{"id":"7c606463-85e4-4065-a303-0c8a4deae8df","arxiv_id":"1908.11840","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves sharp power-law asymptotics, including explicit prefactors, for long exit times near a repelling equilibrium, confirming a conjecture of Mikami.","lead":"This paper proves a 1995 conjecture about how long a noisy dynamical system can linger near an unstable equilibrium before escaping. It shows the probability of an atypically long stay decays as a power of the noise level, and it computes the exact coefficient for box-shaped regions.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Density comparison in Lemma 4.1 rests on [BC14, Thm 2.14.B] with q=0, a case the paper admits the stated theorem does not formally cover; the asserted extension via Thm 3.10 is plausible but unverified, and the sharp prefactor in Theorem 2.2 collapses if it fails.","rationale":"The reader's weakest_assumption is the C^5 linearizing conjugacy, but that is an explicitly stated hypothesis of the theorem rather than an unproven step; it restricts scope without threatening the internal validity of the argument. The reader's rationale does list the [BC14, Theorem 2.14.B] q=0 issue among the main weaknesses, which is why agreement is partial, but the present stress-test identifies that external-theorem gap as the single most load-bearing concern. The internal Malliavin estimates in Sections 5.1 and 5.2 appear internally coherent, including the nontrivial inverse-moment argument in Lemma 5.5, so no clear internal contradiction was found. If the q=0 check succeeds, the central argument would appear solid, and the verdict should remain conditional at most because of the strong smoothness and non-resonance assumptions; if the q=0 check fails, the proof of the prefactor would not stand. Thus the recommended verdict is unchanged: CONDITIONAL, pending verification of the external theorem in the q=0 case.","tokens_in":30922,"tokens_out":22389,"duration_ms":199802,"concrete_test":"Obtain [BC14] and verify directly whether Theorem 3.10 (and the derivation of Theorem 2.14 from Theorem 2.1) genuinely covers q=0 with constants C,a,b,gamma independent of the time horizon T; check that its hypotheses are met by U_T and Z_T with the inverse-moment bounds (5.25). If q=0 is not covered, reconstruct Theorem 5.1 for q=0 from the approximation argument in [BC14, Section 3] and re-run Lemma 4.1; a minimal sanity check is to test the resulting bound against the exact densities of the linear process Z_T, where the discrepancy is explicit.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is Theorem 5.1, an application of Bally-Caramellino [BC14, Theorem 2.14.B] with derivative order q=0. The authors explicitly flag (Section 5, before Theorem 5.1) that q=0 is not formally allowed by that statement, and assert that the result is nevertheless valid because Theorem 2.14 is derived from their Theorem 2.1, part B of which is restated as Theorem 3.10 with q=0 allowed. This is the only mechanism by which the Malliavin estimates of Sections 5.1–5.2 are converted into the density discrepancy in Lemma 4.1; Lemma 4.1 then drives the iteration in Lemma 4.2, Lemma 3.4, and finally Theorem 2.2. If the q=0 extension is not actually covered, then the bound (5.22), the Gaussian comparison in Lemma 3.4, and the explicit prefactor (2.13)/(3.15) are not proved. This is a verification gap rather than an identified error: the pointer to Theorem 3.10 is specific and the assertion is credible, but the paper does not reproduce the argument. The C^5 conjugacy assumption is a strong, explicitly stated hypothesis and restricts scope, but it is not an internal gap; the flagged external-theorem issue is the more serious risk to the central claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies small-noise perturbations (1.1) of a C^5 vector field b on R^d with 0 a hyperbolic repelling equilibrium whose linearization has distinct positive real eigenvalues λ1>...