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The narrow escape problem in arbitrary dimension

T0 review · 2 major / 3 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read For any smooth bounded domain in dimension d≥2, the narrow-escape rate is C_d times the sum of hole capacities, and the exit hole is chosen in proportion to those capacities.

desk verdict A serious arbitrary-dimension narrow escape paper with a correctable but load-bearing prefactor error: the stated C_d is off by ω_d^2, and the paper's own remark and numerics already use the corrected constant. read the letter →

arxiv 2608.02212 v1 pith:QF22OWNT submitted 2026-08-03 math.AP math.PR

Louis Carillo , Tony Lelièvre , Thomas Normand , Gabriel Stoltz , Urbain Vaes This is my paper
classification math.APmath.PR MSC 35J2560J6535P15
keywords narrowescapeproblemquasi-stationarydistributionBrownianmotionmetastabilityeigenvalueasymptoticsboundarysingularitiesexittimehigh-dimensionaldiffusion
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 aims to give asymptotic formulas for the narrow escape problem—a Brownian particle diffusing in a bounded domain whose reflecting boundary is pierced by N small absorbing holes—valid in any dimension and for general smooth domains. It claims that, in the limit where all hole radii tend to zero, the quasi-stationary exit rate is λ_ε^0 = C_d K_ε (1+O(E_d(K_ε))), with K_ε = Σ_k K_ε^(k) the sum of hole capacities (logarithmic in d=2, power-law for d≥3) and C_d = max{d-2,1}/(2|Ω|ω_d) a universal constant. The exit-hole law is equally simple: P(exit through hole k) = K_ε^(k)/K_ε, up to the same error. These results matter because they extend classical 2D/3D narrow-escape formulas to arbitrary dimension and give a quantitative basis for modelling high-dimensional metastable systems and for kinetic Monte Carlo simulations of rare events.

What carries the argument

The quasi-mode φ_ε = 1 + Σ_k K_ε^(k) f_k, where each f_k is the unique mean-zero weak solution of -Δf_k = C_d in Ω, ∂_n f_k = -C_d|Ω| δ_{x^(k)} on ∂Ω. Each f_k is decomposed as the fundamental solution Λ(y) (with Λ = log|y| for d=2, -|y|^{2-d} for d≥3) composed with a boundary-flattening diffeomorphism Ψ_k so that its normal derivative vanishes off the hole, plus a sub-singular term S_k built from the Neumann Green function. The parameters K_ε^(k) are chosen so that K_ε^(k) f_k ≈ -1 on the k-th absorbing boundary, making φ_ε almost vanish on all holes. Comparing this quasi-mode with the true quasi-stationary distribution ν_ε through the Green-type identities in Sections 3.2–3.3 yields the ei

What would settle it

Directly evaluate the constant in Lemma 2.8 in the flat-boundary (half-ball) limit: the paper's own integration in (2.30) gives A_d ω_d/2, whereas Lemma 2.8 asserts |Ω|C_d = A_d/(2ω_d); these differ by a factor ω_d^2. A high-accuracy numerical computation of λ_ε^0 for a ball in d=3 with one small hole would then decide which prefactor is effective.

Watch

Extended reading notes

Core claim

The central discovery is that the narrow-escape problem in arbitrary dimension is governed by a capacity-weighted superposition of the holes. For each hole k, define K_ε^(k) = -(log r_ε^(k))^{-1} if d=2 and (r_ε^(k))^{d-2} if d≥3, and let K_ε be their sum. The paper proves that the first eigenvalue λ_ε^0 of the Laplacian with mixed boundary conditions (Dirichlet on the holes, Neumann on the rest) satisfies λ_ε^0 = C_d K_ε(1+O(E_d(K_ε))) with C_d = max{d-2,1}/(2|Ω|ω_d); hence the quasi-stationary exit time is exponential with mean 1/λ_ε^0. It further proves that the probability of exiting through hole k is K_ε^(k)/K_ε + O(E_d(K_ε)), so in dimension 2 the logarithmic capacities make two holes

Load-bearing premise

The prefactor in the mean-exit-time formula rests on the assertion that the boundary-flux normalization of the singular part of the quasi-mode is exactly |Ω|C_d at each hole point; if this constant is wrong, all prefactors change by a fixed multiplicative factor.

