{"id":"6f380bdd-ca80-4b7a-9a15-4aa5773c43f6","arxiv_id":"2505.03104","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":1,"one_line_summary":"For SDEs with superlinearly growing drift and non-degenerate multiplicative noise, the variable-step tamed Euler-Maruyama scheme converges uniformly in time at rate O(eta_n^alpha) in Wasserstein and total variation distance, for any alpha in (0,1/2).","lead":"This paper proves that a tamed Euler-Maruyama scheme, using decreasing step sizes, approximates the law of a wide class of stochastic differential equations for all times with an explicit convergence rate. The rate is the step size raised to any power below one half, measured both in Wasserstein and total variation distance.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the proof of Theorem 1.1 is internally consistent under A1–A3; A2 is a strong but explicit modeling restriction rather than a flaw.","rationale":"The reader's ACCEPT with moderate confidence is reasonable. I examined the main estimates: Lemma 2.3's Lyapunov bound for the tamed process, Lemma 2.4's one-step error, Lemma 2.8's gradient/Hessian bounds, and the summation in Theorem 1.1. The only point that initially seemed delicate—the use of Lemma 2.2 in Lemma 2.3—is resolved by a scaling argument (apply it to 3ξ), and the constants work for sufficiently small η. The TV result follows from the C^2_b bound with the min of sup and Lipschitz norms via standard density. The theorem is honest about A2; the abstract's phrase 'multiplicative noise' could overstate generality, but the theorem itself is precisely scoped. Hence UNCHANGED.","tokens_in":30790,"tokens_out":35040,"duration_ms":326714,"concrete_test":"Re-derive Lemma 2.3 with σ(x)=I+0.1 sin(x) diag (A2-compliant) and b(x)=-x^3, and numerically evaluate the one-step inequality (2.5) for η=10^{-4}, 10^{-3}, 10^{-2} and α=0.25, checking that E[V(Y_{t_n})^3] stays uniformly bounded (λ'>0). This directly tests the contraction estimate (2.7) that underlies the uniform moment bound and the domino summation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After a careful pass through the proof, I find no load-bearing error. The domino decomposition (3.1), the one-step estimates in Lemma 2.4, the semigroup gradient bounds in Lemma 2.8, and the summation Lemma 3.2 fit together: the |ln η_n| factor from Lemma 3.2(iii) is absorbed by η_n^{1/2}|ln η_n| ≤ C η_n^α for any α<1/2, and the final TV step uses the min of the ||f||∞ and ||∇f|| bounds, so the standard approximation from C^2_b test functions is legitimate. The conditional Gaussian estimates in Lemma 2.3 require applying Lemma 2.2 to the scaled variable 3ξ; this is not spelled out but works with η||Σ||≤1/54, consistent with η1≤η. Assumption A2 is the most restrictive condition and does exclude degenerate or growing diffusion coefficients, but it is stated clearly and is used essentially; that is a scope limitation, not an internal gap.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves uniform-in-time distributional approximation rates for a tamed Euler-Maruyama scheme (1.2) applied to the SDE (1.1) with polynomially growing, dissipative drift and multiplicative noise. Under Assumptions A1-A3 (dissipativity and polynomial growth of b; bounded, uniformly elliptic C^2 diffusion coefficient sigma; non-increasing step sizes satisfying eta_{n-1}-eta_n <= theta eta_n^2), Theorem 1.1 gives W1 and total-variation error bounds of order eta_n^alpha for every alpha in (0,1/2), uniformly in n, and Theorem 1.2 states the additive-noise counterpart. The proof combines a domino decomposition (3.1), one-step error estimates (Lemma 2.4), uniform moment bounds for X and Y (Lemmas 2.1 and 2.3), Bismut-Elworthy-Li gradient estimates for the semigroup of X (Lemma 2.8), and summation estimates for variable step sequences (Lemma 3.2).","tokens_in":30978,"tokens_out":30991,"duration_ms":271538,"significance":"If correct, the result supplies a long-time, uniform-in-time convergence theory for an explicit EM-type scheme in the multiplicative-noise setting, including a total-variation rate, which is considerably stronger than typical finite-time L^p convergence results. The proof is detailed and largely self-contained, with the main estimates assembled cleanly: the rate is stated as an explicit function of the step-size sequence, and the dependence of constants on the problem data is tracked. The principal limitation is the strength of Assumption A2, which globally bounds sigma and requires uniform ellipticity; this excludes degenerate or growing diffusions, but it is explicitly stated and is used essentially in the semigroup gradient estimates and the Gaussian integration-by-parts argument.","major_comments":[],"minor_comments":[{"comment":"In the proof around (2.8), the decomposition of U(Y_{t_n}) uses the ball B(mu,1/9), while Lemma 