{"id":"577a5f0b-3f72-4e35-981e-5ed0c0af2f72","arxiv_id":"2412.10890","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A unified hypocoercivity framework shows adaptive Langevin dynamics is a near-optimal lift, and proves the generalized Langevin equation cannot beat square-root speedup.","lead":"This paper unifies two frameworks for proving exponential convergence of kinetic equations to equilibrium, and applies them to adaptive Langevin dynamics. It also shows that the generalized Langevin equation cannot converge faster than the square root of the overdamped rate, with explicit numbers in a Gaussian benchmark.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"ALD near-optimality rests on the unproved Proposition 3.1 and its omitted boundary conditions; unless the rescaling to [30, Theorem 5] is fully verified, the ALD upper-bound claim is unsupported.","rationale":"The reader's weakest assumption is the correct one. The abstract framework and Theorem 1 are presented transparently, and the GLE lower-bound observation requires only Assumptions 1-2, which are verified in Lemma 4.1; that part of the paper is self-contained and convincing. The ALD upper-bound claim is the place where the manuscript relies on an unproved external statement, Proposition 3.1, and the boundary conditions required by Assumption 5 are not explicitly included in the proposition. This makes the ALD near-optimality claim conditional rather than established, but nothing in the present pass suggests the cited theorem is wrong. A self-contained proof or an explicit verification of the rescaling, constants, and boundary conditions would settle the matter. The appropriate verdict remains conditional acceptance, matching the reader's assessment.","tokens_in":19549,"tokens_out":21475,"duration_ms":195137,"concrete_test":"Re-derive Proposition 3.1 from [30, Theorem 5] for the ALD operator (18), writing z̃ = ε z / √(2d) and exhibiting the map between the divergence equation and the theorem's hypotheses, including the boundary conditions. Verify that the constants (30) follow exactly with c0 = 2T² + 43Px⁻¹ and c1 = 290 + 991T⁻²Px⁻¹ + 43 max(Px⁻¹, T²/π²)M, and that the term ε⁻¹ z v·∇q φ1 in (13) is controlled by (28)-(29). If the cited theorem yields different constants, or if it only provides boundary conditions incompatible with φ0(0)=φ0(T)=0 and Tφ1(0)=Tφ1(T)=0, the ALD claim is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central new upper-bound result is the ALD near-optimality claim in Section 3. It is obtained by applying Theorem 1, whose only non-structural input is Assumption 5: existence of φ0, φ1 for the divergence equation (11) with the boundary conditions φ0(0)=φ0(T)=0 and Tφ1(0)=Tφ1(T)=0 and the bounds (12)-(13). For ALD this is dispatched by Proposition 3.1, which is not proved in the manuscript but imported from [30, Theorem 5] and [20,18]. The proposition as stated only gives the regularity estimates (28)-(29); it does not explicitly assert the boundary conditions needed for the integration by parts in Lemma 2.1. Moreover the constants c0, c1 in (30) depend on the rescaling z̃ = ε z / √(2d) and on Px = min(Pq, 2d/ε²); a mismatch in that rescaling or in the T-dependence of c1 would change the optimized rate by a non-universal factor. If Proposition 3.1 fails, the ALD rate bound and the 'near-optimal second-order lift' conclusion are unsupported. The GLE lower bound in Section 4 is independent of Assumption 5 and is not affected.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper unifies the Albritton–Armstrong–Mourrat–Novack variational hypocoercivity framework with the Eberle–Lörler concept of second-order lifts of reversible diffusions. It states a general Theorem 1 under Assumptions 1–5, giving an explicit exponential decay rate in the untwisted L2 norm, and proves it via a time-averaged energy and an averaging lemma (Lemma 2.1). The main applications are: (i) adaptive Langevin dynamics (ALD), for which the paper claims, for near-quadratic potentials and suitable parameters, near-optimality as a second-order lift of overdamped Langevin dynamics; and (ii) the generalized Langevin equation (GLE), for which a square-root lower bound on acceleration is obtained via the general lift lower bound, with an explicit Gaussian benchmark showing