Editorial extensions

If this is right

  • In any dimension, the mean exit time from the quasi-stationary state is asymptotically 1/(C_d K_ε), showing that escape slows down exponentially in d=2 (logarithmic capacity) and algebraically for d≥3 as holes shrink.
  • The probability of exiting through a given hole is, to leading order, proportional to K_ε^(k); in dimension 2 alone, holes of radii 2ε and ε are asymptotically equally likely—a purely two-dimensional phenomenon.
  • The proof justifies the quasi-stationary distribution as the relevant initial condition: after a short transient the exit time is exponential, independent of the exit point, and the exit-point law is encoded in the normal-derivative measure of the QSD.
  • The additive structure of K_ε means that multiple holes act independently to leading order, with interactions appearing only in the error; this supports mean-field-like models for multi-target exit problems.
  • The dimension-dependent error E_d(K_ε) gives a quantitative warning that the asymptotic regime is harder to reach in higher dimensions, since the relative error scales as r_ε for d≥4 rather than as K_ε.

Reading between the lines

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

  • If the prefactor C_d is universal, a single measurement of the eigenvalue at small hole size directly reveals the total capacity K_ε even when individual hole radii are unknown—a useful inversion for experiments or simulations.
  • The simple exit-probability law suggests that rare-event sampling algorithms can pre-weight each target by its capacity K_ε^(k)/K_ε to accelerate the observation of escape through a chosen hole.
  • The quasi-mode construction via boundary flattening plus sub-singular corrections should extend to non-spherical holes (elliptical, polygonal) as long as the leading singularity of the flux is the same; testing this would be a natural follow-up.
  • The numerical evidence in the paper suggests that the ratio (λ_1-λ_0)/λ_0 must be large for the asymptotic regime to be visible; testing this criterion across different domains would be a useful practical guide.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The paper studies the narrow escape problem for reflected Brownian motion in a smooth bounded domain in any dimension d>=2, with N small absorbing spherical holes on the boundary. The authors use a quasi-stationary distribution (QSD) approach. They construct a quasi-mode φε = 1 + Σ_k K_ε^(k) f_k, where f_k solves a point-source Neumann problem, and derive the asymptotic of the smallest eigenvalue λ_ε^0 (inverse mean exit time) and of the exit-hole probabilities. Main results are Theorem 1.4: λ_ε^0 = C_d K_ε (1+O(E_d(K_ε))) with C_d = max{d-2,1}/(2|Ω|ω_d), and Theorem 1.6: P(exit through hole k) = K_ε^(k)/K_ε + O(E_d(K_ε)). The paper also presents a Walk-on-Spheres Monte Carlo algorithm and numerical tests in dimensions 2–6 and for various two-dimensional domains.

Significance. If the technical content is correct, this is a valuable contribution: it extends the QSD-based spectral approach to narrow escape problems to general smooth domains and arbitrary dimension, with explicit dimension-dependent constants that are fixed by the hole radii and the Laplacian fundamental solution, not by fitted parameters. The paper also gives a numerically practical Monte Carlo method for high-dimensional settings, which is a genuine strength. However, the main theorem as printed contains a prefactor normalization error in Lemma 2.8 and Eq. (1.14); this is a load-bearing inconsistency that must be corrected before the result can be accepted.

