$W_{\bf d}$-convergence rate of EM schemes for invariant measures of supercritical stable SDEs

Lihu Xu; Peng Chen; XiaoLong Zhang; Xicheng Zhang

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$W_{\bf d}$-convergence rate of EM schemes for invariant measures of supercritical stable SDEs

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Pith's one-line read This paper proves that the invariant measure of an Euler–Maruyama discretization of an SDE driven by multiplicative α-stable noise converges to the true invariant measure at an explicit polynomial rate in the step size, for α∈(1/2,2).

desk verdict First quantitative EM invariant-measure rates for supercritical stable SDEs; two repairable but real gaps in the Stein-equation and regularity arguments. read the letter →

arxiv 2411.09949 v1 pith:HIJ3RK6I submitted 2024-11-15 math.PR

Peng Chen , Lihu Xu , Xiaolong Zhang , Xicheng Zhang This is my paper · ORCID

classification math.PR MSC 60H3560J7560G5265C30

keywords stableprocessesEuler-MaruyamaschemeinvariantmeasureStein/PoissonequationweightedHölderspacesWassersteindistancesupercriticalSDEsheavy-tailedsampling

verification ladder T0 review T1 audit T2 compute T3 formal

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 the invariant measure of an Euler–Maruyama discretization of an SDE driven by multiplicative α-stable noise converges to the true invariant measure at an explicit polynomial rate in the step size, for α∈(1/2,2). The metric is W_d with $d(x,y)=|x-y|^\gamma\wedge 1$, a bounded Hölder-type Wasserstein distance, and the rate is $\eta^{\gamma\wedge(2\alpha-1-\varepsilon)}$ for α≤1 and $\eta^{\gamma/\alpha}$ for α>1. A separate result gives rate $\eta^{1/\alpha}$ in Wasserstein-1 under stronger monotone-drift assumptions and α∈(1,2). The proof works by solving a nonlocal Stein/Poisson equation and showing its solution lies in weighted Hölder spaces with only logarithmic growth, which is what makes the small-α, supercritical regime tractable.

What carries the argument

The central object is the nonlocal Stein/Poisson equation $\mathcal{L}f=\mu(g)-g$ for the stable generator $\mathcal{L}f=b\cdot\nabla f+\mathcal{L}^{(\alpha)}_{\sigma(x)}f$. The argument shows that $f=\int_0^\infty P_t\bar g\,dt$ belongs to weighted Hölder spaces with weight $(\log(e+|x|^2))^{-1}$, using two-sided heat-kernel estimates for the SDE density with unbounded drift, commutator estimates, and a third-difference regularity lemma. This regularity is what converts the generator error $\mathbb{E}[\mathcal{L}f(\xi)]$ for a stationary EM step ξ into the explicit powers of η that appear in the rate.

What would settle it

Take the one-dimensional OU process (B.1) for α∈(1/2,1] with the metric $d(x,y)=|x-y|^\gamma\wedge 1$. Proposition B.1 gives the exact invariant distributions of both the SDE and its EM scheme, so $W_d(\mu,\mu_\eta)$ can be computed or simulated to high precision; if its scaling in η is slower than $\eta^{\gamma\wedge(2\alpha-1-\varepsilon)}$ for all small ε, then Theorem 1.2 is false. Alternatively, finding a drift satisfying (Hγ) for which the two-sided heat-kernel bound (1.7)–(1.8) fails would invalidate the regularity lemmas and the rate.

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Extended reading notes

Core claim

Under the dissipativity assumption $\langle x,b(x)\rangle\le -\kappa_1|x|^2+\kappa_2$ and the Hölder regularity assumption (Hγ), the invariant measure μη of the Euler–Maruyama scheme satisfies $W_d(\mu,\mu_\eta)\le C(\eta^{\gamma\wedge(2\alpha-1-\varepsilon)})$ for α∈(1/2,1] and $W_d(\mu,\mu_\eta)\le C\eta^{\gamma/\alpha}$ for α>1, where $d(x,y)=|x-y|^\gamma\wedge 1$ and γ∈((1-α)_+,1]. The discovery is that the Stein/Poisson equation $\mathcal{L}f=\mu(g)-g$ has a solution $f=\int_0^\infty P_t\bar g\,dt$ lying in weighted Hölder spaces with logarithmic growth, $C^1_{\log}$ and $C^{\beta+\gamma}_{\log}$, for every β∈(0,α); this regularity is exactly strong enough to control the one-step generator error along the stationary EM chain and to yield the first explicit discretization rate for supercritical stable SDEs with multiplicative noise.