a 1.93-optimal lift. The abstract theorem and the GLE lower bound are self-contained, while the ALD upper bound relies on an external divergence-equation result (Proposition 3.1 imported from [30] and [20,18]).","tokens_in":19832,"tokens_out":12261,"duration_ms":105722,"significance":"If all claims hold, the paper is a valuable contribution: it gives a clean, fully constructive presentation of variational hypocoercivity through the lens of lifts, recovers sharp rates for Langevin-type dynamics, and provides a surprising negative result for GLE (no more than square-root acceleration). The proof of Theorem 1 is short and self-contained once Assumptions 1–5 are granted, and the Gaussian GLE computation is explicit and reproducible. The GLE observation that Assumptions 1–2 hold with a degenerate dissipation is elegant and rigorous. However, the headline ALD near-optimality claim depends crucially on Proposition 3.1, which is not proved in the manuscript; because Assumption 5 is the only non-structural input to Theorem 1, the ALD upper-bound claim is only as strong as that imported proposition and the asserted boundary conditions.","major_comments":[{"comment":"Proposition 3.1 is the only verification of Assumption 5 for ALD, and Assumption 5 is the only non-structural input to Theorem 1. Yet the proposition is not proved in this manuscript: it is imported from [30, Theorem 5] and [20,18], with only a verbal rescaling z̃ = εz/√(2d). In addition, the statement of Proposition 3.1 gives the regularity estimates (28)–(29) but does not explicitly include the boundary conditions φ0(0,·)=φ0(T,·)=0 and Tφ1(0,·)=Tφ1(T,·)=0 required by Assumption 5; the paragraph preceding the proposition asserts that these follow from [20,18] without proof. Since the integration by parts in the proof of Lemma 2.1, specifically the passage in (23), uses exactly those boundary conditions, the ALD upper-bound estimate is not self-contained as written. A mismatch in the rescaling or in the domain hypotheses of [30, Theorem 5] would alter c0 and c1 in (30) and hence the optimized rate, so this gap is load-bearing for the paper's central ALD near-optimality claim.","section":"Section 3, Proposition 3.1"},{"comment":"The conclusion that ALD is 'up to a universal constant, an optimal second-order lift' for 'convex potentials that are not far from quadratic' is not stated as a quantified theorem. The paper sets M=0 and L≈Pq informally, but 'near-quadratic' is never defined, and the final lower bound λ ≥ Pq/(66334√(Pq+M+L)) depends on L. Remark 3.2 explicitly concedes that the L-dependence may be an artifact of the proof technique and that the authors cannot recover the L-free rates of [20] for standard Langevin dynamics. Unless the class of potentials is precisely specified (for instance, M=0 and L ≤ C Pq for a universal constant C) or the L-dependence is removed, the headline claim in the abstract is stronger than what the displayed inequalities prove.","section":"Section 3, final paragraph and Remark 3.2"},{"comment":"The assertion that (12) and (13) follow from (28)–(29) 'since their left-hand sides only involve at most first derivatives of φ0, first and second derivatives of φ1, and up to eighth moments of v against the standard Gaussian' is not a complete argument. The estimates (12)–(13) need to hold for every zero-mean g ∈ L2(¯µ) with constants uniform in g, and (29) does not control all terms appearing in (13), in particular the mixed terms involving ∇U · ∇qφ1 and zBzφ1 that are later handled using [20, Lemma 2.2] and the structural assumptions on U. Since the authors are already computing the constants in (30) explicitly, the missing step is to verify that the solution produced by [30, Theorem 5] satisfies both (28)–(29) and the boundary conditions in Assumption 5; this should be presented in the manuscript rather than delegated.","section":"Section 3, Proposition 3.1 and estimate (12)-(13)"}],"minor_comments":[{"comment":"There is a typo in the abstract: 'is a also second-order lift' should read 'is also a second-order lift'.","section":"Abstract"},{"comment":"The lower-bound constants are inconsistent: Remark 1.1 states trel(ˆP) ≥ 1/√2 P_x^{-1/2} by [29, Theorem 11], while Corollary 2 states trel(ˆP) ≥ 1/2 P_x^{-1/2}. Since Example 4.2 compares with 0.964 