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Recovering semipermeable barriers from reflected Brownian motion

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Pith's one-line read Discrete tracks of a reflected Brownian motion reveal the hidden barriers that shape them.

desk verdict First real treatment of semipermeable barrier recovery from sampled Brownian motion; solid proofs, heavy mixing assumptions, deserves refereeing. read the letter →

arxiv 2412.14740 v1 pith:E5VNGCLJ submitted 2024-12-19 math.PR math.STstat.TH

Alexander Van Werde , Jaron Sanders This is my paper · ORCID
classification math.PRmath.STstat.TH MSC 60J6562M0562G05
keywords semipermeablebarriersreflectedBrownianmotionsnappingoutsetestimationbarrierrecoverytransitionkerneldiscontinuityHausdorffdistance
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 claims that the location and shape of semipermeable barriers can be recovered from discrete-time samples of a reflected Brownian motion, and it gives explicit algorithms with quantitative guarantees. The guarantees are strongest when the observation period is long, with a polynomial error rate for fixed sampling frequency and an exponential rate for high-frequency data. The authors identify three qualitatively different recovery regimes and show that the difficulty depends on interpretable environmental parameters such as barrier curvature, permeability, stationary density, and mixing time. If correct, this means that animal tracking or particle-tracking data can be used to locate the obstacles that shape the observed movements, even when the barriers are only partially permeable.

What carries the argument

The key machinery is a local coupling between the semipermeably reflected process and a classical reflected Brownian motion reflecting on a straight line. Lemma 3.5 shows that, locally, a smooth barrier is nearly a straight line, and Lemmas 3.7-3.12 bound the error of this approximation in terms of the curvature, permeability, and a spatial scale parameter. This reduces the core estimation problem to one-dimensional reflected Brownian motion computations, and also provides the discontinuity of the transition kernel that the detection algorithms exploit.

What would settle it

Run Algorithm 1 on a simulated process in a domain with a known barrier but with a stationary measure that is exponentially small in some subregion, and check whether the recovered barrier misses the barrier in that subregion even when the observation time exceeds the bound in (2.8).

Watch

Extended reading notes

Core claim

The paper establishes that semipermeable barriers can be recovered from discrete samples of a reflected Brownian motion, provided the process mixes and explores the domain. The central results are Theorem 2.3, which gives complete recovery with error decaying as $T^{-2/3}$ for fixed sampling rate; Theorem 2.5, which gives partial recovery of the barrier pieces that were actually hit, with error of order $\sqrt{t}$; and Theorem 2.7, which shows that in the high-frequency, large-$T$ regime the error decays exponentially as $\exp(-c\sqrt{T})$. The algorithms detect barriers by looking for discontinuities in the one-step transition kernel: the process typically stays on one side of a barrier for a random amount of time, so nearby starting points separated by a barrier lead to very different short-time distributions.

Load-bearing premise

The recovery guarantees require the process to mix and explore the entire domain within the observation period, so the stationary density must be bounded below everywhere and the mixing time must be finite; if part of the domain is visited extremely rarely, no finite observation time is guaranteed to recover a barrier there.

Editorial extensions

If this is right

  • If the estimates are correct, one can locate semipermeable barriers from discrete movement data at a rate that degrades only polynomially with the sampling interval in the fixed-frequency regime and exponentially in the high-frequency regime.
  • The recovery algorithms can be applied to real-life animal tracking data, as illustrated by the reindeer case study in which the method recovers both impermeable (coastline, slopes) and semipermeable (rivers) barriers.
  • The dependence of the error on interpretable parameters such as mixing time, minimum stationary density, and curvature provides a way to predict when recovery is feasible and how accurate it will be.
  • The paper provides a starting point for estimating permeability parameters, since knowing the barrier locations is a prerequisite for estimating how easily the process crosses them.
  • The results imply that the boundary-estimation problem for reflected Brownian motion changes qualitatively with sampling frequency, potentially leading to faster convergence when high-frequency data is available.