major comments (2)
  1. [Lemma 2.8, Eqs. (2.27)–(2.30); Theorem 1.4, Eq. (1.14)] The normalization identity in Lemma 2.8 is false as stated. The proof computes in (2.28)–(2.30) that, for v∈C^1(∂Ω), the flux is -max{d-2,1}ω_d v(x^(k))/2 (after restoring the minus sign in (2.28)). The lemma instead asserts -|Ω|C_d v(x^(k)), and with C_d from (1.14) one has |Ω|C_d = max{d-2,1}/(2ω_d). The two expressions differ by a factor ω_d^2. Consequently the compatibility condition in (2.32) requires C_d = max{d-2,1}ω_d/(2|Ω|), not (1.14). This error propagates through Proposition 3.2(b), Lemma 3.3, and the prefactor in Theorem 1.4, changing the mean exit time by a dimension-dependent factor ω_d^2. Theorem 1.6 is unaffected because it uses only ratios of K_ε^(k).
  2. [Remark 1.5 and Section 4.2] The paper itself uses the corrected constant. In Remark 1.5, for d=3 the authors state Eν[τ] = |Ω_ε|/(2π r_ε), which corresponds to C_3 = ω_3/(2|Ω|), not to the printed C_3=1/(2|Ω|ω_3). Similarly, the prefactors listed in Section 4.2 for the unit hyperball — 1, 1.5, 4, 7.5, 12 in dimensions 2–6 — are exactly A_d ω_d/(2|Ω|) with A_d=max{d-2,1}, not (1.14). These internal checks confirm that Theorem 1.4 as printed is not the formula used or tested in the paper.
minor comments (3)
  1. [Throughout] There are many inconsistent cross-references: Lemma 2.3 is called Theorem 2.3 in the proof of Lemma 2.5; Corollary 2.4 is called Theorem 2.4 in the proof of Lemma 2.6; Lemma 2.5 is called Theorem 2.5 in the proof of Corollary 2.7; Lemma 2.8 is called Theorem 2.8; Lemma 2.10 is called Theorem 2.10; Proposition 2.1 is called Theorem 2.1 in several places; Lemma 2.12 is called Theorem 2.12; Lemma 3.1 is called Theorem 3.1; Proposition 3.2 is called Theorem 3.2; Lemma A.1 is called Theorem A.1. These should be corrected globally.
  2. [Section 4.1] The text refers to 'Theorem 1.2' in the paragraph on the Monte Carlo method; there is no Theorem 1.2, and the intended reference is likely Remark 1.2 or Proposition 1.3. Similarly, Section 4's introduction refers to 'Theorem 1.7', which should be Remark 1.7.
  3. [Figure 5 and Section 4.2] The dashed lines in Figure 5 and the expected constants 1, 1.5, 4, 7.5, 12 use the corrected prefactor C_d = A_d ω_d/(2|Ω|). After the main theorem is corrected, the text should state this explicitly; currently the figure caption attributes the lines to Theorem 1.4 as printed, which would give different slopes.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the asymptotic prefactor and exit-hole law are derived from the Laplacian fundamental solution with no fitted parameters; self-citations are auxiliary.

full rationale

The central derivation is self-contained. The quasi-mode φ_ε = 1 + Σ_k K_ε^{(k)} f_k is an explicit ansatz, and each f_k is constructed as the solution of the point-source Neumann problem (2.6). The leading constant C_d is not fitted to exit-time data; it is fixed by the normalization of the fundamental solution Λ in (2.17) and by the flux computation in Lemma 2.8. Theorems 1.4 and 1.6 then follow from the spectral estimates in Proposition 3.2 and Lemma 3.3, with remainders controlled by explicit bounds (2.33) and (2.56). No displayed equation in the proof reduces to its own conclusion: the quasi-mode's Rayleigh quotient is computed from the PDE, and the exit-hole law is a consequence of the eigenvalue identity and the 'hole removed' quasi-mode (3.17)-(3.21). The citations to the authors' own works ([36], [9]) are used for auxiliary regularity/QSD facts and for context; the core asymptotic claim is proved in this paper. I also flag the known normalization inconsistency in Lemma 2.8: the flux computation (2.30) gives max{d-2,1}ω_d/2 v(x^{(k)}), while Lemma 2.8 asserts -|Ω| C_d v(x^{(k)}) with C_d from (1.14); this is a correctness defect (a factor ω_d^2) in the prefactor of Theorem 1.4, not a circularity. Independent numerical checks in Section 4 target the same constants, so the derivation is externally checkable rather than self-referential.