m^{-1/2} against 1/2 m^{-1/2} to obtain the 1.93-optimality factor, the correct constant from [29] should be used consistently; otherwise the numerical optimality factor may change.","section":"Remark 1.1 and Corollary 2"},{"comment":"The sentence 'By a factorisation argument, the same holds true for arbitrary Gaussian probability measures on Rd' is too terse; a one-sentence justification or a citation would clarify how the scalar computation transfers to the multivariate Gaussian case.","section":"Example 4.2"},{"comment":"In the statement of Theorem 1, the notation R and Pv is used in (14) but these constants are only defined later in Assumption 4 and (9); adding a short parenthetical reminder would improve readability.","section":"Equation (14)"}],"recommendation":"major_revision","confidential_remarks":"The central abstract theorem and the GLE lower bound are sound and well-presented. The ALD section, however, is not self-contained: Proposition 3.1 is imported from a companion preprint by one of the authors ([30]) and from the authors' own prior works ([20,18]), and the boundary conditions required by Assumption 5 are asserted but not proved. This is a novelty and independence concern for the main application, not just a presentation issue. I would encourage the authors to include a complete proof of Proposition 3.1, or at least a detailed verification of the hypotheses of [30, Theorem 5] in the ALD setting, before the paper is considered for publication. The inconsistent lower-bound constant in Remark 1.1 versus Corollary 2 should also be corrected."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The genuinely new content is Sections 3 and 4. Section 3 treats adaptive Langevin dynamics (ALD) as a second-order lift, computes explicit constants, and shows that for near-quadratic potentials with suitable parameters, ALD achieves rate ~ c sqrt(P_q), hence is a near-optimal lift of overdamped Langevin. Section 4 makes a nice observation: generalized Langevin dynamics is also a second-order lift, so its relaxation time cannot beat a square root in the Poincare constant of the position marginal; the Gaussian benchmark shows it is 1.93-optimal, beating standard Langevin on the same class. These are new, and worth citing.\n\nThe abstract framework in Section 2 is a clean, honest unification of the variational hypocoercivity of [2] and the lifts of [29], with a short self-contained proof of Theorem 1. The proof of Lemma 2.1 is clear. The GLE part is independent of Assumption 5 and is the most robust part of the paper. The optimality comparison uses [29, Theorem 11] as an external lower bound, so no circularity in the main claim.\n\nThe soft spot is exactly the ALD section. The near-optimality depends on Proposition 3.1, imported from [30], a preprint sharing an author, and from [20,18]. The proposition as stated gives regularity estimates (28)-(29) but does not explicitly state the boundary conditions required by Assumption 5. The text says those are clear from [20,18], and the constants c0, c1 are stated without derivation. A referee needs to verify the boundary conditions and the rescaling; without them, the integration by parts in Lemma 2.1 fails and the ALD rate collapses. I don't think this is fatal—the claimed estimates are plausible and the subsequent algebra checks—but it is a load-bearing import, not a proof.\n\nMinor: Remark 3.2 concedes an L-dependence; for near-quadratic L ~ P_q this is harmless, but it means the rate is not fully sharp in the general case.\n\nWho is this for? Anyone working on quantitative convergence rates for kinetic MCMC or on non-reversible lifts. It deserves a serious referee. I would send it out with a request that the authors either prove Proposition 3.1 inside the paper or give a precise statement with the boundary conditions verified. If that lands, accept.","headline":"A clean, honest unification with two genuinely new applications—ALD near-optimality and a GLE square-root lower bound—but the ALD part rests on an imported Proposition 3.1 that deserves a hard look.","tokens_in":20394,"tokens_out":3757,"would_cite":true,"duration_ms":33519,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60J60","60J25","82C31"],"pacs":[],"model":"deepseek-v4-flash","headline":"Adaptive Langevin dynamics