>λd>0. Under the additional assumption of a C^5 linearizing diffeomorphism f, it proves (Theorem 2.2) that for rectangular-type domains R=f^{-1}(box), uniformly over initial points X0=εx with |x|≤K(ε), the exit-time tail satisfies P{τ_R>α log ε^{-1}+r(ε)} = ε^{β(α)}ψ(x)(1+o(ε^p)), with β(α)=∑(λ_j α-1)_+ and ψ an explicit Gaussian integral over the covariance C0 of the linearized process; Corollary 2.4 gives logarithmic asymptotics for general domains. The proof transforms to coordinates Y=f(X), writes Y in Duhamel form, compares the density of U to a Gaussian via Malliavin calculus (Theorem 5.1 from Bally-Caramellino), and iterates over short time intervals.","tokens_in":31233,"tokens_out":6127,"duration_ms":56547,"significance":"If the proof is correct, this is a substantial advance: it upgrades Mikami's logarithmic conjecture to sharp asymptotics with an explicit prefactor in the linearizable case. Strong points include the absence of fitted constants, the derivation of both the exponent β(α) and the prefactor ψ from the linearized covariance C0, uniform estimates over initial conditions, and the fact that all main lemmas are proved in the text. The proof is credible but hinges on an external theorem in a form not formally covered (the q=0 case), which is the main risk to the sharp prefactor; the C^5 conjugacy assumption is also a real restriction of scope that should be stated more carefully.","major_comments":[{"comment":"Theorem 5.1 is invoked with Malliavin derivative order q=0, although the manuscript itself states that [BC14, Theorem 2.14.B] as stated does not formally allow q=0 and asserts validity via Theorem 3.10 there. This q=0 case is load-bearing: Lemma 4.1(1) is the only bridge converting the Malliavin estimates of Sections 5.1–5.2 into the density discrepancy used in Lemma 4.2, Lemma 3.4, Proposition 3.2, and Theorem 2.2. Without it, the Gaussian comparison and the explicit prefactor ψ in (2.13) are not proved. Please provide a self-contained proof of the q=0 case or reproduce the argument from [BC14, Theorem 3.10] with enough detail to verify the hypotheses and constants; a pointer is not sufficient.","section":"Section 5, paragraph before Theorem 5.1; Lemma 4.1"},{"comment":"The main theorem is conditional on the existence of a C^5 linearizing diffeomorphism f, which is not a consequence of Hartman–Grobman and can fail under resonances. Remark 2.5(4) acknowledges this, but the abstract and introduction describe the result for smooth vector fields without this hypothesis. Since the proof transfers the exit problem to the box R'=f(R) and needs bounded third-order Malliavin derivatives of the transformed coefficients, the theorem applies only to the Sternberg-type linearizable subclass. Please state this restriction explicitly in the abstract and in the statements of the corollaries that are described as proving Mikami's conjecture.","section":"Section 2, assumption before Remark 2.1; Remark 2.5(4)"}],"minor_comments":[{"comment":"The reference to 'Proposition 2.2' should be 'Theorem 2.2'; no Proposition 2.2 exists in the paper.","section":"Remark 2.5(1)"},{"comment":"In the lower-bound argument, the first display reads P{τ > log ε^{-1}+r(ε)} and drops the factor α; it should be P{τ > α log ε^{-1}+r(ε)}.","section":"Section 4.1, text above (4.5)"},{"comment":"In the display after (5.18), the expression appears to contain 'U^2_i' where 'U^i' is meant.","section":"Section 5.1.3, display after (5.18)"},{"comment":"The phrase 'on both both sides' contains a duplicated word.","section":"After Corollary 2.4"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the paper is within scope and the central result is strong if the proof is correct. The main risk is the q=0 use of [BC14]; this should be verified or supplied before acceptance. The C^5 conjugacy assumption should also