Reading between the lines

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

  • The discontinuity in the transition kernel that the algorithms exploit is likely to be useful beyond the specific process considered: any stochastic process that spends a positive amount of local time on a barrier should produce a detectable short-time effect, so the method might extend to other reflected or switching diffusions.
  • If the polynomial $T^{-2/3}$ rate is indeed optimal, as the paper expects, then the exponential rate in the high-frequency regime implies that increasing the sampling rate is fundamentally more powerful than merely increasing the observation period.
  • A testable extension is to verify the sensitivity of the algorithm to the mixing time: in domains with a low-permeability barrier that traps the process, the recovery guarantee weakens dramatically, so one could design a simulation study to measure how the error depends on the trapping strength.
  • The case study suggests that the method could be used to compare the permeability of different types of linear infrastructure (roads, rivers, fences) in ecological studies, which would complement existing simulation-based approaches.
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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

0 major / 6 minor

Summary. The paper studies the statistical recovery of the union of the outer boundary and m interior semipermeable barriers of a reflected Brownian motion with semipermeable barriers in a bounded planar domain, from samples {X_{jt}: j=0,...,⌊T/t⌋}. The main results are three regimes: Theorem 2.3 gives complete recovery with Hausdorff error ε when T ≳ (tmix/πmin)√(κ/ε^3) log(...), i.e. ε ~ T^{-2/3} up to logarithms, using Algorithms 1 and 2; Theorem 2.5 gives partial recovery of the visited barrier pieces with error O(√t log(T/t)) for fixed T and high-frequency sampling; and Theorem 2.7, for m=0, gives exponential convergence exp(-c√T) for the trace of the boundary in the high-frequency regime. The proofs use local approximation of barriers by straight lines, coupling with reflected Brownian motion on a line, Markov-chain concentration inequalities, and covering-time estimates. The paper also presents an application to reindeer movement data.

Significance. If the paper's claims hold, it supplies the first statistical recovery guarantees for semipermeable barrier locations from discrete trajectory data. The proofs are unusually transparent: the algorithms are explicit, no fitted parameters enter the rates, and the dependence on geometry, permeability, mixing, and sampling is quantified through interpretable constants. The coupling reduction to one-dimensional reflected Brownian motion in Section 3.2 is a genuine technical contribution. The main caveats are that the stated T^{-2/3} optimality is an analogy with boundary estimation from i.i.d. samples rather than a proved lower bound, Theorem 2.7 is limited to the outer-barrier case, and the guarantees in Theorem 2.3 inherit the exploration/mixing requirement encoded in tmix/πmin. These are limitations, not internal inconsistencies, and the authors are appropriately explicit about them.

minor comments (6)
  1. [Appendix F (proof of Proposition 4.1)] The proof of Proposition 4.1 does not explicitly cover the outer boundary B0: Corollary 4.9 applies to two D-valued processes starting on opposite sides of a barrier, while the negative side of B0 lies outside D. Since the paper has already defined the extension W_u1(P,o)=u, the missing case can be handled by comparing the transition kernel of an interior box adjacent to B0 with the zero measure; adding this one-sentence argument would make the proof complete.
  2. [Appendix G.4] In the count of rectangles before equation (G.28), the area of R(j,k,n,h) is stated as ϵℓ; from the definition in Algorithm 2 the rectangle has side lengths ϵ and ℓ/5, so the area is ϵℓ/5. The factor can be absorbed into the constants, but the displayed formula is incorrect.
  3. [Section 2.2 / Theorem 2.3] The parameters πmin and tmix appear in the denominator of (2.8), but the statements do not explicitly require πmin>0 and tmix<∞. For the class of processes considered these are true, but the hypotheses should state this to make the conditional guarantee non-vacuous.
  4. [Lemma 4.10] The symbol PS in Lemma 4.10 is introduced as P(Xt∈·|X0∈S), but the proof and (E.1) use the stationary conditional distribution π(S)^{-1}∫_S P(Xt∈·|X0=x0)dπ(x0); the two definitions coincide only under that interpretation and should be stated.
  5. [Sections 2.1 and 2.3.3] The notation X0 denotes both the initial value of the process and the set of boundary points visited by time T in (2.10) and (2.14); this overload is confusing and should be resolved.
  6. [Appendix D (proof of Lemma 4.8)] The sentence 'note that rδ/2 = rδ/2' is tautological; it should say that one applies Corollary 3.11 with the radius rδ/2, which yields the displayed bound with δ/2 and four times the t/rδ^2 term.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the recovery guarantees are derived from the assumed reflected-Brownian model through explicit coupling, concentration, and external covering-time results, with no fitted parameter presented as a prediction.

