Long exit times near a repelling equilibrium

Hong-Bin Chen; Yuri Bakhtin

REVIEW 2 major objections 4 minor 1 cited by

Long exit times near a repelling equilibrium

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 1908.11840 v2 pith:TQ3WIY6K submitted 2019-08-30 math.PR

Yuri Bakhtin , Hong-Bin Chen This is my paper · ORCID

classification math.PR MSC 60H0760H1060J60

keywords vanishingnoiselimitunstableequilibriumexitproblempolynomialdecayMalliavincalculusheteroclinicnetworks

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 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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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 $o(\varepsilon^{\beta(\alpha)})$.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the 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.

The 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.

The 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.

Who 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.

Referee Report

2 major / 4 minor

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.

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 (2)

[Section 5, paragraph before Theorem 5.1; Lemma 4.1] 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 2, assumption before Remark 2.1; Remark 2.5(4)] 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.

minor comments (4)

[Remark 2.5(1)] The reference to 'Proposition 2.2' should be 'Theorem 2.2'; no Proposition 2.2 exists in the paper.
[Section 4.1, text above (4.5)] 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 5.1.3, display after (5.18)] In the display after (5.18), the expression appears to contain 'U^2_i' where 'U^i' is meant.
[After Corollary 2.4] The phrase 'on both both sides' contains a duplicated word.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the Gaussian prefactor and power exponent are computed from the linearized coefficients and the exit geometry, not fitted or imported from the paper's own conclusions.

full rationale

The paper's central quantity, the tail probability P{τ_R > α log ε^{-1} + r(ε)}, is reduced through Proposition 3.2 to Lemma 3.3 and Lemma 3.4. Lemma 3.3 rewrites the exit event using the explicit representation Y_t^j = ε e^{λ_j t}(y_j + U_t^j), where U_t is defined through stochastic integrals in (3.7)–(3.10); no unknown parameter is fitted or chosen to match the target tail probability. Lemma 3.4 then compares the law of U_{T_0} with a centered Gaussian Z whose covariance C_0 is computed in (2.11) directly from σ(0) and the eigenvalues λ_j. The prefactor ψ in (2.13) is the resulting Gaussian integral over the rectangle A±, again with no fitted constants. The Malliavin estimates in Section 5 prove that the density of U_T is close to that of Z_T, with an explicit ε^δ error bound; they do not presuppose the theorem. The cited external result [BC14] is used as a density-comparison tool, and the authors explicitly flag that their use with q=0 goes beyond the literal statement of the cited theorem and relies on a stated extension in that paper. This is a verification gap in the proof, not circular reasoning. Self-citations such as [BPG19a] and [BPG18] appear as context for the one-dimensional case and as motivation, not as load-bearing inputs for the higher-dimensional theorem. The smooth-conjugacy assumption is a genuine hypothesis, not an input that already contains the conclusion. Overall, the derivation is self-contained enough: the target asymptotics are not equivalent by construction to any fitted input or to any earlier claim of the same authors.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim is derived from the linearized Gaussian model; there are no fitted constants and no invented physical entities. The main external inputs are the smooth linearizing conjugacy, the uniform non-degeneracy of the transformed diffusion coefficient, the standard Malliavin calculus framework, and the density discrepancy theorem of Bally-Caramellino, which is used at q=0 despite the stated conditions.

assumptions (5)

