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This paper proves that a truncated Euler–Maruyama scheme achieves a uniform-in-time 1/2-order strong error and a 1/2-order L1-Wasserstein error for the invariant measure, for SDEs with drift that is only dissipative at infinity and has poly

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The numerical invariant measure of an explicit truncated Euler–Maruyama scheme converges to the true invariant measure at order h^{1/2} in the L1-Wasserstein distance, uniformly in time, for drifts that are dissipative only at long distances.

T0 review reviewed 2026-08-03 challenge →

load-bearing objection Real extension of coupling-based invariant-measure error analysis to locally Lipschitz drifts, but Lemma 4.4 has an exponent error that currently breaks the h^{1/2} claim; fixable with the already-available s parameter. the 2 major comments →

arxiv 2511.12124 v2 pith:NWVGCYYN submitted 2025-11-15 math.NA cs.NA

Discretization, Uniform-in-Time Estimations and Approximation of Invariant Measures for Nonlinear Stochastic Differential Equations with Non-Uniform Dissipativity

Shan Huang , Xiaoyue Li This is my paper
classification math.NA cs.NA MSC 65C3060H10
keywords invariant measuretruncated Euler-Maruyamauniform-in-time errorergodicityWasserstein distancecouplingcontractivity at infinitynon-uniform dissipativity
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

This paper proves that for a broad class of stochastic differential equations with drift that is only dissipative at infinity, a simple explicit truncation of the Euler–Maruyama scheme preserves ergodicity and converges at order 1/2 uniformly in time. Specifically, for every time step k, the expected strong error is bounded by C h^{1/2} with constants independent of h and k, and the same rate holds for the L1-Wasserstein distance between the scheme's invariant measure and the exact invariant measure. The result matters because it gives rigorous quantitative guarantees for using explicit schemes in long-time simulation and sampling for systems with non-convex, nonlinear drifts, such as bistable oscillators. The proof combines a truncation map that creates a globally Lipschitz drift with a carefully constructed Wasserstein coupling that is proved to contract per step under a concave distance function.

Core claim

Under Assumptions 2.1 (local Lipschitz continuity plus contractivity at infinity) and 4.1 (polynomial-growth Lipschitz drift), the truncated Euler–Maruyama scheme (3.5) is shown to be exponentially ergodic in the Lp-Wasserstein distance for all p≥1, and its strong error satisfies sup_{k≥0} E|X_k − x_{kh}| ≤ C h^{1/2}. Because both the exact diffusion and the numerical chain converge exponentially fast to their invariant measures, the triangle inequality then yields W1(μ_h, μ) ≤ C h^{1/2} for the numerical invariant measure. The proof does not rely on the numerical chain being exponential contractive; instead it uses a coupling between the numerical solution and a suitably coupled exact solut

What carries the argument

The central object is the one-step coupling defined in (4.11), a mixture of synchronous and reflection coupling, parameterized by a truncated drift difference and thresholds. The key new element is a specially chosen concave function f, constructed from a decreasing weight φ, which defines a Wasserstein-type metric in which the expected one-step distance contracts by a factor (1 − c h). The proof splits the state space into three distance regimes and uses separate estimates for the second moments of the coupling increments. The truncation map π_h is what permits the local Lipschitz drift to be treated globally with Lipschitz constant M h^{−θ}, which is essential for the truncation error boun

Load-bearing premise

The whole uniform-in-time bound collapses if the drift's local growth exceeds polynomial, because the truncation-error estimates are then no longer O(h^{3/2}).

What would settle it

Run the scheme on a one-dimensional SDE with drift b(x) = −x − x^3 log(1+x^2), which is dissipative at infinity but has super-polynomial local growth, and check whether sup_{k} E|X_k − x_{kh}| remains bounded by C√h for small h over long horizons. If it does not, Assumption 4.1 is genuinely needed.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • For any step size below an explicit upper bound, the numerical invariant measure is guaranteed to be within a fixed multiple of √h of the exact invariant measure in the L1-Wasserstein distance, uniformly over all initial values.
  • The strong error of the scheme is bounded by C√h for all times, so long-run averages and ergodic estimates computed from a single trajectory inherit a non-asymptotic accuracy bound.
  • The truncation scheme reduces to classical Euler–Maruyama when the drift is globally Lipschitz, so the uniform-in-time rates apply in that setting as well.
  • Numerical experiments on a sine-perturbed linear-drift SDE and on the cubic (Duffing-like) SDE confirm the predicted rates and the convergence of the empirical stationary densities.
  • The proof provides explicit formulas for the parameters (M, H, m) and the maximal step size h, which makes the theoretical bounds usable in practice.