full rationale

The paper's central claims are self-contained in the sense required here. The identification of barriers with discontinuities of the transition kernel is not assumed by definition: Proposition 4.6 proves continuity of the transition kernel on a fixed side of the barriers, while Lemma 4.7, Lemma 4.8, and Corollary 4.9 prove that crossing a barrier creates a genuine discontinuity. These results are derived from the coupling estimates in Lemmas 3.5-3.10 and Corollary 3.12, which locally approximate the curved barrier by a straight reflecting line and control the error via curvature, permeability, and spacing parameters. The statistical guarantees of Algorithms 1-3 are then obtained by concentration arguments (Lemma 4.10 and its appendices) over the empirical transition kernels, not by inserting the true barrier locations into the algorithm. No parameter in the theorems is fitted to a subset of the data and then reported as a prediction; the sample size bounds contain tmix/pi_min, which is an explicit and honest exploration/mixing constraint, and the paper states that the guarantee degrades when this quantity is unfavorable. The external citations (Lejay for the process, Matthews and Chen-Friedman for covering times, Paulin for Markov-chain concentration) are standard tools used independently of the recovery conclusion, and there are no self-citations carrying a load-bearing argument. The conjectures about optimality are explicitly flagged as expectations rather than theorems. I therefore find no circular step and no reduction, by the paper's own equations, of any output to an input.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central results rest on modeling assumptions (smooth separated barriers, stationarity, πmin>0, finite tmix) plus standard external theorems. The theory itself introduces no fitted parameters and no new physical entities; semipermeable barriers and local times are adopted from Lejay's snapping out Brownian motion.

assumptions (6)
  • domain assumption Existence and pathwise uniqueness of reflected Brownian motion with semipermeable barriers, Definition 1
    Proved in Proposition 3.1 by gluing classical reflected Brownian motions; the modeling assumption that observed movement follows this process is the starting point.
  • domain assumption Barriers B0,...,Bm are C∞ smooth, nonintersecting, simple closed curves; D is bounded and simply connected
    Used throughout, including the curvature bound κ in (2.3), the separation parameter ρ in (2.4), and the local straight-line approximation Lemma 3.5.
  • domain assumption The process starts in stationarity for Theorem 2.3, with X0 ~ π
    Assumed in Theorem 2.3 and Lemma 4.10; the empirical transition kernel estimates require the chain to be in its stationary regime.
  • domain assumption The stationary measure has πmin > 0 and finite mixing time tmix for the complete-recovery results
    The observation time bound (2.8) is inversely proportional to πmin and linear in tmix; if either degenerates, no finite-sample guarantee is stated.
  • standard math Narrow escape asymptotics of Chen and Friedman [15] and covering time estimates of Matthews [41] for reflected Brownian motion in smooth domains
    Used in Lemma 6.1 and the proof of Theorem 2.7; both are external theorems taken as background.
  • standard math Bernstein-type concentration for Markov chains from Paulin [44] and standard Gaussian or reflection-principle bounds
    Used in Lemma 4.10, Appendix E, and Lemma 5.6 to control deviations of empirical transition counts.

