{"id":"4ddbad05-5d1d-4cba-9d47-09c22ead53b8","arxiv_id":"2411.09949","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Euler-Maruyama schemes for supercritical α-stable SDEs converge to the invariant measure at explicit rates in a bounded Hölder Wasserstein metric when the drift is dissipative and Hölder continuous.","lead":"This paper proves explicit convergence rates for Euler-Maruyama approximations of the invariant measure of supercritical stable SDEs with Hölder drift and multiplicative noise. The rates are given in a Wasserstein metric with a bounded Hölder cost, covering stability indices α in (1/2,2).","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 2.15's C0 resolvent argument cannot hold: f=∫P_t ḡ dt has logarithmic growth and is not in D(A)⊂C0, so the proof of the Stein equation L f = -ḡ has a repairable gap.","rationale":"The reader's conditional verdict is appropriate, but the load-bearing weakness is not the one the reader emphasized. In §2.4, Proposition 2.15 tries to solve the Poisson equation in C0, yet the paper itself establishes only logarithmic growth for f, and explicit OU examples show f∉C0. Since Theorem 2.16 uses (2.45) for the compactly supported approximations, the pointwise equation L f = -ḡ is not proven as written. This is an internal inconsistency, not a reliance on an external conjecture. It is nonetheless repairable: one can differentiate P_t ḡ under the integral, use Theorem 2.14's uniform regularity (C^1_log and C^{ζ+γ}_log for ζ+γ>α) and the exponential ergodicity (2.20) to justify the interchange, and obtain L f = -ḡ without invoking C0 generator theory. The integer Hölder issue the reader flagged in Lemma 2.13 is genuine but also repairable by choosing a non-integer β0∈(α,β+γ) in the Volterra argument, so it does not threaten the stated rate. The reliance on the Menozzi–Zhang heat kernel estimates appears consistent with the hypotheses (Lipschitz b and σ∈C^γ with γ>1-α), so I do not treat that as the primary concern. The main theorem should still be accepted conditionally, pending a corrected proof of the Stein equation.","tokens_in":30492,"tokens_out":55316,"duration_ms":524973,"concrete_test":"Take the OU process of Appendix B with d=1, α∈(1,2), and a compactly supported g∈C0 with μ(g)≠0. Set ḡ=g-μ(g) and compute f(x)=∫_0^∞ P_t ḡ(x)dt using the explicit semigroup of (B.1); show f(x)≍log|x| as |x|→∞. This directly contradicts Proposition 2.15's assertion that f∈D(A)⊂C0. If Theorem 2.16 can instead be re-proved by direct differentiation under the integral for the cutoffs g_n, the central claim survives; otherwise the Stein/Poisson equation is not established and the main theorem lacks proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In §2.4, Proposition 2.15 claims that f(x):=∫_0^∞ P_t ḡ(x)dt belongs to D(A) and satisfies A f = -ḡ, where A is the generator of (P_t) in C0, the space of continuous functions vanishing at infinity. This is internally inconsistent: by construction D(A)⊂C0, but the paper's own Theorem 2.14 shows f has only logarithmic growth (|f(x)| ≲ log(e+|x|^2)). For the OU process of Appendix B with compactly supported g and μ(g)≠0, f(x)∼c log|x| as |x|→∞, so f∉C0. Moreover, ḡ=g-μ(g) generally has a nonzero limit at infinity, so the right-hand side of the Poisson equation is not in C0 either. The Hille-Yosida passage from U_ε ḡ to f in C0 therefore fails. Theorem 2.16 invokes (2.45) for the compactly supported cutoffs g_n, so the pointwise identity L f = -ḡ, which is the foundation of the Stein method in Theorem 1.2, is not rigorously established as written. The gap is likely repairable by a direct differentiation under the integral combined with the regularity estimates of Theorem 2.14 and exponential ergodicity, but it is a genuine missing step in the current proof.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":30798,"tokens_out":11678,"duration_ms":105292,"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":[{"comment":"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.","section":"§2.4, Proposition 2.15"},{"comment":"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.","section":"§2.3, Lemma 2.13 and equation (2.32)"},{"comment":"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.","section":"§2.3, Lemma 2.12"}],"minor_comments":[{"comment":"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).","section":"§3, Proof of Theorem 1.1"},{"comment":"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.","section":"§2.4, Theorem 2.14"},{"comment":"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').","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The two gaps identified above are technical and localized: the C0-resolvent argument in Proposition 2.15 is not valid as stated, and the norm-equivalence step in Lemma 2.13 has an integer-index problem. Both are plausibly repairable within the manuscript's scope, and I would not reject on the basis of the current text. The revision must, however, supply a rigorous proof of the Stein equation identity Lf=-ḡ and a corrected C^{β0} argument in Lemma 2.13. The paper is otherwise a strong contribution to the quantitative sampling literature for supercritical stable SDEs."