Where Pith is reading between the lines

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

  • The 1/2 rate is likely not sharp; a finer coupling analysis might yield a higher-order rate (e.g., 1) under stronger smoothness, but the paper does not pursue this.
  • The uniform-in-time strong error could directly yield non-asymptotic bounds for empirical measures over finite horizons, going one step beyond the invariant-measure result.
  • Because the polynomial-growth Lipschitz condition is the sole restriction beyond contractivity at infinity, one might test whether the same scheme handles drifts with log-polynomial growth by adjusting the truncation exponent, a plausible but unproven extension.
  • The coupling construction depends on the Gaussian and isotropic structure of the noise; for correlated or non-Gaussian noise, a different reflection mechanism would be needed.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper studies explicit truncated Euler–Maruyama (TEM) approximations for SDEs with drift satisfying local Lipschitz continuity and contractivity at infinity (Assumption 2.1). It proposes the TEM scheme (3.5), proves existence and uniqueness of its invariant measure plus exponential ergodicity in Wasserstein distance (Theorem 3.6), then adds a polynomial-growth Lipschitz condition (Assumption 4.1) and derives a uniform-in-time strong error of order 1/2 (Theorem 4.19) via a customised coupling that combines synchronous, mirror, and reflection couplings. This is used to transfer the error to invariant measures: W_1(μ_h, μ) ≤ C h^{1/2} (Theorem 4.20). Two numerical experiments, including a Duffing-type system and a cubic-drift SDE, are reported in support of the theory.

Significance. If the proof pipeline is correct, the paper makes a useful extension of the non-asymptotic Wasserstein framework of Majka–Mijatović–Szpruch [48] from globally Lipschitz drifts to non-globally Lipschitz drifts with contractivity at infinity. The explicit construction of the coupling, the deterministic parameter choices (M, H, m), and the uniform-in-time strong error estimate are nontrivial and potentially significant for sampling and for numerical ergodic theory. The paper also benefits from using the exponential ergodicity of the exact solution as an external ingredient rather than re-proving it. However, the manuscript contains a load-bearing exponent error in the key truncation-error estimate Lemma 4.4, and an omitted proof for the moment bound Lemma 3.3. These issues are local and repairable, but until they are fixed the central claims should not be regarded as established.

major comments (2)
  1. [Lemma 4.4, Eq. (4.6)] The bound on J2 is mis-derived. After Cauchy–Schwarz, (E|π_h(x)−x|^2)^{1/2} = O(h^{sθ/(2ℓ)}) by Lemma 4.2 with n=2, so J2 = O(h^{1+sθ/(2ℓ)}), not O(h^{1+sθ/ℓ}). With the stated condition s ≥ ℓ/(2θ), this is only O(h^{5/4}) when ℓ=2θ. Inserting this into the geometric-sum argument in Theorem 4.19 yields E|X_k−x_{kh}| = O(h^{1/4}), not O(h^{1/2}), and Theorem 4.20 would collapse to O(h^{1/4}). The flaw is repairable by taking s ≥ ℓ/θ (a condition already weaker than the s ≥ 3ℓ/(2θ) used for J1), but the proof as written is incomplete at a load-bearing point.
  2. [Lemma 3.3] Lemma 3.3 is asserted without proof ('To avoid repetition, we omit the proof') and is used as a black box in Theorem 3.4 (numerical ergodicity), Lemma 4.4, and Theorem 4.19 for uniform-in-time moment bounds of the TEM scheme. Since this is not a standard published theorem but an adaptation of [3, Lemma 4.2]/[42, Theorem 5.5], the omission is a gap in the proof of the central claims. Please either include the proof or state precisely which theorem in [3] or [42] covers the exact TEM iteration (3.5) with contractivity at infinity and provide the corresponding statement.
minor comments (5)
  1. [Section 4.4, proof of Theorem 4.19] The line 'E|X_k−x_{kh}| ≤ ... = 2/φ(r_2) Ef(...)' contains an undefined symbol r_2 and an unjustified equality; presumably r_1 and a law-of-X_k argument are intended. Please correct.
  2. [Lemma 4.4] Eq. (4.4) contains the typo 'Combing'; in (4.7) the reference to Lemma 3.3 for the exact solution appears to be a slip for Lemma 2.4.
  3. [Remark 4.17 and Eq. (4.100)] The expression for m contains what looks like a typographical artifact: 'c_2^3/(8Φ(1)^2 c_* c_2)' should simplify to 'c_2^2/(8Φ(1)^2 c_*)', and the lower bound is not consistent with the preceding constraints. Please re-check.
  4. [Section 5, Example 5.1] The stated marginal stationary density p_y(u) = e^{u^2 − u^4/2}/Z is not the stationary density of the Ornstein–Uhlenbeck component dy_t = −y_t dt + dB_t^2; that density is Gaussian, ∝ e^{−y^2}. The formula appears to be the stationary density of Example 5.2. This affects the interpretation of Figures 4–5 and should be corrected.
  5. [Introduction/Assumption 4.1] Assumption 4.1 is a genuine strengthening of Assumption 2.1; the abstract and introduction could more prominently state that the uniform-in-time and invariant-measure rates require this polynomial-growth Lipschitz condition.