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Pith. "Pith review of Recovering semipermeable barriers from reflected Brownian motion." pith.science (2026). https://pith.science/paper/E5VNGCLJ

@misc{pith2026241214740,
  author       = {Pith},
  title        = {Pith review of: Recovering semipermeable barriers from reflected Brownian motion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E5VNGCLJ}},
  note         = {Machine review of arXiv:2412.14740}
}
abstract

We study the recovery of one-dimensional semipermeable barriers for a stochastic process in a planar domain. The considered process acts like Brownian motion when away from the barriers and is reflected upon contact until a sufficient but random amount of interaction has occurred, determined by the permeability, after which it passes through. Given a sequence of samples, we wonder when one can determine the location and shape of the barriers. This paper identifies several different recovery regimes, determined by the available observation period and the time between samples, with qualitatively different behavior. The observation period $T$ dictates if the full barriers or only certain pieces can be recovered, and the sampling rate significantly influences the convergence rate as $T\to \infty$. This rate turns out polynomial for fixed-frequency data, but exponentially fast in a high-frequency regime. Further, the environment's impact on the difficulty of the problem is quantified using interpretable parameters in the recovery guarantees, and is found to also be regime-dependent. For instance, the curvature of the barriers affects the convergence rate for fixed-frequency data, but becomes irrelevant when $T\to \infty$ with high-frequency data. The results are accompanied by explicit algorithms, and we conclude by illustrating the application to real-life data.

Figures

Figures reproduced from arXiv: 2412.14740 by the authors.

Figure 1
Figure 1. A simulated sample path of a reflected Brownian motion with semipermeable barriers, and the same data observed at a finite sampling rate. Our goal is to recover the underlying barriers given a finite number of samples. The current paper studies the statistical problem which aims to recover the location of the barriers based on a sequence of samples {Xit : i = 0, 1, . . . , ⌊T /t⌋}. Our goal is to construct estimator… view at source ↗
Figure 2
Figure 2. Visualization for our recovery notions. Partial recovery aims to recover the parts of the barrier which were hit by the continuous-time process based on discrete-time samples. Complete recovery is more demanding and asks to recover the barriers completely. The parameters T and t have a pronounced effect in our results, giving fundamental regimes where qualitatively different guarantees become possible. The regimes m… view at source ↗
Figure 3
Figure 3. Visualization of Lemma 3.5. Locally in B(x0, rδ), the barrier is contained in a strip of points y with ⟨y, nˆ⟩ ≈ c, and the normal vectors ⃗ni are well-approximated by nˆ. 3.2.2. Coupling with a process reflecting on a straight line. Let c ∈ R and nˆ ∈ R 2 be as in Lemma 3.5 and define a straight line by A + := {y ∈ R 2 : ⟨y, si(0)ˆn⟩ = si(0)c + 4δrδ}, (3.4) Then, we will compare Xt with a process Y + t which reflec… view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Visualization of the process Y + t from Section 3.2.2. The figures on the left and right correspond to the first and second case of (3.5), respectively. In particular, the function f(t) := ⟨Xmin{t,τ} − Y + min{t,τ} , si(0)nˆ⟩ can only increase at times when Xmin{t,τ} ∈…
Figure 5
Figure 5. Figure 5: Visualization for the coupling used in the proof of Proposition 4.6. probabilities from (4.7) and (4.8), P [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]
Figure 6
Figure 6. Figure 6: together with one of the employed tracks. We recognize both impermeable and semipermeable barriers in the algorithm’s output. The coastline and the slopes bordering the valley yield impermeable barriers, constraining the animals, while semipermeable barriers arise from…
Figure 7
Figure 7. Figure 7: Visualization of the construction which is used in the proof of existence for Proposition 3.1. The process X starts by following a classical reflected Brownian motion with initial condition X0, which we denote by Y [0] . When a random variable τ0 indicates that the loc…
Figure 8
Figure 8. Figure 8: Visualization of the event from (D.1) in the proof of Lemma 4.8. Note that every point in the blue spherical cap is at distance ≥ c2 √ t from ∪ m i=0Bi . Appendix D. Proof of Lemma 4.8 Proof of Lemma 4.8. We rely on the tools from Section 3.2. Let c1, c2 be as in Lemma…
Figure 9
Figure 9. Figure 9: Visualization of the binning operation used in Appendix E. C2/c3 ≤ η/3 and 4C5/c2 3 ≤ η/3, and subsequently takes c1, c2 to be sufficiently small to ensure that the contribution of all remaining terms is ≤ η/3 and that the conditions hold. It remains to consider the ca…
Figure 10
Figure 10. Figure 10: Visualization of the sets R+(p, ⃗v ) and Rb+ considered in Lemma 3.5. By the Cauchy–Schwarz inequality and the fact that q ∈ B(x0, rδ), |⟨q − x0, ⃗v − si(0)ˆn⟩| ≤ rδ(∥⃗v − si(0)⃗ni(x0)∥ + ∥si(0)ˆn − si(0)⃗ni(x0)∥). (J.6) Taking c2 sufficiently small in (J.2), it can b…
Figure 11
Figure 11. Figure 11: Full image for the barriers uncovered by Algorithm 1. 62 [PITH_FULL_IMAGE:figures/full_fig_p062_11.png]