is a near-optimal second-order lift of overdamped Langevin dynamics.","keywords":["hypocoercivity","second-order lifts","kinetic Fokker-Planck equations","adaptive Langevin dynamics","generalised Langevin equation","relaxation time","convergence rates","Poincaré inequality"],"falsifier":"A concrete check is numerical: simulate ALD for convex near-quadratic potentials, for example $U(q)=q^2/2+\\alpha q^4$ with small $\\alpha$, tune $\\epsilon$ and $\\gamma$ as prescribed, and measure the $L^2$ decay rate as $P_q\\to 0$; if the best observed rate is $o(\\sqrt{P_q})$, the near-optimality claim is false. For the Gaussian GLE, the relaxation time $0.964\\,m^{-1/2}$ at the optimal parameters $\\lambda=2\\sqrt{2m}$ and $\\gamma=3\\sqrt{3m}$ is directly checkable, and any second-order lift with relaxation time below $2^{-1/2}P_x^{-1/2}$ would contradict the paper's Corollary 2.","tokens_in":19317,"feed_emoji":"⚡","tokens_out":11864,"duration_ms":100436,"temperature":0.7,"pith_summary":"This paper draws a direct line between two previously separate ideas: variational hypocoercivity, which proves exponential decay for kinetic equations by averaging in time, and second-order lifts, which describe when a non-reversible process is a kinetic acceleration of a reversible diffusion. Its main theorem turns five structural assumptions into an explicit rate $\\lambda = 2P_v/(1+(C_{1,T}+C_{0,T}\\sqrt{RP_v})^2)$ in the untwisted norm $L^2(\\hat\\mu)$, so the constants can be compared with lower bounds instead of being hidden by a norm change. On that basis the paper claims that adaptive Langevin dynamics, for near-quadratic potentials with tuned parameters, is a near-optimal second-order lift of overdamped Langevin dynamics, converging at order $\\sqrt{P_q}$ up to a universal constant. It further claims that the generalised Langevin equation is also a second-order lift, so it cannot converge faster than the square root of the overdamped rate; in a Gaussian benchmark the optimal parameters give a relaxation time about $0.964\\,m^{-1/2}$, within a factor of $1.93$ of the theoretical floor.","feed_headline":"Adaptive Langevin dynamics nearly attains optimal lift speed","feed_subtitle":"Generalized Langevin equation also hits the square-root barrier; explicit rates inside.","key_machinery":"The carrying object is the notion of a second-order lift: a semigroup $(\\hat P_t)$ on $X\\times V$ is a second-order lift of the reversible diffusion semigroup $(P_{t/2})$ when $\\Pi T\\Pi=0$ and $-(T\\Pi)^*(T\\Pi)$ is the generator of the overdamped diffusion, with spectral gap $P_x$. This structure identifies the two intermediate inequalities, the space-time-velocity Poincaré–Lions inequality and the averaging lemma, as consequences of the lift, and it also supplies the lower bound $t_{\\rm rel}(\\hat P)\\ge 2^{-1/2}P_x^{-1/2}$ on relaxation time. The averaging lemma is powered by Assumption 5, the solvability of $-\\partial_t\\varphi_0 - L_x\\varphi_1 = g$ with boundary conditions and the regularity estimates (12)-(13); the proof then runs through a time-averaged $L^2$ energy and an adapted Poincaré inequality. The same machinery, applied to the Gaussian generalised Langevin equation, reduces the computation to the spectral analysis of a $3\\times 3$ matrix and the propagator norm of a matrix exponential.","core_discovery":"The paper's central discovery is a single abstract theorem that makes the variational hypocoercivity machinery constructive and simultaneously interprets it through the lift structure. For any linear kinetic equation $\\partial_t f + T f = L_v f$ satisfying Assumptions 1 through 5, every zero-mean solution converges in $L^2(\\hat\\mu)$ with the explicit rate above and an explicit constant; the proof uses a time-averaged energy and an averaging lemma whose constants come from a divergence equation. The genuinely new applications are two. First, adaptive Langevin dynamics satisfies the assumptions once the potential meets quadratic-growth conditions, and after optimizing $T$, $\\epsilon$, and $\\gamma$, the paper obtains $\\lambda \\ge P_q/(66334\\sqrt{P_q+M+L})$; for convex near-quadratic potentials this is of order $\\sqrt{P_q}$, which is the best possible for a second-order lift, so ALD is claimed to be near-optimal. Second, the generalised