be reflected in the abstract so that the claims do not overstate the scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper proves what it claims: for a smooth vector field with simple positive real eigenvalues, the small-noise diffusion has exit-time tails that match the linearized Gaussian approximation up to an explicit prefactor, uniformly over initial points in a shrinking neighborhood. This goes well beyond Mikami's logarithmic asymptotics and the authors' earlier one-dimensional results. The iteration scheme over logarithmic-length blocks is a genuine technical step, and the explicit formula for the prefactor in terms of the Gaussian covariance C0 is a real contribution. The proof is substantial and mostly self-contained: the Malliavin derivative bounds in Section 5 are worked out in detail, and the reduction to the box domain is handled carefully.\n\nThe soft spot is exactly the one flagged in the paper itself: Theorem 5.1 applies Bally–Caramellino's density comparison at derivative order q = 0, a case their stated theorem does not formally cover. The authors assert that the result remains valid because their Theorem 2.14 is derived from Theorem 2.1, part B of which is restated as Theorem 3.10 with q = 0 allowed. That pointer is specific and credible, but the paper does not reproduce the argument or give the precise wording of Theorem 3.10, so the reader cannot verify without going to the source. Since Lemma 4.1 drives the whole iteration, this is a verification gap rather than an identified error. It should be fixed before final acceptance, either by stating the q = 0 version as a lemma with proof sketch or by citing the precise statement.\n\nThe C^5 conjugacy assumption is strong but explicitly given and is a scope restriction, not an internal gap. The authors also note the non-resonance condition from Sternberg. For applications to heteroclinic networks this may be limiting, but the paper is honest about it.\n\nWho should read this: anyone working on small-noise diffusions near unstable equilibria, rare-event tails, or metastability in heteroclinic networks. The proof deserves referee time and careful checking, especially the external-theorem dependency. I would send it to peer review and expect the q = 0 issue to be resolved in revision.","headline":"Sharp exit-time asymptotics with explicit prefactor in higher dimensions, proving Mikami's conjecture in the simple-real-eigenvalue case; the load-bearing density comparison rests on a plausible but not fully documented extension of a Bally–Caramellino theorem.","tokens_in":31734,"tokens_out":884,"would_cite":true,"duration_ms":10512,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H07","60H10","60J60"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that diffusion exit-time tails near a repelling equilibrium decay as an explicit power of the noise amplitude, with a computable Gaussian prefactor.","keywords":["vanishing noise limit","unstable equilibrium","exit problem","polynomial decay","Malliavin calculus","heteroclinic networks"],"falsifier":"Take $d=2$ with eigenvalues $\\lambda_1=2,\\lambda_2=1$, unit noise $\\sigma=I$, and a small nonlinear term such as $b(x)=(2x_1+x_1^2,\\,x_2+x_2^2)$; compute or simulate $P\\{\\tau_R>\\alpha\\log\\varepsilon^{-1}\\}$ for $\\alpha=0.75$ and $\\alpha=1$, and check whether $\\varepsilon^{-\\beta(\\alpha)}P$ converges to the Gaussian integral (2.13) with $C_0=\\mathrm{diag}(1/4,1/2)$. A deviation larger than $o(\\varepsilon^p)$ as $\\varepsilon\\to0$ would disprove Theorem 2.2; similarly, constructing a resonant smooth field where the conjugacy is only continuous and observing a non-Gaussian prefactor would show the $C^5$ assumption is load-bearing.","tokens_in":30683,"feed_emoji":"⏱️","tokens_out":5429,"duration_ms":46600,"temperature":0.7,"pith_summary":"This paper studies a diffusion obtained from a smooth flow near a repelling equilibrium