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Works this paper leans on

62 extracted references · 51 canonical work pages

  1. [1]

    Aamari, C

    E. Aamari, C. Aaron, and C. Levrard. Minimax boundary estimation and estimation with boundary.Bernoulli, 2023. doi:10.3150/23-BEJ1585

    work page doi:10.3150/23-bej1585 2023

  2. [2]

    Albrecht, C.M

    D. Albrecht, C.M. Winterflood, M. Sadeghi, T. Tschager, F. Noé, and H. Ewers. Nanoscopic compartmentalization of membrane protein motion at the axon initial segment. Journal of Cell Biology, 2016. doi:10.1083/jcb.201603108

    work page doi:10.1083/jcb.201603108 2016

  3. [3]

    Aldous, L

    D. Aldous, L. Lovász, and P. Winkler. Mixing times for uniformly ergodic Markov chains. Stochastic Processes and their Applications, 1997. doi:10.1016/S0304-4149(97) 00037-9

    work page doi:10.1016/s0304-4149(97 1997

  4. [4]

    Anderson and S

    R.F. Anderson and S. Orey. Small random perturbation of dynamical systems with reflecting boundary. Nagoya Mathematical Journal, 1976. doi:10.1017/ S0027763000017232

    work page 1976

  5. [5]

    Y. Bai, Y. Wang, H. Zhang, and X. Zhuo. Bayesian estimation of the skew Ornstein-Uhlenbeck process. Computational Economics, 2022. doi:10.1007/ s10614-021-10156-z

    work page 2022

  6. [6]

    Barahona, L

    M. Barahona, L. Rifo, M. Sepúlveda, and S. Torres. A simulation-based study on Bayesian estimators for the skew Brownian motion. Entropy, 2016. doi:10.3390/ e18070241

    work page 2016

  7. [7]

    Bardou and M

    O. Bardou and M. Martinez. Statistical estimation for reflected skew processes.Statis- tical Inference for Stochastic Processes, 2010. doi:10.1007/s11203-010-9047-6

    work page doi:10.1007/s11203-010-9047-6 2010

  8. [8]

    Beyer, E

    H.L. Beyer, E. Gurarie, L. Börger, M. Panzacchi, M. Basille, I. Herfindal, B. Van Moorter, S. R. Lele, and J. Matthiopoulos. ‘You shall not pass!’: quanti- fying barrier permeability and proximity avoidance by animals.Journal of Animal Ecology, 2014. doi:10.1111/1365-2656.12275

    work page arXiv 2014

Show all 62 references
  1. [9]

    J.P.N. Bishwal. Parameter estimation in stochastic differential equations. Springer, 2007

    2007

  2. [10]

    Bressloff

    P.C. Bressloff. A probabilistic model of diffusion through a semi-permeable barrier. Proceedings of the Royal Society A, 2022. doi:10.1098/rspa.2022.0615

