ngo, On the infinite time horizon approximation for Lévy-driven McKean–Vlasov SDEs with non-globally lipschitz continuous and super-linearly growth drift and diffusion coefficients, Journal of Mathematical Analysis and Applications 543 (2025) 128982. 37

    2025

  24. [24]

    J. Zhu, Y. Zhao, S. Gan, Euler-type methods for Lévy-driven McKean–Vlasov SDEs with super-linear coefficients: Mean-square error analysis, Communications in Nonlinear Science and Numerical Simulation 161 (2026) 110093

    2026

  25. [25]

    Wang, Distribution dependent SDEs for Landau type equations, Stochastic Processes and their Applications 128 (2018) 595–621

    F.-Y. Wang, Distribution dependent SDEs for Landau type equations, Stochastic Processes and their Applications 128 (2018) 595–621

    2018

  26. [26]

    J. Bao, J. Hao, Uniform-in-time estimates for mean-field type SDEs and applica- tions, Journal of Differential Equations 440 (2025) 113445

    2025

  27. [27]

    J. Bao, J. Wang, Long-time behavior of one-dimensional McKean–Vlasov SDEs with common noise, Journal of Mathematical Analysis and Applications 552 (2025) 129819

    2025

  28. [28]

    X. Chen, G. dos Reis, W. Stockinger, Wellposedness, exponential ergodicity and numerical approximation of fully super-linear McKean–Vlasov SDEs and associated particle systems, Electronic Journal of Probability 30 (2025) 1 – 50

    2025

  29. [29]

    Liang, M

    M. Liang, M. B. Majka, J. Wang, Exponential ergodicity for SDEs and McK- ean–Vlasov processes with Lévy noise, Annales de l’Institut Henri Poincaré, Prob- abilités et Statistiques 57 (2021) 1665 – 1701

    2021

  30. [30]

    J. Bao, J. Wang, Stationary distributions of McKean-Vlasov SDEs with jumps: Existence, multiplicity and uniqueness, Stochastic Processes and their Applications 198 (2026) 104967

    2026

  31. [31]

    F. Bai, Z. Chen, X. Sun, Existence and approximation of measure attractors and invariant measures for McKean-Vlasov stochastic lattice system with Lévy noise, Journal of Differential Equations 453 (2026) 113784

    2026

  32. [32]

    Soni, Neelima, C

    S. Soni, Neelima, C. Kumar, G. dos Reis, Tamed Euler approximation for fully superlinear growth McKean–Vlasov SDE and their particle systems: sharp rates for strong propagation of chaos, convergence and ergodicity, arXiv preprint arXiv:2510.16427 (2025)

    work page arXiv 2025

  33. [33]

    Y.Cui, X.Li, Y.Liu, F.Wang, Numericalapproximationtotheinvariantmeasureof McKean-Vlasov stochastic differential equations, arXiv preprint arXiv:2401.02878 (2025)

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  34. [34]

    J. Bao, M. B. Majka, J. Wang, Geometric ergodicity of modified Euler schemes for SDEs with super-linearity, arXiv preprint arXiv:2412.19377 (2024)

    work page arXiv 2024

  35. [35]

    Gyöngy, N

    I. Gyöngy, N. V. Krylov, On stochastic equations with respect to semimartingales I, Stochastics 4 (1980) 1–21

    1980

  36. [36]

    Dareiotis, C

    K. Dareiotis, C. Kumar, S. Sabanis, On tamed Euler approximations of SDEs driven by Lévy noise with applications to delay equations, SIAM Journal on Numerical Analysis 54 (2016) 1840–1872. 38

    2016

  37. [37]

    P. D. Hinds, A. Sharma, M. V. Tretyakov, Well-posedness and approximation of reflected McKean–Vlasov SDEs with applications, Mathematical Models and Methods in Applied Sciences 35 (2025) 1845–1887

    2025

  38. [38]

    Zhang, M

    Y. Zhang, M. Song, J. Zhu, Dimension-independentLp convergence rate of prop- agation of chaos and numerical analysis for McKean-Vlasov stochastic differential equations, Communications in Nonlinear Science and Numerical Simulation 152 (2026) 109472

    2026

  39. [39]

    H. P. Rosenthal, On the subspaces ofL p spanned by sequences of independent random variables, Israel Journal of Mathematics 8 (1970) 273–303. 39

    1970