Circularity Check

0 steps flagged

No circularity: the uniform-in-time and invariant-measure rates are derived from explicit assumptions and independent external ergodicity results; self-citations are not load-bearing reductions.

full rationale

The derivation chain is not circular. The central estimate Theorem 4.19 is obtained from a coupling contractivity bound (Theorem 4.16) proved in this paper under Assumption 2.1, a one-step local error estimate (Lemma 4.4) proved under the explicitly added polynomial Lipschitz Assumption 4.1, and the triangle inequality; no fitted parameter is introduced and no prediction is a renamed input. Theorem 4.20 combines Theorem 4.19 with exact-process exponential ergodicity from Luo-Wang [47] (not a self-citation) and numerical exponential ergodicity proved from the Meyn-Tweedie criterion (Lemma 2.7). The TEM scheme and exact moment bounds are taken from the authors' earlier paper [42]; that citation is transparent ('Borrowing the idea from [42]') and the cited result is an independent published theorem whose assumptions do not include the claimed invariant-measure rate, so it does not make the argument circular. The omitted proof of Lemma 3.3 (citing [42, Theorem 5.5]) and the possible exponent issue in Lemma 4.4 flagged by the reviewer are proof-completeness/correctness concerns, not circular reductions: Lemma 4.4 does not define its conclusion into its hypotheses. Hence no circular step is established.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

The central derivation pulls exact ergodicity and moment bounds from cited sources, adds Assumption 4.1, and chooses the scheme parameters M, θ, m, H by hand. No free constants are fitted to numerical data, and no new physical or mathematical entities are postulated.

free parameters (3)
  • M (truncation base) = M = max{|b(0)|, 2L*, 1, sqrt(K), 1/(512 R |σ|)} (Remark 4.17)
    Hand-chosen to satisfy the inequalities (4.33), (4.60) and (4.100); it controls the truncation threshold M h^{-θ} and enters the step-size bound h1. Not fitted to data.
  • θ (truncation exponent) = any θ ∈ (0, θbar], with θbar ∈ (0,1/2)
    Scheme parameter in the truncation map (3.2); the truncation-error rate h^{sθ/ℓ} in Lemma 4.2 depends on it. Not fitted to data.
  • coupling parameters m and H = H = 2R, m = 8 ∨ ( ... − R ) (Remark 4.17)
    Chosen by hand so that the contractivity theorem 4.16 holds across the three distance regimes r ∈ (0,√h], (√h,r1], and (r1,∞).
axioms (5)
  • domain assumption Exact SDE (1.2) has a unique strong solution with sup_t E|x_t|^q < ∞ for all q (Lemma 2.4).
    Imported from [42, Theorems 2.3 and 5.2]; used throughout for exact-solution moment bounds.
  • domain assumption Under Assumption 2.1, the exact semigroup converges exponentially in W_q with rate λ (Lemma 2.5).
    Imported from [47, Corollary 1.8]; essential for the final W1(μ_h, μ) bound.
  • standard math Meyn–Tweedie Harris ergodicity theorem (Lemma 2.7).
    Used to prove existence, uniqueness and exponential ergodicity of the numerical chain.
  • domain assumption Assumption 4.1: polynomial-growth Lipschitz continuity of b.
    Introduced in Section 4.1; stronger than Assumption 2.1 and needed for the truncation-error bounds in Corollary 4.3 and Lemma 4.4.
  • standard math Gaussian increments of Brownian motion and rotational invariance of isotropic normals.
    Used throughout the coupling proofs, e.g., Lemma 4.5 and the reduction to one-dimensional integrals in Lemma 4.7.