    2022

  3. [11]

    Bressloff

    P.C. Bressloff. Renewal equations for single-particle diffusion in multilayered media. SIAM Journal on Applied Mathematics, 2023. doi:10.1137/23M1545835

    2023 doi

  4. [12]

    V.-E. Brunel. Methods for estimation of convex sets.Statistical Science, 2018. doi: 10.1214/18-STS669

    2018 doi

  5. [13]

    Burdzy, Z.-Q

    K. Burdzy, Z.-Q. Chen, and D.E. Marshall. Traps for reflected Brownian motion. Mathematische Zeitschrift, 2006. doi:10.1007/s00209-005-0849-y

    2006 doi

  6. [14]

    Burdzy, Z.-Q

    K. Burdzy, Z.-Q. Chen, and J. Sylvester. The heat equation and reflected Brownian motion in time-dependent domains.Annals of Probability, 2004. doi:10.1214/aop/ 1079021464

    2004 doi

  7. [15]

    Chen and A

    X. Chen and A. Friedman. Asymptotic analysis for the narrow escape problem.SIAM journal on mathematical analysis, 2011. doi:10.1137/090775257

    2011 doi

  8. [16]

    Cholaquidis, R

    A. Cholaquidis, R. Fraiman, and M. Hernández-Banadik. Home-range estimation under a restricted sample scheme. Journal of Nonparametric Statistics, 2024. doi: 10.1080/10485252.2023.2280003

    2024

  9. [17]

    Cholaquidis, R

    A. Cholaquidis, R. Fraiman, G. Lugosi, and B. Pateiro-López. Set estimation from reflected Brownian motion.Journal of the Royal Statistical Society Series B: Statistical Methodology, 2016. doi:10.1111/rssb.12149

    2016 doi

  10. [18]

    Cholaquidis, R

    A. Cholaquidis, R. Fraiman, E. Mordecki, and C. Papalardo. Level set and drift estimation from a reflected Brownian motion with drift.Statistica Sinica, 2021. doi: 10.5705/ss.202018.0211. 26

    2021

  11. [19]

    A. Cuevas. Set estimation: Another bridge between statistics and geometry.Boletín de Estadística e Envestigación Operativa, 2009

    2009

  12. [20]

    S. Dineen. Multivariate calculus and geometry. Springer, 2014. doi:10.1007/ 978-1-4471-6419-7

    2014

  13. [21]

    Erhard, T

    D. Erhard, T. Franco, and D.S. da Silva. The slow bond random walk and the snapping out Brownian motion.Annals of Applied Probability, 2021. doi:10.1214/20-AAP1584

    2021 doi

  14. [22]

    R. Forien. Gene flow across geographical barriers—scaling limits of random walks with obstacles. Stochastic Processes and their Applications, 2019. doi:10.1016/j.spa.2018. 10.006

    2019 doi

  15. [23]

    Harrison and L.A

    J.M. Harrison and L.A. Shepp. On skew Brownian motion.The Annals of Probability,

  16. [24]

    Höfling and T

    F. Höfling and T. Franosch. Anomalous transport in the crowded world of biological cells. Reports on Progress in Physics, 2013. doi:10.1088/0034-4885/76/4/046602

    2013 doi

  17. [25]

    Holcman and Z

    D. Holcman and Z. Schuss. Escape through a small opening: receptor traffick- ing in a synaptic membrane. Journal of Statistical Physics, 2004. doi:10.1007/ s10955-004-5712-8

    2004

  18. [26]

    Karr.Point processes and their statistical inference

    A. Karr.Point processes and their statistical inference. Marcel Dekker, New York, 1991

    1991

  19. [27]

    Kusumi, C

    A. Kusumi, C. Nakada, K. Ritchie, K. Murase, K. Suzuki, H. Murakoshi, R.S. Kasai, J. Kondo, and T. Fujiwara. Paradigm shift of the plasma membrane concept from the two-dimensional continuum fluid to the partitioned fluid: high-speed single-molecule tracking of membrane molecul...