2.2 is then applied with B(mu,1/3). Please clarify that Lemma 2.2 is applied to the scaled variable 3Y_{t_n}; after scaling, the relevant ball is B(3mu,1/3) and the smallness condition becomes eta_n ||Sigma|| <= 1/54, not merely 1/6. The current text is confusing and the radius mismatch should be corrected.","section":"Lemma 2.3"},{"comment":"The condition in the statement reads 'theta < c e^{-c/beta}', which is ambiguous. The proof requires theta < c e^{-c}/beta, that is, beta appears in the denominator of the bound, not in the exponent. Please restate the hypothesis precisely.","section":"Lemma 3.2"},{"comment":"The absorption of eta_n^{1/2}|ln eta_n| into eta_n^alpha uses eta_n <= 1 and alpha < 1/2. It would be helpful to state explicitly that finitely many early indices, for which eta_n is not small, are absorbed into the constant C.","section":"Proof of Theorem 1.1, after (3.6)"},{"comment":"The 'standard approximation method' that passes from the C^2_b test-function estimate (3.8) to the W1 and total-variation bounds is only mentioned in one sentence. A short explanation of how bounded measurable and Lipschitz test functions are approximated by C^2_b functions with controlled norms would make the argument fully self-contained.","section":"Section 3, beginning"},{"comment":"There are several typographical and OCR-style artifacts, such as 'TAMED EULER-MARUY AMA' in the running title, 'F or ad-dimensional' at the start of Lemma 2.2, and assorted broken ligatures in the displayed text. A careful copyedit is needed.","section":"Throughout"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is within the scope of the journal and the main theorems appear correct. Assumption A2 is strong, but it is explicitly stated and used essentially, so it is a scope limitation rather than a flaw. I see no citation-pattern or novelty concerns. The remaining issues are local presentation problems, hence minor revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Punchline: this is a genuinely new uniform-in-time W1/TV convergence result for a known explicit tamed scheme, and the proof is in good shape; the real cost is a restrictive but clearly stated uniform ellipticity/boundedness assumption on the diffusion.\n\nWhat is new: the tamed EM scheme (1.2) is Sabanis's, but the infinite-horizon analysis with decreasing step sizes is new. Theorems 1.1 and 1.2 give W1 and TV rates O(η_n^α) for any α∈(0,1/2), uniformly in n, under partial dissipation. That is exactly the gap left by the finite-time Lp result [13] and the divergence result [8]. The proof is a domino decomposition plus Bismut-Elworthy-Li semigroup estimates plus careful summation (Lemma 3.2); all the pieces fit. I read the proof and found no load-bearing error. The stress-test note is right: A2 is a scope restriction, not a hidden gap.\n\nSoft spots: (1) A2 is strong: sigma globally bounded with bounded derivatives and sigma^{-1} uniformly bounded. Degenerate or growing noise is excluded. That should be front and center. (2) The rate is η_n^α with α<1/2, and the same α appears in the taming; it does not reach the usual finite-time order 1/2. Fine, but calibrate expectations. (3) Minor: in Lemma 2.3 the application of Lemma 2.2 to the scaled variable 3ξ is not fully spelled out; the implied condition η‖Σ‖≤1/54 holds for η small but should be stated. The passage from C^2_b test functions to Lip(1)/B_b is routine but terse. (4) No numerical illustrations; this is a pure proof paper, so no data to check. The citation pattern is healthy: no self-citations, and the contrast with [13], [8], and [7] is accurate.\n\nWho this is for: anyone analyzing explicit schemes for superlinear SDEs over long horizons, especially for sampling/ergodic applications. I would send it to a serious referee and expect acceptance after minor revisions.","headline":"Genuinely new infinite-horizon W1/TV rates for tamed EM with multiplicative noise; proof holds up, main caveat is the strong but explicit uniform ellipticity assumption on sigma.","tokens_in":31520,"tokens_out":4258,"would_cite":true,"duration_ms":39744,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H10","65C30","60J60"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that a tamed Euler–Maruyama scheme for SDEs with non-globally Lipschitz drift and multiplicative noise has uniform-in-time convergence rates of order $\\eta_n^\\alpha$ under the $L^1$-Wasserstein and total-variation…","keywords":["SDEs with polynomially growing drift","tamed Euler-Maruyama scheme with decreasing step","Wasserstein distance","total variation distance","convergence rate","multiplicative noise","uniform-in-time convergence"],"falsifier":"Check whether the claimed uniform-in-time rate persists when the assumption on $\\sigma$ is minimally violated: take a one-dimensional SDE with a dissipative polynomial