Load-bearing premise

The proof requires the two-sided heat-kernel estimate (1.7)–(1.8) to hold for the whole coefficient class (Hγ) with unbounded drift; if that estimate fails for some admissible drift and diffusion, the Stein-solver regularity and the stated rate would collapse.

Editorial extensions

If this is right

For α∈(1/2,1], the invariant measure of the EM chain converges under $W_d$ at an explicitly polynomial rate, so sampling from heavy-tailed targets by Euler–Maruyama is justified with a rate.
For α∈(1,2) with monotone drift, the stronger Wasserstein-1 rate $\eta^{1/\alpha}$ holds, matching the time-scaling of stable increments.
The Stein/Poisson regularity estimates apply to the whole supercritical class with multiplicative noise and unbounded drift, so the same machinery can bound discretization errors for other observables beyond the Wasserstein distance.
Choosing γ=1 when α>1 recovers a bounded-Lipschitz-type metric, while choosing γ close to $(1-\alpha)_+$ covers the finest Hölder metric available to the method.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: the logarithmic-growth weight mechanism suggests the same $W_d$ rate should extend to the Pareto-type EM scheme mentioned in Remark 1.4, since only the small-time scaling of the increments is used.
Editorial inference: the $\varepsilon$ loss in the α∈(1/2,1] rate looks like an artifact of taking a limit β↑α in the proof, not an intrinsic obstruction; the one-dimensional OU example in Appendix B offers a testbed for sharpness.
Editorial inference: for α≤1/2 the short-time gradient estimate diverges, so a genuinely new estimate would be needed; if a rate were sought in an unweighted metric such as $W_1$, the logarithmic-growth regularity would not suffice.
Editorial inference: the same Stein/Poisson regularity could be used to prove rates for other discretizations or for time-averaged functionals, because the central bound is about the semigroup, not about the specific EM recursion.

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Editorial analysis

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

Desk Editor's Note

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Short version: this is a genuine extension, and the main theorem is probably right, but two spots in the proof need fixing before I'd trust the written derivation.

What's new: [8] covers additive noise, alpha in (1,2), and smooth drift. Here they get W_d rates for multiplicative stable noise, alpha in (1/2,2), gamma-Holder drift, and dissipativity, with d=|x-y|^gamma wedge 1. The rates are explicit, and the regularity theory for the nonlocal Stein/Poisson equation in weighted Holder spaces with logarithmic growth is the core new input. That's a real contribution. The paper is mostly honest about limitations: Remark 1.3 says the rates are not sharp, Appendix B gives the OU benchmark, and Remark 2.10 explains why alpha in (0,1/2] is out of reach. The citations to [29] and [48] overlap with the authors, but those are published external theorems with proofs, not restatements of the target result. No circularity.

Soft spots, in order:

1. Proposition 2.15 is not valid as written. The resolvent argument runs in C0, but for typical g the solution f = integral P_t gbar dt has logarithmic growth and is not in C0; moreover gbar = g - mu(g) is generally not in C0 either. So f is not in D(A), and the Hille-Yosida passage in C0 does not give Af = -gbar. The pointwise identity is very likely true—differentiate under the integral using Theorem 2.14 and exponential ergodicity—but that step is missing.

2. Lemma 2.13 applies Theorem 2.3(i) with beta0 = (beta+gamma) wedge 2, which can be the integer 2, while that difference characterization is stated for non-integer exponents. So the C^2 bound in (2.32) is not justified. This looks repairable, for instance by taking beta0 = 2-epsilon and absorbing the loss into the epsilon in Theorem 1.2, but as written it is a gap.

Both gaps are technical rather than conceptual, and neither makes me doubt the main theorem. But they are load-bearing in the proof of the Stein-equation regularity, so the paper needs a corrected version before acceptance.

Who this is for: people working on numerical approximation of invariant measures for stable-driven SDEs, and anyone using nonlocal Stein equations. It deserves a serious referee; the novelty and scope justify the time.