    2005

  20. [28]

    Kutoyants.Statistical inference for ergodic diffusion processes

    Y.A. Kutoyants.Statistical inference for ergodic diffusion processes. Springer Science & Business Media, 2013

    2013

  21. [29]

    A. Lejay. On the constructions of the skew Brownian motion.Probability Surveys, 2006. doi:10.1214/154957807000000013

    2006 doi

  22. [30]

    A. Lejay. The snapping out Brownian motion.The Annals of Applied Probability, 2016. doi:10.1214/15-AAP1131

    2016 doi

  23. [31]

    A. Lejay. Estimation of the bias parameter of the skew random walk and application to the skew Brownian motion. Statistical Inference for Stochastic Processes, 2018. doi:10.1007/s11203-017-9161-9

    2018 doi

  24. [32]

    A. Lejay. A Monte Carlo estimation of the mean residence time in cells surrounded by thin layers.Mathematics and Computers in Simulation, 2018. doi:10.1016/j.matcom. 2017.05.008

    2018 doi

  25. [33]

    Lejay, E

    A. Lejay, E. Mordecki, and S. Torres. Is a Brownian motion skew? Scandinavian Journal of Statistics, 2014. doi:10.1111/sjos.12033

    2014 doi

  26. [34]

    Lejay, E

    A. Lejay, E. Mordecki, and S. Torres. Two consistent estimators for the skew Brownian motion. EEAIM: Probability and Statistics, 2019. doi:10.1051/ps/2018018

    2019

  27. [35]

    Lejay and P

    A. Lejay and P. Pigato. Statistical estimation of the Oscillating Brownian Motion. Bernoulli, 2018. doi:10.3150/17-BEJ969

    2018 doi

  28. [36]

    Lejay and P

    A. Lejay and P. Pigato. Maximum likelihood drift estimation for a threshold diffusion. Scandinavian Journal of Statistics, 2020. doi:10.1111/sjos.12417

    2020 doi

  29. [37]

    Levin and Y

    D.A. Levin and Y. Peres.Markov chains and mixing times. American Mathematical Society, second edition, 2017

    2017

  30. [38]

    Lions and A.-S

    P.-L. Lions and A.-S. Sznitman. Stochastic differential equations with reflecting boundary conditions. Communications on Pure and Applied Mathematics, 1984. doi:10.1002/cpa.3160370408

    1984 doi

  31. [39]

    Loe, B.B

    L.E. Loe, B.B. Hansen, A. Stien, S.D. Albon, R. Bischof, A. Carlsson, R J. Irvine, M. Meland, I.M. Rivrud, E. Ropstad, V. Verbjørn, and A. Mysterud. Behavioral 27 buffering of extreme weather events in a high-Arctic herbivore.Ecosphere, 2016. doi: 10.1002/ecs2.1374

    2016 doi

  32. [40]

    OnaBrownianmotionwithahardmembrane

    V.MandrekarandA.Pilipenko. OnaBrownianmotionwithahardmembrane. Statistics & Probability Letters, 2016. doi:10.1016/j.spl.2016.02.005

    2016 doi

  33. [41]

    Matthews

    P. Matthews. Covering problems for Brownian motion on spheres. The Annals of Probability, 1988. doi:10.1214/aop/1176991894

    1988

  34. [42]

    Pankrashkin

    K. Pankrashkin. An inequality for the maximum curvature through a geometric flow. Archiv der Mathematik, 2015. doi:10.1007/s00013-015-0804-z

    2015 doi

  35. [43]

    Paquette and F.-J

    S.R. Paquette and F.-J. Lapointe. A statistical procedure to assess the significance level of barriers to gene flow.Journal of Genetics and Genomics, 2009. doi:10.1016/ S1673-8527(08)60161-7

    2009

  36. [44]