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

The derivation introduces no fitted parameters: K_ε^(k) are determined by the hole radii, and C_d is derived from the fundamental solution, though printed incorrectly. The mathematical infrastructure (W^{1,p} boundary-Dirac theory, Green's function bounds, Zaremba regularity, uniform spectral gap) is imported from the literature.

assumptions (5)
  • domain assumption Ω is C∞ bounded and the holes are disjoint balls centered at boundary points x^(k).
    Setting (1.1)-(1.3); the boundary-flattening Lemma 2.3 requires C∞ boundary regularity.
  • standard math The W^{1,p} weak formulation of the Neumann problem with a boundary Dirac mass is well-posed.
    Definition 2.2 relies on Aramaki [2, Prop. 2.3]; p<d/(d-1) ensures δ_{x^(k)} ∈ W^{-1/p,p}(∂Ω).
  • standard math Pointwise bounds for the Neumann Green's function of the Laplacian.
    Used in Prop. 2.9 to bound the subsingular terms S_k; cited from Hoff [28] and Taylor-Kim-Brown [52].
  • standard math Zaremba mixed-boundary elliptic regularity and a uniform spectral gap λ_1^ε ≥ C > 0 independent of ε.
    Appendix B and Lemma A.3 depend on Savaré [46], Shamir [47], Mitrea-Mitrea [38], and Poincaré constants [43].
  • domain assumption The holes shrink to points with K_ε → 0 and remain disjoint for small ε.
    Needed for the boundary error estimates in Lemma 3.1 and for the asymptotic expansion to make sense.

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Cite this review

Pith. "Pith review of The narrow escape problem in arbitrary dimension." pith.science (2026). https://pith.science/paper/QF22OWNT

@misc{pith2026260802212,
  author       = {Pith},
  title        = {Pith review of: The narrow escape problem in arbitrary dimension},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QF22OWNT}},
  note         = {Machine review of arXiv:2608.02212}
}
read the original abstract

The narrow escape problem is a prototypical example for studying entropic metastability, motivated by the analysis of biological and chemical systems. The problem concerns the determination of the exit time and position of a Brownian particle trapped in a domain with a reflecting boundary pierced by narrow holes. Our goal is to investigate this problem in a general domain in any dimension (greater than or equal to two), using the quasi-stationary distribution approach to metastability. In particular, we derive the asymptotic expansion of the mean exit time and the law of the exit position in the limit where the hole sizes tend to zero. Our analytical predictions are illustrated by numerical simulations, using dedicated Monte Carlo techniques.

Figures

Figures reproduced from arXiv: 2608.02212 by the authors.

