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arxiv: 2604.14543 · v1 · submitted 2026-04-16 · 🧮 math.PR

The Euler-Maruyama method for invariant measures of McKean-Vlasov stochastic differential equations

Zhen Wang , Mingyan Wu This is my paper

Pith reviewed 2026-05-10 10:20 UTC · model grok-4.3

classification 🧮 math.PR
keywords Euler-Maruyama methodMcKean-Vlasov SDEsinvariant measuresWasserstein metricnumerical approximationparticle systemmonotonicity conditionstochastic differential equations
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The pith

The Euler-Maruyama scheme for McKean-Vlasov SDEs produces a numerical solution with a unique invariant measure that converges to the true one at an explicit rate in the Wasserstein metric.

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

The paper studies the long-term behavior of the Euler-Maruyama discretization applied to McKean-Vlasov stochastic differential equations whose coefficients obey a monotonicity condition. It first proves that the discrete-time process converges to the continuous solution, then shows that the discrete process itself admits a unique invariant measure. The authors derive the rate at which this numerical invariant measure approaches the continuous one when measured in the Wasserstein distance. Parallel results are obtained for the finite-particle system that approximates the McKean-Vlasov equation. These facts supply a rigorous justification for using the scheme to compute stationary expectations of interacting particle systems.

Core claim

Under the stated monotonicity condition, the Euler-Maruyama discretization of a McKean-Vlasov SDE yields a numerical solution that converges strongly to its continuous-time counterpart; this discrete solution possesses a unique invariant measure, and the Wasserstein distance between the numerical invariant measure and the true invariant measure tends to zero at a rate determined by the time-step size. The same convergence and uniqueness properties hold for the associated interacting particle system.

What carries the argument

The Euler-Maruyama time-stepping scheme together with the monotonicity condition on the drift and diffusion coefficients, which produces contractivity in the Wasserstein metric for both the continuous and discrete dynamics.

If this is right

  • Stationary expectations of the McKean-Vlasov dynamics can be approximated by long-time averages of the Euler-Maruyama trajectory.
  • The Wasserstein error bound supplies a concrete criterion for choosing the time step to achieve a prescribed accuracy in the invariant measure.
  • The particle-system version validates the use of finite-N simulations to compute mean-field stationary quantities.
  • The same monotonicity assumption simultaneously guarantees both well-posedness of the continuous equation and stability of its numerical approximation.

Where Pith is reading between the lines

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

  • The result suggests that higher-order schemes could be analyzed by the same contractivity argument once a suitable monotonicity or dissipativity condition is identified.
  • Because the proof relies only on Wasserstein contraction, the technique may extend directly to other mean-field models arising in neuroscience or statistical mechanics.
  • Quantitative rates open the possibility of adaptive time-stepping algorithms that automatically refine the mesh until the invariant-measure error falls below a tolerance.

Load-bearing premise

The coefficients of the McKean-Vlasov SDE satisfy a monotonicity condition that guarantees contractivity of the flow in the Wasserstein metric.

What would settle it

An explicit McKean-Vlasov SDE whose coefficients meet the monotonicity condition but for which the Euler-Maruyama scheme either fails to possess a unique invariant measure or produces a Wasserstein convergence rate slower than the one derived in the paper.

Figures

Figures reproduced from arXiv: 2604.14543 by Mingyan Wu, Zhen Wang.

Figure 1
Figure 1. Figure 1: Left: Empirical density functions at different time points. Right: Empirical density functions at t=30 with different initial val￾ues. figure (2) Strong convergence of the EM scheme [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Strong convergence of the Euler-Maruyama scheme. figure (3) Convergence of the particle system. We provide numerical evidence for the convergence rate predicted in Theorem 2.3 which quantifies the relationship between the interacting particle system and its mean-field limit [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Left: Chaos error vs N. Right: Discretization error vs h. figure empirical distributions from the two different initial conditions are indistinguishable, confirming that the invariant measure exists and is unique. F-4 [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Evolution of the particle system toward the invariant mea￾sure. Left: Initial configuration. Center: Intermediate stage. Right: Steady state. figure 6. Data Availability Statement All data, models, and code generated or used during the study appear in the submitted article. References BSY [1] J. Bao, M. Scheutzow, C. Yuan, Existence of invariant probability measures for functional McKean-Vlasov SDEs. Elect… view at source ↗
read the original abstract

This paper investigates the approximation of invariant measures for McKean-Vlasov stochastic differential equations (SDEs) using the Euler-Maruyama (EM) scheme under a monotonicity condition. Firstly, the convergence of the numerical solution from the EM scheme to its continuous-time counterpart is established. Secondly, we show that the numerical solution admits a unique invariant measure and derive its convergence rate under the Wasserstein metric. In parallel, it is demonstrated that the associated particle system also possesses these properties.

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 / 2 minor

Summary. The paper investigates the Euler-Maruyama (EM) discretization of McKean-Vlasov SDEs under a monotonicity condition on the coefficients. It first establishes convergence of the numerical solution to the continuous-time process, then proves that the EM scheme admits a unique invariant measure and derives the rate at which this numerical invariant converges to the true invariant measure in the Wasserstein metric. Analogous results are shown for the associated interacting particle system.