drift, e.g. $b(x)=-x^3-x$, and a bounded but not uniformly elliptic diffusion such as $\\sigma(x)=e^{-x^2}$, run the tamed scheme with $\\eta_n=n^{-1/2}$, and measure $\\sup_n W_1(\\mathcal L(X_{t_n}),\\mathcal L(Y_{t_n}))//\\eta_n^\\alpha$ for a fixed $\\alpha\\in(0,1/2)$; if the ratio diverges with $n$, the uniform-in-time claim genuinely depends on uniform ellipticity, while if it stays bounded, the assumption is stronger than necessary.","tokens_in":30566,"feed_emoji":"🎯","tokens_out":8408,"duration_ms":75081,"temperature":0.7,"pith_summary":"This paper establishes that a simple, explicit tamed Euler–Maruyama scheme—where the drift increment is divided by $1+\\eta_n^\\alpha\\|\\nabla b\\|_{\\mathrm{op}}$—approximates the true process of an SDE with a superlinearly growing, dissipative drift and multiplicative noise not only on finite intervals but uniformly over the whole infinite time horizon. The main theorems give, for any $\\alpha\\in(0,1/2)$, a constant $C$ such that both the $L^1$-Wasserstein distance and the total-variation distance between the law of the true solution at time $t_n$ and the law of the scheme are at most $C\\eta_n^\\alpha$ for every $n$. The rates are obtained for variable step sizes satisfying a mild decay condition, and the additive-noise case is covered as a separate theorem. The significance is that the tamed scheme is explicit and easy to implement, so it offers a practical way to simulate the long-time distributional behavior of SDEs whose drift lies outside the globally Lipschitz class.","feed_headline":"Tamed Euler scheme is uniformly accurate in time for stiff SDEs","feed_subtitle":"Both Wasserstein and total-variation errors decay like $C\\eta_n^\\alpha$ at every step, forever.","key_machinery":"The proof is carried by the domino decomposition, which telescopes the total discrepancy into a sum over steps: $P_{0,t_n}f(x_0)-Q_{0,t_n}f(x_0)=\\sum_{k=1}^n Q_{0,t_{k-1}}(P_{t_{k-1},t_k}-Q_{t_{k-1},t_k})P_{t_k,t_n}f(x_0)$. The paper combines this decomposition with gradient estimates for the continuous semigroup obtained from the Bismut–Elworthy–Li integration-by-parts identity, where the uniform ellipticity of $\\sigma$ gives a bound on $\\sigma^{-1}$ used in that identity, and with a Gaussian one-step error estimate for the tamed step. The gradient estimates decay exponentially in $t_n-t_k$, so the sum over $k$ is controlled by a new summation lemma (Lemma 3.2) that converts the time-decaying terms into an overall $\\eta_n^\\alpha$. Moment estimates for both processes at all orders, via the exponential Lyapunov function $V(x)=e^{|x|}$ near infinity, keep the unbounded polynomial factors under control at every time.","core_discovery":"The paper's central claim is that the taming denominator $1+\\eta_n^\\alpha\\|\\nabla b(Y_{t_{n-1}})\\|_{\\mathrm{op}}$ in the Euler increment is enough to control the superlinear drift for all times, not just up to a fixed horizon. Under the three assumptions (dissipative polynomial drift, uniformly bounded and uniformly elliptic diffusion with bounded first and second derivatives, and variable step sizes that are nonincreasing, tend to zero, have divergent sum, and satisfy $\\eta_{n-1}-\\eta_n\\le \\theta\\eta_n^2$), and with a $C^2$ drift whose Hessian grows at most polynomially, the discrepancy between the true Markov semigroup and the discrete semigroup is bounded by $C\\eta_n^\\alpha$ in both $W_1$ and total variation, uniformly in $n$. The additive-noise version relaxes the one-step error but keeps the same conclusion.","pith_inferences":["Combining the paper's bounds with the exponential ergodicity of the true SDE (which its lemmas establish) implies that the law of $Y_{t_n}$ reaches a neighborhood of the invariant measure at the same rate $\\eta_n^\\alpha$; the paper does not frame the result this way, but it makes the tamed scheme a viable explicit sampler for the ergodic limit.","If the uniform ellipticity of $\\sigma$ were relaxed to a hypoelliptic or degenerate setting, the total-variation conclusion would likely fail because the Gaussian integration-by-parts step in Lemma 3.1 uses $\\sigma^{-1}$; testing whether the Wasserstein rate survives under weaker nondegeneracy would separate the two metrics' requirements.","The same domino decomposition with an integration-by-parts gradient estimate should transfer to other explicit locally Lipschitz schemes, such as truncated, adaptive, or implicit Euler–Maruyama methods, giving uniform-in-time distributional rates under the same dissipativity and ellipticity conditions."],"forward_implications":["The tamed scheme is uniformly accurate in distribution over the infinite horizon: at any fixed large $n$, the law of the discrete process is within $C\\eta_n^\\alpha$ of the law of the SDE in both Wasserstein and total-variation distance, with a constant independent of $n$.","Variable