Figure 1
Figure 1. Example of a domain Ωε with N = 3 holes. The blue parts of the boundary are reflecting, while the red parts are absorbing. to refer to the reflecting Neumann region and the absorbing Dirichlet region of the boundary ∂Ωε, respectively. Note that ∂Ωε = Γ ε N ∪ Γ ε D. Let (Xt)t⩾0 be the reflected Brownian motion in Ωε initialized according to a distribution µ0 [51, 55]: dXt = √ 2 dWt − 1∂Ωε (Xt)n(Xt) dLt, with X0 ∼ µ0 … view at source ↗
Figure 2
Figure 2. Visualization of the boundary-flattening diffeomorphism Ψk from Theo￾rem 2.3. Its inverse Φk is given by (2.10). This figure represents the general case where x (k) is not at the origin and nk is not aligned with one of the coordinate axes. Notice first that δx(k) ∈ W− 1 p ,p(∂Ω) for any p ∈ (1, d/(d − 1)), as  W− 1 p ,p(∂Ω)′ ,→ C 0 (∂Ω) for such p. Secondly, since R Ω g = Cd|Ω| and ⟨h, 1⟩ = −Cd|Ω|, the compatibil… view at source ↗
Figure 3
Figure 3. This figure illustrates the domain Uk near the boundary at x (k) , introduced in (2.18). Notice that Uk has been represented as a tube, as it has been constructed using the tubular neighbourhood theorem (see proof of Theorem 2.3). The grey area represents the support of Λe k. Since Λe k = 0 outside of it, there are no continuity issues on the boundary ∂Uk ∩ Ω. Using this function, we construct the function Λe k as (… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Trajectories from a naive Euler–Maruyama algorithm (left) and the Algo￾rithm 1 (right), for a two-dimensional disk with two holes of radius rε = 10−1 . As an illustration, the time step h is set to 10−4 . The Walk on Sphere method. Algorithm 1 uses the WoS method when …
Figure 5
Figure 5. Figure 5: First eigenvalue λ 0 ε = (E[τε])−1 as a function of Kε in dimensions 2 to 6. The black dashed lines are the theoretical estimates from Theorem 1.4, which predict λ 0 ε to be asymptotically linear in Kε with a known prefactor Cd. 4.2. Mean exit time. We first illustrate…
Figure 6
Figure 6. Figure 6: Various domains used to illustrate Theorem 1.4 in dimension 2. The red area represents the holes, of radius 0.1 here [PITH_FULL_IMAGE:figures/full_fig_p027_6.png]
Figure 7
Figure 7. Figure 7: (Left) First eigenvalue λ 0 ε in terms of rε = ε for the various shapes. The grey dashed line is the theoretical slope from Theorem 1.4. (Right) The first eigengap λ 1 ε − λ 0 ε in terms of rε. These results have been obtained with FEM in Gridap (see first paragraph of…
Figure 8
Figure 8. Figure 8: Probability to exit by the large hole in terms of Kε. The black dashed lines are the theoretical predictions of Theorem 1.6. by our results when initializing the Brownian dynamics at the center of the domain, we need: λ 1 ε − λ 0 ε λ0 ε ≫ 1, rε ≪ 1. The second conditio…

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Works this paper leans on

57 extracted references · 24 canonical work pages

  1. [36]

    Leli` evre, M

    T. Leli` evre, M. Rachid, and G. Stoltz. A spectral approach to the narrow escape problem in the disk, 2024. arXiv:2401.06903 [math.AP]

    arXiv 2024

  2. [9]

    Carillo, T

    L. Carillo, T. Leli` evre, T. Normand, and U. Vaes. A spectral approach to the narrow escape problem in two-dimensional domains. Preprint, 2026

    2026

  3. [1]

    Ammari, K

    H. Ammari, K. Kalimeris, H. Kang, and H. Lee. Layer potential techniques for the narrow escape problem. J. Math. Pures Appl. (9), 97(1):66–84, 2012.doi:10.1016/j.matpur.2011.09.011

    doi:10.1016/j.matpur.2011.09.011 2012

  4. [2]

    J. Aramaki. Existence and regularity for the Neumann problem to the Poisson equation and an application to the Maxwell-Stokes type equation. Commun. Math. Anal., 21(1):54–66, 2018

    2018

  5. [3]

    Badia and F

    S. Badia and F. Verdugo. Gridap: An extensible Finite Element toolbox in Julia. Journal of Open Source Software, 5(52):2520, 2020.doi:10.21105/joss.02520

    doi:10.21105/joss.02520 2020

  6. [4]

    B´ enichou and R

    O. B´ enichou and R. Voituriez. Narrow-Escape Time Problem: Time Needed for a Particle to Exit a Confining Domain through a Small Window. Phys. Rev. Lett., 100:168105, 16, Apr. 2008.doi: 10.1103/PhysRevLett.100.168105

    work page doi:10.1103/physrevlett.100.168105 2008

  7. [5]

    D. J. Bicout and A. Szabo. Entropic barriers, transition states, funnels, and exponential protein folding kinetics: a simple model. Protein Sci., 9(3):452–465, 2000.doi:10.1110/ps.9.3.452

    work page doi:10.1110/ps.9.3.452

  8. [6]

    H. Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011, page 613. REFERENCES 39

    2011

Show all 57 references
  1. [7]

    Burdzy and Z.-Q

    K. Burdzy and Z.-Q. Chen. Discrete approximations to reflected Brownian motion. Ann. Probab., 36(2):698–727, 2008.doi:10.1214/009117907000000240