Referee Report

3 major / 3 minor

Summary. The paper studies the Euler-Maruyama (EM) approximation of the invariant measure of supercritical stable SDEs with multiplicative noise. Under Hölder regularity of the coefficients and a dissipativity condition, the authors prove Wasserstein convergence rates for the invariant measures of the EM scheme: for α∈(1/2,2) and the bounded-Hölder-type distance d(x,y)=|x-y|^γ∧1, Theorem 1.2 gives a rate of η^{γ∧(2α-1-ε)} for α∈(1/2,1] and η^{γ/α} for α>1, up to logarithmic factors. The proof is based on Stein's method: the authors establish weighted Hölder regularity for solutions of nonlocal Stein/Poisson equations, using two-sided heat-kernel bounds for stable SDEs with unbounded drift, and then estimate the discrepancy between the EM semigroup and the continuous semigroup. An appendix gives matching first-order rates for Ornstein-Uhlenbeck processes, showing that the rates are at least sharp in that model.

Significance. If correct, the main result is the first explicit discretization rate for invariant measures of supercritical stable SDEs with multiplicative noise, going substantially beyond the subcritical case treated by Chen, Deng, Schilling and Xu. The paper also provides a useful Stein-equation regularity theory for nonlocal generators with unbounded coefficients and only Hölder-regular data. Strengths include the explicit nature of the estimates, the use of external heat-kernel and ergodicity theorems rather than fitted constants, and the sharp OU example in Appendix B that calibrates the rate. The two technical gaps identified below concern the rigorous derivation of the Stein equation and one norm-equivalence step; both appear repairable without changing the main claims, but they currently prevent the proof from being fully valid as written.

major comments (3)

[§2.4, Proposition 2.15] The Hille–Yosida argument proving (2.45) is not valid as written. Since A is the generator of a C0-semigroup on C0, one has D(A)⊂C0; however, Theorem 2.14, specifically (2.40), shows that f(x)=∫_0^∞ P_t ḡ(x)dt has logarithmic growth. For non-centered g∈C0, the function ḡ=g-μ(g) does not vanish at infinity, so f is not in C0 and the limits lim_{ε↓0}U_ε ḡ=f and lim_{ε↓0}g_ε=-ḡ are not C0-limits. Thus the closure argument cannot be applied. Because Theorem 2.16 invokes (2.45) for the compactly supported truncations g_n, the pointwise identity Lf=-ḡ — the foundation of the Stein method in Theorems 1.1 and 1.2 — is not rigorously established as written. This gap is likely repairable by differentiating under the integral and using the regularity of Theorem 2.14 together with exponential ergodicity (2.20), but the argument must be supplied.
[§2.3, Lemma 2.13 and equation (2.32)] In the small-time part of Lemma 2.13, Theorem 2.3(i) is applied with β0=(β+γ)∧2. The equivalent-characterization statement in Theorem 2.3 is stated for non-integer β0 only. When β+γ≥2, which can occur for α>1 with γ=1 and β close to α, one has β0=2, an integer, and the inequality sup_{|v|≤1}‖δ^3_v P_t g‖∞/|v|^2 does not by itself justify the C^2-norm bound (2.32). Since (2.32) is used in the large-time estimate (2.34) and in Theorem 2.14, this is a load-bearing gap. The fix is local: replace β0=(β+γ)∧2 by β0=(β+γ)∧(2-ε) with ε small so that β0>α and non-integer, and adjust the subsequent constants and τ accordingly.
[§2.3, Lemma 2.12] In the proof of Lemma 2.12, the estimate E|f(X^{x±v-z}_t)| ≲ (1+|x±v-z|^m)‖f‖_{L∞_m} for m∈[0,α) is asserted as 'standard' without proof or reference. This estimate is used with f equal to [δ_v,L]P_{s+t0}g in Lemma 2.13, so it is needed in a form that supplies the weighted L∞ norm of the commutator term. A short proof or a precise citation should be provided.

minor comments (3)

[§3, Proof of Theorem 1.1] The references '(3.5)' after the estimates for I3 and I4 point to an equation that is introduced only later in the proof of Theorem 1.2; the intended reference appears to be (3.1).
[§2.4, Theorem 2.14] The conclusion ‖f‖_{C^{β+γ}_{log}} is stated for every β∈(0,α), but the proof via Lemma 2.13 and Theorem 2.3(i) requires β+γ to be non-integer. Either state the result for non-integer β+γ only, or add an interpolation argument for the exceptional integer values such as β+γ=2.
[Throughout] The displayed text contains numerous formatting artifacts, such as '/BD' and misplaced inequality signs, which should be cleaned in the production version. The abstract also contains a typo ('Stei n').