by adding small noise of amplitude $\\varepsilon$, and asks how long it takes to escape a surrounding domain. The central claim is that the probability of an atypically long exit, of the form $\\{\\tau > \\alpha\\log\\varepsilon^{-1}+r(\\varepsilon)\\}$, decays as a pure power of $\\varepsilon$ with an explicit prefactor: $\\varepsilon^{-\\beta(\\alpha)}P\\{\\tau_R>\\alpha\\log\\varepsilon^{-1}+r(\\varepsilon)\\}\\to\\psi(x)$, where $\\beta(\\alpha)=\\sum_j(\\lambda_j\\alpha-1)_+$ and $\\psi$ is an explicit Gaussian integral over the exit box. This proves a conjecture of Mikami that had previously been known only up to logarithmic equivalence or in one dimension. The practical upshot is that the rare slow escapes controlling long-term noisy dynamics near unstable points can be quantified exactly, not just by an exponent.","feed_headline":"Long exits from a repeller follow a precise power law","feed_subtitle":"The rare chance of staying past α log(1/ε) equals an explicit Gaussian prefactor times ε^β.","key_machinery":"The argument is carried by the linearizing conjugacy $f$, assumed to be a $C^5$ diffeomorphism, which sends the neighborhood of $0$ to a box $R'=[-L,L]^d$ and converts the SDE into $dY_t^j=\\lambda_j Y_t^j\\,dt+\\varepsilon F^j_l(Y_t)\\,dW^l_t+\\varepsilon^2 G^j(Y_t)\\,dt$. Duhamel's formula writes $Y_t^j=\\varepsilon e^{\\lambda_j t}(y^j+U_t^j)$, so the exit event becomes a condition on $U_t$, and $U_t$ is a small perturbation of a Gaussian martingale $Z_t$ with covariance $C_0$. The proof uses Malliavin calculus to compare densities: a density-discrepancy theorem bounds $|\\rho_{U_T}-\\rho_{Z_T}|$ in terms of Sobolev norms and Malliavin determinants, and because this estimate is only valid for times $T\\leq\\theta\\log\\varepsilon^{-1}$, the time $\\alpha\\log\\varepsilon^{-1}$ is split into $N$ small steps and the Gaussian approximation is iterated. The explicit prefactor $\\psi$ emerges from the Gaussian density integrated over the box-shaped exit set.","core_discovery":"The paper proves that for a $C^5$ vector field whose linearization at $0$ is diagonal with real eigenvalues $\\lambda_1>\\cdots>\\lambda_d>0$, and for domains whose preimages under the linearizing conjugacy are boxes, the exit-time tail has the same polynomial asymptotics as the linearized Gaussian process. The exact statement is Theorem 2.2: uniformly in initial points $X_0=\\varepsilon x$ with $|x|\\leq K(\\varepsilon)$, one has $|\\varepsilon^{-\\beta(\\alpha)}P\\{\\tau_R>\\alpha\\log\\varepsilon^{-1}+r(\\varepsilon)\\}-\\psi(x)|=o(\\varepsilon^p)$ for some $p>0$, with $\\psi$ given by an integral of a Gaussian density with covariance $C_0$ from (2.11). A corollary for general domains gives upper and lower bounds whose gap is a travel-time correction, and taking logarithms yields $\\log P\\{\\tau_D>\\alpha\\log\\varepsilon^{-1}+r(\\varepsilon)\\}/\\log\\varepsilon\\to\\beta(\\alpha)$, confirming the conjecture stated by Mikami. Thus the nonlinear diffusion inherits, to polynomial precision, the tail of its tangent Gaussian approximation, including a computable prefactor.","pith_inferences":["The technique suggests that the same iterative Gaussian comparison should yield sharp asymptotics for the exit location on the box boundary, since the prefactor $\\psi$ already encodes the Gaussian exit distribution.","One can test numerically whether the prefactor $\\psi$ is universal: any two vector fields with the same linear part and the same $\\sigma(0)$ should produce the same leading constant, independent of the nonlinear terms.","If the smooth-conjugacy assumption is dropped, the exponent may survive via large deviations, but the Gaussian prefactor would likely fail or become non-explicit; this would pinpoint where smoothness is economically used."],"forward_implications":["Mikami's conjectured exponent $\\mu(h)=\\sum_j((h\\lambda_j/\\lambda_1-1)_+)$ is confirmed for