step sizes are allowed: any nonincreasing sequence $\\eta_n\\downarrow 0$ with divergent sum and step-difference bound $\\eta_{n-1}-\\eta_n\\le\\theta\\eta_n^2$, such as $\\eta_n=\\eta/n^\\gamma$, inherits the same rate.","For additive noise $\\sigma\\equiv\\sigma_0$, the same result holds, and the proof uses the sharper one-step error $O(\\eta_k^{2+2\\alpha})$ for intermediate steps.","The uniform bound applies at every discrete time $t_n$, so long-time simulations with the tamed scheme remain distributionally close to the true solution at every step rather than drifting away as $n$ grows."],"supporting_citations":[{"why":"Introduces the tamed Euler approximation and its finite-time $L^p$ convergence, which is the starting point for the long-time analysis.","marker":"[13]"},{"why":"Shows that explicit Euler may diverge for superlinearly growing drift, motivating the need for tamed schemes.","marker":"[8]"},{"why":"Supplies the Bismut–Elworthy–Li formula used to bound derivatives of the SDE semigroup.","marker":"[1]"},{"why":"Provides the derivative formula for heat semigroups used in the gradient estimates.","marker":"[3]"},{"why":"Gives the ergodicity estimate used to obtain exponentially decaying gradient bounds for large times.","marker":"[7]"},{"why":"Gives existence and uniqueness of strong solutions under the stated assumptions.","marker":"[12]"},{"why":"Provides the Kantorovich–Rubinstein duality used to reduce Wasserstein and total-variation estimates to test functions.","marker":"[16]"}],"fun_headline_variants":["Tamed Euler yields uniform-in-time Wasserstein and TV rates","Explicit tamed method: uniform error decay for stiff SDEs","Uniformly accurate tamed Euler for superlinear SDEs","Stiff SDEs? Tamed Euler stays accurate at every step","Tamed Euler handles non-Lipschitz drift with uniform rates"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof needs the diffusion matrix $\\sigma$ to be globally bounded, uniformly elliptic ($\\sigma^{-1}$ uniformly bounded), and twice continuously differentiable; if the multiplicative noise can vanish, grow, or have unbounded derivatives, the gradient estimates and the total-variation control collapse.","fun_headline_variants_meta":{"raw":{"variants":["Tamed Euler yields uniform-in-time Wasserstein and TV rates","Explicit tamed method: uniform error decay for stiff SDEs","Uniformly accurate tamed Euler for superlinear SDEs","Stiff SDEs? Tamed Euler stays accurate at every step","Tamed Euler handles non-Lipschitz drift with uniform rates"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000289,"raw_usage":{"total_tokens":1661,"prompt_tokens":882,"completion_tokens":779,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":498,"completion_tokens_details":{"reasoning_tokens":688}},"tokens_in":498,"tokens_out":779,"duration_ms":6384,"temperature":1.0,"reasoning_tokens":688,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:01:37.934017+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check whether the claimed uniform-in-time rate persists when the assumption on $\\sigma$ is minimally violated: take a one-dimensional SDE with a dissipative polynomial drift, e.g. $b(x)=-x^3-x$, and a bounded but not uniformly elliptic diffusion such as $\\sigma(x)=e^{-x^2}$, run the tamed scheme with $\\eta_n=n^{-1/2}$, and measure $\\sup_n W_1(\\mathcal L(X_{t_n}),\\mathcal L(Y_{t_n}))//\\eta_n^\\alpha$ for a fixed $\\alpha\\in(0,1/2)$; if the ratio diverges with $n$, the uniform-in-time claim genuinely depends on uniform ellipticity, while if it stays bounded, the assumption is stronger than necessary.","supporting_citations":[{"cited_title":"1905, Springer, 2007","cited_arxiv_id":null,"evidence_quote":"Gives existence and uniqueness of strong solutions under the stated assumptions."},{"cited_title":"Notes Pure Appl","cited_arxiv_id":null,"evidence_quote":"Gives the ergodicity estimate used to obtain exponentially decaying gradient bounds for large times."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the tamed Euler approximation and its finite-time $L^p$ convergence, which is the starting point for the long-time analysis."},{"cited_title":"2130, 1563–1576","cited_arxiv_id":null,"evidence_quote":"Shows that explicit Euler may diverge for superlinearly growing drift, motivating the need for tamed schemes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Bismut–Elworthy–Li formula used to bound derivatives of the SDE semigroup."},{"cited_title":"1, 252–286","cited_arxiv_id":null,"evidence_quote":"Provides the derivative formula for heat semigroups used in the gradient estimates."},{"cited_title":"58, American Mathematical Society, Providence, RI, 2003","cited_arxiv_id":null,"evidence_quote":"Provides the Kantorovich–Rubinstein duality used to reduce Wasserstein and total-variation estimates to test functions."}],"review_version":1}