    2008 doi

  2. [8]

    Caginalp and X

    C. Caginalp and X. Chen. Analytical and numerical results for an escape problem. Arch. Ration. Mech. Anal., 203(1):329–342, 2012.doi:10.1007/s00205-011-0455-6

    2012 doi

  3. [10]

    Chen and A

    X. Chen and A. Friedman. Asymptotic analysis for the narrow escape problem. SIAM J. Math. Anal., 43(6):2542–2563, 2011.doi:10.1137/090775257

    2011 doi

  4. [11]

    Chevalier, O

    C. Chevalier, O. B´ enichou, B. Meyer, and R. Voituriez. First-passage quantities of Brownian motion in a bounded domain with multiple targets: a unified approach. J. Phys. A, 44(2):025002, 24, 2011. doi:10.1088/1751-8113/44/2/025002

    2011 doi

  5. [12]

    Collet, S

    P. Collet, S. Mart´ ınez, and J. San Mart´ ın.Quasi-stationary distributions. Probability and its Ap- plications (New York). Springer, Heidelberg, 2013, pages xvi+280.doi:10 . 1007 / 978 - 3 - 642 - 33131-2

    2013

  6. [13]

    Crozet and E

    S. Crozet and E. Bopp. Nalgebra: Linear algebra library for the Rust programming language, 2024

    2024

  7. [14]

    Di Ges` u, T

    G. Di Ges` u, T. Leli` evre, D. Le Peutrec, and B. Nectoux. Jump Markov models and transition state theory: the quasi-stationary distribution approach. Faraday Discuss., 195:469–495, 2016.doi: 10.1039/C6FD00120C

    2016 doi

  8. [15]

    Di Ges` u, T

    G. Di Ges` u, T. Leli` evre, D. Le Peutrec, and B. Nectoux. Sharp asymptotics of the first exit point density. Ann. PDE, 5(1):Paper No. 5, 174, 2019.doi:10.1007/s40818-019-0059-2

    2019 doi

  9. [16]

    C. Dietze. Weyl’s law for Neumann Schr¨ odinger operators on H¨ older domains.S´ eminaire Laurent Schwartz — EDP et applications:1– 9, 2022-2023.doi:10.5802/slsedp.158

    2022 doi

  10. [17]

    J. J. Duistermaat and J. A. C. Kolk. Distributions. Cornerstones. Birkh¨ auser Boston, Inc., Boston, MA, 2010, pages xvi+445.doi:10.1007/978-0-8176-4675-2. Theory and applications, Translated from the Dutch by J. P. van Braam Houckgeest

    2010 doi

  11. [18]

    L. C. Evans. Partial differential equations, volume 19 of Graduate Studies in Mathematics. Amer- ican Mathematical Society, Providence, RI, 1998, pages xviii+662

    1998

  12. [19]

    Felli, B

    V. Felli, B. Noris, and R. Ognibene. Eigenvalues of the Laplacian with moving mixed boundary conditions: the case of disappearing Neumann region. J. Differential Equations, 320:247–315, 2022. doi:10.1016/j.jde.2022.02.052

    2022 doi

  13. [20]

    M. I. Freidlin and A. D. Wentzell. Diffusion processes on graphs and the averaging principle. Ann. Probab., 21(4):2215–2245, 1993

    1993

  14. [21]

    D. Frenkel. Entropy-driven phase transitions. Physica A: Statistical Mechanics and its Applications, 263(1):26–38, 1999.doi:https://doi.org/10.1016/S0378- 4371(98)00501- 9. Proceedings of the 20th IUPAP International Conference on Statistical Physics

    1999 doi

  15. [22]

    Geuzaine and J.-F

    C. Geuzaine and J.-F. Remacle. Gmsh: A 3-D Finite Element Mesh Generator with Built-in Pre- and Post-Processing Facilities. International Journal for Numerical Methods in Engineering, 79:1309– 1331, Sept. 2009.doi:10.1002/nme.2579

    2009 doi

  16. [23]

    Gilbarg and N

    D. Gilbarg and N. S. Trudinger. Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin, 2001, pages xiv+517. Reprint of the 1998 edition

    2001

  17. [24]

    Gol’dshtein, I

    V. Gol’dshtein, I. Mitrea, and M. Mitrea. Hodge decompositions with mixed boundary conditions and applications to partial differential equations on Lipschitz manifolds. In volume 172, number 3, pages 347–400. 2011.doi:10.1007/s10958-010-0200-y. Problems in mathematical analysi...