this class of systems, including the prefactor, not just the logarithmic rate.","The probability of staying longer than $\\alpha\\log\\varepsilon^{-1}$ is asymptotically $\\psi(x)\\varepsilon^{\\beta(\\alpha)}$, so rare long stays are controlled by the leading eigenvalue spectrum and the noise covariance at the equilibrium.","For general domains, exit probabilities are sandwiched between two explicit prefactors differing only by the deterministic travel time between nested domains.","If initial conditions are random, $X_0=\\varepsilon\\xi_\\varepsilon$, the tail probability converges to $\\mathbb{E}\\psi(\\xi)$ whenever $\\xi_\\varepsilon$ has tail $o(\\varepsilon^{\\beta(\\alpha)})$."],"supporting_citations":[{"why":"States the conjecture on the logarithmic asymptotics of long exit times that this paper proves.","marker":"[Mik95]"},{"why":"Establishes that typical exit times are of order $\\lambda_1^{-1}\\log\\varepsilon^{-1}$, the baseline around which atypical long exits are measured.","marker":"[Kif81]"},{"why":"Provides the limiting distribution of $\\tau-T_\\varepsilon$, which the present result refines with polynomial-rate asymptotics.","marker":"[Day95]"},{"why":"Supplies the one-dimensional Malliavin-calculus predecessor whose iteration scheme is extended here to higher dimensions.","marker":"[BPG19a]"},{"why":"Gives the density-discrepancy estimate between two diffusion laws that is the core quantitative tool of the proof.","marker":"[BC14]"},{"why":"Provides the Malliavin calculus framework, including derivative formulas and negative-moment estimates used throughout Section 5.","marker":"[Nua95]"},{"why":"Gives the smoothness and non-resonance conditions under which the linearizing conjugacy is $C^5$.","marker":"[Ste57]"},{"why":"Supplies the Hartman-Grobman theorem giving the topological conjugacy that is upgraded to a $C^5$ diffeomorphism.","marker":"[KH95]"}],"fun_headline_variants":["Exit tail from repeller matches Gaussian power law","Repeller exit times get polynomial asymptotics","Precise power law for long escapes from a repeller","Long exit times from repeller: exact polynomial tail","Noise-led escapes from repeller obey power law"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes that the nonlinear flow can be straightened by a $C^5$ coordinate change that is a diffeomorphism; if only a homeomorphism exists, the Gaussian density comparison and the explicit prefactor no longer follow.","fun_headline_variants_meta":{"raw":{"variants":["Exit tail from repeller matches Gaussian power law","Repeller exit times get polynomial asymptotics","Precise power law for long escapes from a repeller","Long exit times from repeller: exact polynomial tail","Noise-led escapes from repeller obey power law"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000269,"raw_usage":{"total_tokens":1559,"prompt_tokens":817,"completion_tokens":742,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":433,"completion_tokens_details":{"reasoning_tokens":667}},"tokens_in":433,"tokens_out":742,"duration_ms":6714,"temperature":1.0,"reasoning_tokens":667,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T10:05:51.187570+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $d=2$ with eigenvalues $\\lambda_1=2,\\lambda_2=1$, unit noise $\\sigma=I$, and a small nonlinear term such as $b(x)=(2x_1+x_1^2,\\,x_2+x_2^2)$; compute or simulate $P\\{\\tau_R>\\alpha\\log\\varepsilon^{-1}\\}$ for $\\alpha=0.75$ and $\\alpha=1$, and check whether $\\varepsilon^{-\\beta(\\alpha)}P$ converges to the Gaussian integral (2.13) with $C_0=\\mathrm{diag}(1/4,1/2)$. A deviation larger than $o(\\varepsilon^p)$ as $\\varepsilon\\to0$ would disprove Theorem 2.2; similarly, constructing a resonant smooth field where the conjugacy is only continuous and observing a non-Gaussian prefactor would show the $C^5$ assumption is load-bearing.","supporting_citations":[],"review_version":1}