domain assumption The linearizing conjugacy f is a C^5 diffeomorphism (sufficient Sternberg conditions stated in Remark 2.1).
Invoked in Section 2 before Remark 2.1 and in Remark 2.5(4); it guarantees the transformed SDE has C^3 coefficients and bounded higher Malliavin derivatives, a load-bearing regularity condition.
ad hoc to paper Uniform ellipticity of F after modification: min_{|u|=1} |u^T F(x)|^2 >= c0 for all x in R^d (equation (3.6)).
Needed for the negative moments of the Malliavin covariance matrix in Section 5.2; the modification outside the box [-L0,L0]^d is constructed by the authors.
domain assumption The density discrepancy estimate [BC14, Theorem 2.14.B] holds for q=0, as argued via Theorem 3.10 of the same paper.
Theorem 5.1 imports this result at q=0, a case the authors acknowledge is not covered by the theorem's stated conditions; the transfer relies on their meta-argument.
standard math Standard Malliavin calculus framework, including Sobolev spaces D^{k,p}, Meyer inequalities, and Malliavin covariance formulas for SDE solutions.
Used pervasively in Section 5 for density comparison and moment estimates.
standard math Exponential martingale inequality, Gronwall's inequality, and Burkholder-Davis-Gundy inequalities.
Used in Sections 4 and 5 to control martingale increments and moments.

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Pith. "Pith review of Long exit times near a repelling equilibrium." pith.science (2026). https://pith.science/paper/TQ3WIY6K

@misc{pith2026190811840,
  author       = {Pith},
  title        = {Pith review of: Long exit times near a repelling equilibrium},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TQ3WIY6K}},
  note         = {Machine review of arXiv:1908.11840}
}

read the original abstract

For a smooth vector field in a neighborhood of a critical point with all positive eigenvalues of the linearization, we consider the associated dynamics perturbed by white noise. Using Malliavin calculus tools, we obtain polynomial asymptotics for probabilities of atypically long exit times in the vanishing noise limit.

Figures

Figures reproduced from arXiv: 1908.11840 by the authors.

**Figure 1.** The diffeomorphism f : O → f(O) maps R onto f(R) which is a box containing 0. To simplify the notation, we often suppress the dependence on ǫ. In particular, we often write Xt instead of Xǫ t . We need some definitions to state our main result. We start by describing the exit event: — for a measurable set A ⊂ R d , we define the exit time (2.7) τA = inf{t > 0 : Xt 6∈ A}; — for L j −, Lj + ∈ R, j = 1, 2, . . . , d, w… view at source ↗

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Works this paper leans on

16 extracted references · 14 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Normal forms approach to diffusion near hyperbolic equilibria

Sergio Angel Almada Monter and Yuri Bakhtin. Normal forms approach to diffusion near hyperbolic equilibria. Nonlinearity , 24(6):1883--1907, 2011

work page 1907

[2] [2]

Small noise limit for diffusions near heteroclinic networks

Yuri Bakhtin. Small noise limit for diffusions near heteroclinic networks. Dyn. Syst. , 25(3):413--431, 2010

work page 2010

[3] [3]

Noisy heteroclinic networks

Yuri Bakhtin. Noisy heteroclinic networks. Probability Theory and Related Fields , 150(1):1--42, Jun 2011

work page 2011

[4] [4]

Richard F. Bass. Stochastic Processes . Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, 2011

work page 2011

[5] [5]

On the distances between probability density functions

Vlad Bally and Lucia Caramellino. On the distances between probability density functions. Electron. J. Probab. , 19:33 pp., 2014

work page 2014

[6] [6]

Tails of exit times from unstable equilibria on the line

Yuri Bakhtin and Zsolt Pajor-Gyulai. Tails of exit times from unstable equilibria on the line . Accepted at Journal of Applied Probability; available at arXiv e-prints , page arXiv:1810.05341, Oct 2018

arXiv 2018

[7] [7]

Malliavin calculus approach to long exit times from an unstable equilibrium

Yuri Bakhtin and Zsolt Pajor-Gyulai. Malliavin calculus approach to long exit times from an unstable equilibrium. Ann. Appl. Probab. , 29(2):827--850, 04 2019

work page 2019

[8] [8]

Scaling limit for escapes from unstable equilibria in the vanishing noise limit: Nontrivial jordan block case

Yuri Bakhtin and Zsolt Pajor-Gyulai. Scaling limit for escapes from unstable equilibria in the vanishing noise limit: Nontrivial jordan block case. Stochastics and Dynamics , 19(03):1950022, 2019

work page 2019