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"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.\n\nWhat'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.\n\nSoft spots, in order:\n\n1. 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.\n\n2. 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.\n\nBoth 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.\n\nWho 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.","headline":"First quantitative EM invariant-measure rates for supercritical stable SDEs; two repairable but real gaps in the Stein-equation and regularity arguments.","tokens_in":31351,"tokens_out":2819,"would_cite":true,"duration_ms":28111,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H35","60J75","60G52","65C30"],"pacs":[],"model":"deepseek-v4-flash","headline":"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).","keywords":["stable processes","Euler-Maruyama scheme","invariant measure","Stein/Poisson equation","weighted Hölder spaces","Wasserstein distance","supercritical SDEs","heavy-tailed sampling"],"falsifier":"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.","tokens_in":30296,"feed_emoji":"","tokens_out":5176,"duration_ms":51240,"temperature":0.7,"pith_summary":"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.","feed_headline":"Explicit error rate for stable-noise EM sampling","feed_subtitle":"Euler–Maruyama invariant measures converge to the true target at order η^{gamma∧(2α−1−ε)}, even for supercritical noise.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the two-sided heat-kernel estimates (1.7)–(1.8) for the SDE density with unbounded drift, which underpin the short-time regularity Lemmas 2.9, 2.12, and 2.13.","marker":"[29]"},{"why":"Establishes existence and uniqueness of the invariant measure μ and the exponential ergodicity bound (2.20) used to control large-time semigroup decay.","marker":"[48]"},{"why":"Provides the Stein-method framework for bounding Wasserstein distance between invariant measures through the equation $\\mathcal{L}f=\\mu(g)-g$.","marker":"[18]"},{"why":"Gives the prior Euler–Maruyama invariant-measure analysis for additive-noise subcritical stable SDEs; this paper extends it to supercritical multiplicative noise.","marker":"[8]"},{"why":"Supplies the gradient estimate (2.12) for the subcritical monotone-drift case, used in proving Theorem 2.8 and the Wasserstein-1 rate.","marker":"[27]"},{"why":"Supplies the discrete-time stability and ergodicity criterion used in Lemma A.2 to show the EM chain has a unique invariant measure with exponential convergence.","marker":"[30]"}],"fun_headline_variants":["EM sampling of stable SDEs: explicit invariant-measure rate","Explicit EM rates for supercritical stable SDE invariant measures","Hölder regularity yields EM convergence rates for stable noises","First explicit discretization rate for supercritical stable SDEs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["EM sampling of stable SDEs: explicit invariant-measure rate","Explicit EM rates for supercritical stable SDE invariant measures","Hölder regularity yields EM convergence rates for stable noises","First explicit discretization rate for supercritical stable SDEs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001709,"raw_usage":{"total_tokens":6747,"prompt_tokens":911,"completion_tokens":5836,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":527,"completion_tokens_details":{"reasoning_tokens":5765}},"tokens_in":527,"tokens_out":5836,"duration_ms":42000,"temperature":1.0,"reasoning_tokens":5765,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T20:09:38.615977+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Menozzi, and X","cited_arxiv_id":null,"evidence_quote":"Supplies the two-sided heat-kernel estimates (1.7)–(1.8) for the SDE density with unbounded drift, which underpin the short-time regularity Lemmas 2.9, 2.12, and 2.13."},{"cited_title":"Zhang and X","cited_arxiv_id":null,"evidence_quote":"Establishes existence and uniqueness of the invariant measure μ and the exponential ergodicity bound (2.20) used to control large-time semigroup decay."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Stein-method framework for bounding Wasserstein distance between invariant measures through the equation $\\mathcal{L}f=\\mu(g)-g$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the prior Euler–Maruyama invariant-measure analysis for additive-noise subcritical stable SDEs; this paper extends it to supercritical multiplicative noise."},{"cited_title":"Liang and J","cited_arxiv_id":null,"evidence_quote":"Supplies the gradient estimate (2.12) for the subcritical monotone-drift case, used in proving Theorem 2.8 and the Wasserstein-1 rate."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the discrete-time stability and ergodicity criterion used in Lemma A.2 to show the EM chain has a unique invariant measure with exponential convergence."}],"review_version":1}