Significance. If the central claims hold with complete proofs, the work provides a useful theoretical justification for applying the standard EM scheme to approximate invariant measures of mean-field SDEs, including explicit rates in Wasserstein distance. This is relevant for long-time simulation of interacting particle systems arising in statistical physics and mean-field models, and the parallel treatment of the particle system strengthens the applicability.

major comments (2)
  1. [Theorem on unique invariant measure for EM scheme] The abstract asserts that the EM scheme admits a unique invariant measure under the monotonicity condition with no mention of a step-size restriction. For one-sided Lipschitz/monotone drifts, the EM transition kernel is typically contractive in Wasserstein distance only when h satisfies h < 1/(2C) for the monotonicity constant C; without this, uniqueness or moment bounds can fail. The proof of existence/uniqueness for the numerical invariant measure (likely in the section following the convergence result) must explicitly derive and state any such h0 threshold or demonstrate that the result holds for arbitrary h>0 under the given assumptions.
  2. [Rate derivation for invariant-measure convergence] The claimed Wasserstein convergence rate between the numerical and continuous invariant measures relies on the monotonicity condition; the derivation should be checked to confirm that the rate remains valid when the step-size restriction (if needed) is imposed and that no hidden dependence on the monotonicity constant appears in the final bound.
minor comments (2)
  1. [Abstract] The abstract is brief and could usefully state the precise order of the convergence rate (e.g., O(h^α)) and list the exact assumptions on the drift and diffusion coefficients.
  2. [Notation and preliminaries] Notation for the empirical measure, the law of the process, and the Wasserstein distance should be introduced once and used consistently; minor inconsistencies in subscripts or superscripts can be corrected in a revision.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and insightful comments on our manuscript. The points raised regarding the step-size restriction for the invariant measure and the validity of the convergence rate are well-taken. We address each major comment below and will make the necessary revisions to clarify the assumptions and strengthen the presentation.

read point-by-point responses

  1. Referee: [Theorem on unique invariant measure for EM scheme] The abstract asserts that the EM scheme admits a unique invariant measure under the monotonicity condition with no mention of a step-size restriction. For one-sided Lipschitz/monotone drifts, the EM transition kernel is typically contractive in Wasserstein distance only when h satisfies h < 1/(2C) for the monotonicity constant C; without this, uniqueness or moment bounds can fail. The proof of existence/uniqueness for the numerical invariant measure must explicitly derive and state any such h0 threshold or demonstrate that the result holds for arbitrary h>0 under the given assumptions.

    Authors: We agree with the referee that under the one-sided Lipschitz (monotonicity) condition, the EM scheme requires a sufficiently small step size to ensure contractivity of the transition kernel in the Wasserstein metric and thus uniqueness of the invariant measure. Our proof establishes contractivity for h small enough depending on the monotonicity constant C, but this threshold was not explicitly stated in the theorem or abstract. We will revise the manuscript to derive the explicit condition h < 1/(2C) in the relevant section, add it to the theorem statement, and update the abstract to reflect the restriction. This clarification does not change the validity of the results but improves precision. revision: yes

  2. Referee: [Rate derivation for invariant-measure convergence] The claimed Wasserstein convergence rate between the numerical and continuous invariant measures relies on the monotonicity condition; the derivation should be checked to confirm that the rate remains valid when the step-size restriction (if needed) is imposed and that no hidden dependence on the monotonicity constant appears in the final bound.

    Authors: The Wasserstein convergence rate between the numerical invariant measure and the true one is derived by combining the convergence of the EM scheme to the continuous process (which holds for small h) with the contractivity properties under monotonicity. With the step-size restriction h < 1/(2C) imposed, the rate remains valid, and the constants in the bound depend on C but are independent of h for admissible h. We will carefully re-examine the proof to confirm there is no unintended hidden dependence on C in the final rate expression and will add a remark clarifying the dependence on the monotonicity constant. If any adjustment to the bound is required, it will be incorporated. revision: partial

Circularity Check

0 steps flagged

No circularity: direct analytic proof under stated monotonicity assumption

full rationale

The paper establishes convergence of the EM scheme to the continuous McKean-Vlasov SDE, then proves existence and uniqueness of an invariant measure for the numerical scheme together with its Wasserstein convergence rate, all under the given one-sided Lipschitz/monotonicity condition on the coefficients. No parameter fitting, no 'prediction' that reduces to a fitted input by construction, and no load-bearing self-citation chain that substitutes for an independent argument. The derivation chain is self-contained against the external benchmark of standard SDE theory for monotone coefficients; any small-h restriction, if needed, would be a correctness issue rather than a circularity reduction. Steps array left empty per instructions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Central claims rest on a monotonicity assumption for the SDE coefficients, which is a domain-standard condition rather than an ad-hoc invention; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Monotonicity condition on the drift and diffusion coefficients of the McKean-Vlasov SDE
    Explicitly required for all stated convergence and uniqueness results.

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