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the W_d-rate theorem is a genuine derivation from external heat-kernel and ergodicity estimates; self-citations in [29] and [48] are published theorems with independent proofs and do not smuggle in the target result.

full rationale

The central claim, Theorem 1.2, is not an input or a renamed fit. The proof obtains the rate by combining the Stein/Poisson representation (1.13)-(1.15), the regularity estimate for the Stein solution in Theorem 2.14, and direct EM local-error estimates in Section 3. No parameter is fitted to the target quantity, and the Wasserstein metric d(x,y)=|x-y|^gamma ∧ 1 is chosen with the same gamma as the coefficient Holder exponent, but this is a metric choice, not a reduction of the conclusion to the assumption. The cited results [29] and [48] are external published theorems with proofs under assumptions (H_gamma) and (1.9); those assumptions do not include the W_d-convergence rate, so the citations are independent support rather than circular self-citation. The main derivation is self-contained given those external inputs. The reviewer-identified issue in Proposition 2.15 and Theorem 2.16 is a proof gap, not circularity: f = ∫ P_t g-bar dt is shown by the paper itself to have only logarithmic growth, so it need not lie in C0 = D(A), and the Hille-Yosida resolvent passage in C0 is not justified as written. This is a genuine correctness concern, but it does not equate the theorem with its inputs or relabel a fitted value as a prediction; therefore it does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim depends on several external results taken as black boxes: the heat kernel estimates of Menozzi and Zhang [29], the ergodicity estimates of Zhang and Zhang [48], and the gradient estimate of Liang and Wang [27] in the monotone case. Each is a published theorem with a proof; none is a restatement of the target result. No free parameters are fitted to data, and no new entities are postulated. The authors' own previous results are used as tools, not as circular substitutes for the conclusion.

assumptions (4)

domain assumption Two-sided heat kernel estimates (1.7)-(1.8) from Menozzi and Zhang [29] hold for the SDE (1.2) under (Hγ).
Invoked in Lemma 2.9, Lemma 2.12 and Lemma 2.13 to control the derivative and second-difference of the semigroup; the regularity of the Stein solution in Section 2 depends on it.
domain assumption Exponential ergodicity with m-th moment bounds (2.20) from Zhang and Zhang [48] holds for the semigroup under dissipativity (2.19).
Used in Theorem 2.14 to bound the time integral and in Lemma A.2 to prove the EM chain is ergodic with moment control.
domain assumption Gradient estimate (2.12) from Liang and Wang [27] holds in the monotone case (H1) and (2.11).
Used in Theorem 2.8 to obtain exponential decay of ||∇P_t ϕ||∞, which yields the ||∇f||∞ bound.
standard math Interpolation and norm equivalence for weighted Hölder spaces in Theorem 2.3, taken from Hao, Zhang, Zhu and Zhu [21].
Used throughout Section 2 to transfer bounds between different Hölder exponents; the disputed application with integer β0=2 is part of this reliance.

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Pith. "Pith review of $W_{\bf d}$-convergence rate of EM schemes for invariant measures of supercritical stable SDEs." pith.science (2026). https://pith.science/paper/HIJ3RK6I

@misc{pith2026241109949,
  author       = {Pith},
  title        = {Pith review of: $W_\bf d$-convergence rate of EM schemes for invariant measures of supercritical stable SDEs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HIJ3RK6I}},
  note         = {Machine review of arXiv:2411.09949}
}

abstract

By establishing the regularity estimates for nonlocal Stein/Poisson equations under $\gamma$-order H\"older and dissipative conditions on the coefficients, we derive the $W_{\bf d}$-convergence rate for the Euler-Maruyama schemes applied to the invariant measure of SDEs driven by multiplicative $\alpha$-stable noises with $\alpha \in (\frac{1}{2}, 2)$, where $W_{\bf d}$ denotes the Wasserstein metric with ${\bf d}(x,y)=|x-y|^\gamma\wedge 1$ and $\gamma \in ((1-\alpha)_+, 1]$.

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Discretization, Uniform-in-Time Estimations and Approximation of Invariant Measures for Nonlinear Stochastic Differential Equations with Non-Uniform Dissipativity
math.NA 2025-11 conditional novelty 6.0 of 10

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 dissipativ...

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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