    2011 doi

  18. [25]

    Gomez and A

    D. Gomez and A. F. Cheviakov. Asymptotic analysis of narrow escape problems in nonspherical three-dimensional domains. Phys. Rev. E, 91:012137, 1, Jan. 2015.doi:10.1103/PhysRevE.91. 012137

    2015 doi

  19. [26]

    Grisvard

    P. Grisvard. Elliptic problems in nonsmooth domains, volume 69 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011, pages xx+410.doi:10 . 1137 / 1 . 9781611972030 . ch1. Reprint of the 1985 original [MR0775683], Wit...

    2011

  20. [27]

    Hoel and S

    H. Hoel and S. Ragunathan. Higher-order adaptive methods for exit times of Itˆ o diffusions. IMA J. Numer. Anal., 44(5):2821–2863, 2024.doi:10.1093/imanum/drad077

    2024 doi

  21. [28]

    D. Hoff. Cancellation properties and pointwise bounds for the Green’s functions for the Laplace operator. J. Differential Equations, 427:601–640, 2025.doi:10.1016/j.jde.2025.01.083

    2025 doi

  22. [29]

    Holcman and Z

    D. Holcman and Z. Schuss. Escape Through a Small Opening: Receptor Trafficking in a Synaptic Membrane. Journal of Statistical Physics, 117(5):975–1014, 2004.doi:10 . 1007 / s10955 - 004 - 5712-8

    2004

  23. [30]

    W.-T. Hsu. Asymptotic analysis on narrow tubes: narrow escape problems and diffusion processes,

  24. [31]

    W.-T. Hsu. Metastability of diffusion processes in narrow tubes, 2025. Preprint, arXiv:2511.21548

    2025

  25. [32]

    Jakab, I

    T. Jakab, I. Mitrea, and M. Mitrea. On the regularity of differential forms satisfying mixed boundary conditions in a class of Lipschitz domains. Indiana Univ. Math. J., 58(5):2043–2071, 2009.doi: 10.1512/iumj.2009.58.3678

    doi

  26. [33]

    Le Bris, T

    C. Le Bris, T. Leli` evre, M. Luskin, and D. Perez. A mathematical formalization of the parallel replica dynamics. Monte Carlo Methods Appl., 18(2):119–146, 2012.doi:10.1515/mcma-2012-0003

    2012 doi

  27. [34]

    J. M. Lee. Introduction to smooth manifolds, volume 218 of Graduate Texts in Mathematics. Springer, New York, second edition, 2013, pages xvi+708

    2013

  28. [35]

    Leli` evre, D

    T. Leli` evre, D. Le Peutrec, and B. Nectoux. Eyring-Kramers exit rates for the overdamped Langevin dynamics: the case with saddle points on the boundary, 2022. arXiv:2207.09284 [math.PR]

    2022 arXiv

  29. [37]

    X. Li. Matched asymptotic analysis to solve the narrow escape problem in a domain with a long neck. J. Phys. A, 47(50):505202, 18, 2014.doi:10.1088/1751-8113/47/50/505202

    2014 doi

  30. [38]

    Mitrea and M

    I. Mitrea and M. Mitrea. The Poisson problem with mixed boundary conditions in Sobolev and Besov spaces in non-smooth domains. Trans. Amer. Math. Soc., 2007.doi:10.1090/S0002-9947- 07-04146-3

    2007 doi

  31. [39]

    J. Moser. A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differ- ential equations. Comm. Pure Appl. Math., 13:457–468, 1960.doi:10.1002/cpa.3160130308