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Pith. "Pith review of Discretization, Uniform-in-Time Estimations and Approximation of Invariant Measures for Nonlinear Stochastic Differential Equations with Non-Uniform Dissipativity." pith.science (2026). https://pith.science/paper/NWVGCYYN

@misc{pith2026251112124,
  author       = {Pith},
  title        = {Pith review of: Discretization, Uniform-in-Time Estimations and Approximation of Invariant Measures for Nonlinear Stochastic Differential Equations with Non-Uniform Dissipativity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NWVGCYYN}},
  note         = {Machine review of arXiv:2511.12124}
}
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read the original abstract

The approximation of invariant measures for nonlinear ergodic stochastic differential equations (SDEs) is a central problem in scientific computing, with important applications in stochastic sampling, physics, and ecology. We first propose an easily applicable explicit Truncated Euler-Maruyama (TEM) scheme and prove its numerical ergodicity in the $L^p$-Wasserstein distance ($p\geqslant 1$). Furthermore, by combining truncation techniques with the coupling method, we establish a uniform-in-time $1/2$-order convergence rate in moments for the TEM scheme. Additionally, leveraging the exponential ergodicity of both the numerical and exact solutions, we derive a $1/2$-order convergence rate for the invariant measures of the TEM scheme and the exact solution in the $L^1$-Wasserstein distance. Finally, two numerical experiments are conducted to validate our theoretical results.

Figures

Figures reproduced from arXiv: 2511.12124 by Shan Huang, Xiaoyue Li.

Figure 1
Figure 1. Figure 1: Functions with outside-of-sphere dissipativity or dissipativity at infinity. [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Trajectories of Ecos(p x 2 t + y 2 t ): Left—fixed step size with different initial conditions; Right—fixed initial condition with different step sizes. fig1 References IM1 [1] V. Bally and Y. Qin, Approximation for the invariant measure with applications for jump processes (convergence in total variation distance), Stochastic Process. Appl. 176 (2024), Paper No. 104416, 29 pp.; MR4764490. Bao [2] J. H. Ba… view at source ↗
Figure 3
Figure 3. Figure 3: Strong error between the numerical solution and the reference solution at termi [PITH_FULL_IMAGE:figures/full_fig_p046_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Joint density plots of the numerical solutions starting from the initial condition [PITH_FULL_IMAGE:figures/full_fig_p047_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Empirical cumulative marginal density plots of the numerical solutions start [PITH_FULL_IMAGE:figures/full_fig_p048_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Sample means of cos(X x (i) 0 ,hj k ),(i = 1, 2, 3; j = 1, 2) with different initial value x (i) 0 ∈ {1, 0.1, −1.5} in the interval [0, 4] for 5000 sample points and different step sizes h = 2−12 , 2 −10 . fi3 2002 [29] D. J. Higham, X. Mao and A. M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal. 40 (2002), no. 3, 1041–1063; MR1949404.… view at source ↗
Figure 7
Figure 7. Figure 7: The approximation error in the L 1 -Wasserstein distance for 2000 sample points between the distribution of exact solution (xt)t≥0 of SDE (5.2) and the distribution of numerical solutions (Xk)k≥0 by the truncated EM scheme as functions of runtime for h ∈ {2 −14 , 2 −13 , 2 −12 , 2 −11 , 2 −10}. fi4 [PITH_FULL_IMAGE:figures/full_fig_p049_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Empirical cumulative distribution functions and kernel density functions of [PITH_FULL_IMAGE:figures/full_fig_p049_8.png] view at source ↗

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This paper was first reviewed by deepseek-v4-flash on August 3, 2026.