Langevin equation satisfies the two lift assumptions despite violating the microscopic coercivity Assumption 4, so the general lower bound $t_{\\rm rel} \\ge \\tfrac12 P_x^{-1/2}$ applies; in the Gaussian case the paper computes the sharpest relaxation time for optimised parameters and finds it exceeds the lower bound only by the factor $1.93$.","pith_inferences":["A natural next step would be to verify Assumption 5 directly for the generalised Langevin equation under Assumption 6; the paper states this is ongoing, and success would put an upper bound on GLE's rate matching its lower bound.","The paper's identification of GLE as a second-order lift suggests that adding extra degenerate hidden variables, such as higher-order memory or thermostat variables, cannot itself produce more than the square-root acceleration; breaking the barrier would require a structural change such as state-dependent or non-reversible forcing, not just more dimensions.","The explicit factor 1.93 in the Gaussian benchmark gives a concrete target for parameter tuning in GLE-based samplers: the optimal friction and coupling are $\\gamma=3\\sqrt{3m}$ and $\\lambda=2\\sqrt{2m}$, not arbitrary large values.","The paper's rate formula predicts how ALD degrades as the potential's non-quadraticity parameters $M$ and $L$ grow, which is a quantitatively testable prediction for potentials beyond the near-quadratic class."],"forward_implications":["For near-quadratic convex potentials, adaptive Langevin dynamics has convergence rate of order $\\sqrt{P_q}$ up to a universal constant, matching the square-root speed-up that is the theoretical ceiling for second-order lifts.","The generalised Langevin equation cannot accelerate convergence beyond ballistic speed: its relaxation time is at least $\\tfrac12 P_x^{-1/2}$, so adding the memory variable does not beat standard Langevin by a higher-order margin.","In the Gaussian case with $\\lambda=2\\sqrt{2m}$ and $\\gamma=3\\sqrt{3m}$, the generalised Langevin equation has relaxation time $\\approx 0.964 m^{-1/2}$, which is within a factor 1.93 of the lower bound and sharper than the standard Langevin factor 5.46.","Because Assumptions 1 through 4 also hold for randomised Hamiltonian Monte Carlo and the Zig-Zag process, the same explicit-rate framework applies to those samplers without further work.","The optimisation of $T$ is decoupled from the final constant $C=\\exp(T\\lambda)$, so rates can be tuned without worsening the norm constants."],"supporting_citations":[{"why":"Introduces the variational hypocoercivity method, time-averaged $L^2$ energy and adapted Poincaré inequality, that Theorem 1 generalises and makes constructive.","marker":"[2]"},{"why":"Defines second-order lifts of reversible diffusions and gives the relaxation-time lower bound used for the GLE conclusion.","marker":"[29]"},{"why":"Provides the explicit $L^2$ convergence-rate analysis for Langevin dynamics and the divergence-equation constants used as a template for ALD.","marker":"[20]"},{"why":"Develops the divergence-equation construction for kinetic Fokker-Planck equations whose rescaling helps verify Assumption 5 for ALD.","marker":"[18]"},{"why":"Supplies the space-time divergence lemma and regularity theorem that Proposition 3.1 rescales to verify Assumption 5 for adaptive Langevin dynamics.","marker":"[30]"},{"why":"Introduces the adaptive Langevin dynamics model and its hypocoercivity properties; the paper's rate optimisation is compared against this earlier scaling.","marker":"[41]"},{"why":"Gives the corrected scaling of the ALD rate; the paper's parameter optimisation reproduces and refines that scaling.","marker":"[42]"},{"why":"Identifies the spectral gap of an Ornstein-Uhlenbeck generator with the spectral gap of its drift matrix, used in the Gaussian GLE computation.","marker":"[45]"},{"why":"Supplies the propagator-norm estimate for matrix exponentials, used to compute the non-asymptotic relaxation time of the Gaussian GLE.","marker":"[5]"}],"fun_headline_variants":["Near-optimal convergence for adaptive Langevin dynamics","Square-root barrier applies to generalized Langevin equation","Unifying hypocoercivity and lifts yields sharp rates","Constructive theory: ALD nearly optimal, GLE hits barrier","New theorem: adaptive Langevin near-optimal, GLE