    1960 doi

  32. [40]

    M. E. Muller. Some continuous Monte Carlo methods for the Dirichlet problem. Ann. Math. Statist., 27:569–589, 1956.doi:10.1214/aoms/1177728169

    1956

  33. [41]

    Nursultanov, J

    M. Nursultanov, J. C. Tzou, and L. Tzou. On the mean first arrival time of Brownian particles on Riemannian manifolds. J. Math. Pures Appl. (9), 150:202–240, 2021.doi:10.1016/j.matpur. 2021.04.006

    2021 doi

  34. [42]

    I. Y. Popov. Extension theory and localization of resonances for domains of trap type. Math. USSR-Sb., 71(1):209–234, 1992.doi:10.1070/SM1992v071n01ABEH001394

    1992 doi

  35. [43]

    D. Ruiz. On the uniformity of the constant in the Poincar´ e inequality. Adv. Nonlinear Stud., 12(4):889–903, 2012.doi:10.1515/ans-2012-0413

    2012 doi

  36. [44]

    Rupprecht, O

    J.-F. Rupprecht, O. B´ enichou, D. Grebenkov, and R. Voituriez. Exit Time Distribution in Spher- ically Symmetric Two-Dimensional Domains. Journal of Statistical Physics, 158, Sept. 2014.doi: 10.1007/s10955-014-1116-6

    2014 doi

  37. [45]

    M. A. Sadybekov, B. T. Torebek, and B. K. Turmetov. Representation of Green’s function of the Neumann problem for a multi-dimensional ball. Complex Var. Elliptic Equ., 61(1):104–123, 2016. doi:10.1080/17476933.2015.1064402

    2016

  38. [46]

    Savar´ e

    G. Savar´ e. Regularity and perturbation results for mixed second order elliptic problems. Comm. Partial Differential Equations, 22(5-6):869–899, 1997.doi:10.1080/03605309708821287

    1997 doi

  39. [47]

    E. Shamir. Regularization of mixed second-order elliptic problems. Israel J. Math., 6:150–168, 1968. doi:10.1007/BF02760180

    1968 doi

  40. [48]

    Silbergleit, I

    A. Silbergleit, I. Mandel, and I. Nemenman. Potential and field singularity at a surface point charge. J. Math. Phys., 44(10):4460–4466, 2003.doi:10.1063/1.1605497

    2003 doi

  41. [49]

    Singer, Z

    A. Singer, Z. Schuss, and D. Holcman. Narrow escape. III. Non-smooth domains and Riemann surfaces. J. Stat. Phys., 122(3):491–509, 2006.doi:10.1007/s10955-005-8028-4. REFERENCES 41

    2006 doi

  42. [50]

    Stone and N

    J. Stone and N. Matsakis. Rayon: a data parallelism library for Rust, 2024

    2024

  43. [51]

    H. Tanaka. Stochastic differential equations with reflecting boundary condition in convex regions. Hiroshima Math. J., 9(1):163–177, 1979

    1979

  44. [52]

    J. L. Taylor, S. Kim, and R. M. Brown. The Green function for elliptic systems in two dimen- sions. Comm. Partial Differential Equations, 38(9):1574–1600, 2013.doi:10 . 1080 / 03605302 . 2013.814668

    2013

  45. [53]

    T. R. Team. The Rust programming language, 2020

    2020

  46. [54]

    N. S. Trudinger. Linear elliptic operators with measurable coefficients. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), 27:265–308, 1973

    1973

  47. [55]

    Wang and J

    Y. Wang and J. Li. From reflecting brownian motion to reflected stochastic differential equations: a systematic survey and complementary study. SSRN Electronic Journal, Oct. 2020.doi:10.2139/ ssrn.3688563

    2020

  48. [56]

    M. J. Ward and J. B. Keller. Strong localized perturbations of eigenvalue problems. SIAM J. Appl. Math., 53(3):770–798, 1993.doi:10.1137/0153038

    1993 doi

  49. [2025]

    Preprint, arXiv:2508.15060

    arXiv

Pith tools

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