limited"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole rate bound rests on Assumption 5: for every zero-mean function $g$, the divergence equation $-\\partial_t\\varphi_0 - L_x\\varphi_1 = g$ must have solutions with boundary conditions and the regularity bounds (12)-(13). For adaptive Langevin dynamics the paper verifies this through Proposition 3.1, which is obtained by rescaling a theorem from another work rather than by a self-contained proof; if that theorem does not cover the rescaled ALD equation, or if the regularity constants there are wrong, the ALD rate bound and the near-optimality claim collapse.","fun_headline_variants_meta":{"raw":{"variants":["Near-optimal convergence for adaptive Langevin dynamics","Square-root barrier applies to generalized Langevin equation","Unifying hypocoercivity and lifts yields sharp rates","Constructive theory: ALD nearly optimal, GLE hits barrier","New theorem: adaptive Langevin near-optimal, GLE limited"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000205,"raw_usage":{"total_tokens":1434,"prompt_tokens":1026,"completion_tokens":408,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":642,"completion_tokens_details":{"reasoning_tokens":327}},"tokens_in":642,"tokens_out":408,"duration_ms":3880,"temperature":1.0,"reasoning_tokens":327,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T15:32:17.955028+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete check is numerical: simulate ALD for convex near-quadratic potentials, for example $U(q)=q^2/2+\\alpha q^4$ with small $\\alpha$, tune $\\epsilon$ and $\\gamma$ as prescribed, and measure the $L^2$ decay rate as $P_q\\to 0$; if the best observed rate is $o(\\sqrt{P_q})$, the near-optimality claim is false. For the Gaussian GLE, the relaxation time $0.964\\,m^{-1/2}$ at the optimal parameters $\\lambda=2\\sqrt{2m}$ and $\\gamma=3\\sqrt{3m}$ is directly checkable, and any second-order lift with relaxation time below $2^{-1/2}P_x^{-1/2}$ would contradict the paper's Corollary 2.","supporting_citations":[{"cited_title":"Variational methods for the kinetic Fokker–Planck equation","cited_arxiv_id":null,"evidence_quote":"Introduces the variational hypocoercivity method, time-averaged $L^2$ energy and adapted Poincaré inequality, that Theorem 1 generalises and makes constructive."},{"cited_title":"Non-reversible lif ts of reversible diffusion processes and relaxation times","cited_arxiv_id":null,"evidence_quote":"Defines second-order lifts of reversible diffusions and gives the relaxation-time lower bound used for the GLE conclusion."},{"cited_title":"On explicit L2-convergence rate estimate for underdamped Langevin dynamics","cited_arxiv_id":null,"evidence_quote":"Provides the explicit $L^2$ convergence-rate analysis for Langevin dynamics and the divergence-equation constants used as a template for ALD."},{"cited_title":"Space-time divergence lemmas and optimal non-reversible lifts of diffusions on Riemannian manifolds with boundary","cited_arxiv_id":"2412.16710","evidence_quote":"Supplies the space-time divergence lemma and regularity theorem that Proposition 3.1 rescales to verify Assumption 5 for adaptive Langevin dynamics."},{"cited_title":"Hypocoercivity properties of adaptive Langevin dy- namics","cited_arxiv_id":null,"evidence_quote":"Introduces the adaptive Langevin dynamics model and its hypocoercivity properties; the paper's rate optimisation is compared against this earlier scaling."},{"cited_title":"Hypocoercivity properties of adaptive Langevin dynamics","cited_arxiv_id":"1908.09363","evidence_quote":"Gives the corrected scaling of the ALD rate; the paper's parameter optimisation reproduces and refines that scaling."},{"cited_title":"Sp ectrum of Ornstein-Uhlenbeck Operators in Lp Spaces with Respect to Invariant Measures","cited_arxiv_id":null,"evidence_quote":"Identifies the spectral gap of an Ornstein-Uhlenbeck generator with the spectral gap of its drift matrix, used in the Gaussian GLE computation."},{"cited_title":"Propagator norm and sharp decay estimates for Fokker-Planck equations with linear drift","cited_arxiv_id":null,"evidence_quote":"Supplies the propagator-norm estimate for matrix exponentials, used to compute the non-asymptotic relaxation time of the Gaussian GLE."}],"review_version":1}