    D. Paulin. Concentration inequalities for Markov chains by Marton couplings and spectral methods.Electronic Journal of Probability, 2015. doi:10.1214/EJP.v20-4039

    2015 doi

  37. [45]

    Paviolo, F.N

    C. Paviolo, F.N. Soria, J.S. Ferreira, A. Lee, L. Groc, E. Bezard, and L. Cognet. Nanoscale exploration of the extracellular space in the live brain by combining single carbon nanotube tracking and super-resolution imaging analysis.Methods, 2020. doi: 10.1016/j.ymeth.2019.03.005

    2020 doi

  38. [46]

    G Peyré and M. Cuturi. Computational optimal transport: With applications to data science. Foundations and Trends in Machine Learning, 2019. doi:10.1561/ 2200000073

    2019

  39. [47]

    Pommerenke.Boundary behaviour of conformal maps

    C. Pommerenke.Boundary behaviour of conformal maps. Springer Science & Business Media, 2013. doi:10.1007/978-3-662-02770-7

    2013 doi

  40. [48]

    Remon, E

    J. Remon, E. Chevallier, J.G. Prunier, M. Baguette, and S. Moulherat. Estimating the permeability of linear infrastructures using recapture data.Landscape Ecology, 2018. doi:10.1007/s10980-018-0694-0

    2018 doi

  41. [49]

    Ringbauer, A

    H. Ringbauer, A. Kolesnikov, D.L. Field, and N.H. Barton. Estimating barriers to gene flow from distorted isolation–by–distance patterns.Genetics, 2018. doi:10.1534/ genetics.117.300638

    2018

  42. [50]

    Sadegh, J.L

    S. Sadegh, J.L. Higgins, P.C. Mannion, M.M. Tamkun, and D. Krapf. Plasma membrane is compartmentalized by a self-similar cortical actin meshwork.Physical Review X, 2017. doi:10.1103/PhysRevX.7.011031

    2017 doi

  43. [51]

    Sawyer, Matthew J

    H. Sawyer, Matthew J. Kauffman, A.D. Middleton, T.A. Morrison, R.M. Nielson, and T.B. Wyckoff. A framework for understanding semi-permeable barrier effects on migratory ungulates. Journal of Applied Ecology, 2013. doi:10.1111/1365-2664. 12013

    2013 doi

  44. [52]

    Schumm and P.C

    R.D. Schumm and P.C. Bressloff. A numerical method for solving snapping out Brownian motion in 2D bounded domains.Journal of Computational Physics, 2023. doi:10.1016/j.jcp.2023.112479

    2023

  45. [53]

    Ślęzak and S

    J. Ślęzak and S. Burov. From diffusion in compartmentalized media to non-Gaussian random walks. Scientific Reports, 2021. doi:10.1038/s41598-021-83364-0

    2021 doi

  46. [54]

    Su and K.-S

    F. Su and K.-S. Chan. Quasi-likelihood estimation of a threshold diffusion process. Journal of econometrics, 2015. doi:10.1016/j.jeconom.2015.03.038

    2015 doi

  47. [55]

    C. Villani. Optimal transport: old and new . Springer, 2009. doi:10.1007/ 978-3-540-71050-9

    2009

  48. [56]

    Zhao and X

    L. Zhao and X. Xue. The Voter Model with a Slow Membrane.Journal of Theoretical Probability, 2024. doi:10.1007/s10959-024-01321-9. 28 Figure 7. Visualization of the construction which is used in the proof of existence for Proposition 3.1. The process X starts by following a cl...

    2024 arXiv

  49. [58]

    Let E be the event where concentration occurs: E := {ω : W u 1 ( ˆPR(j,k,n,h), PR(j,k,n,h)) ≤ c 100 √ t, ∀j, k, n, hwith R(j, k, n, h) ⊆ D}

    Let us fix C5 at a value which is ≤ 1/ √ 2 and sufficiently small to ensure that the upper bounds onℓ in Lemma G.1 and Lemma G.2 are satisfied, and that the lower bound onE in Lemma G.2 is satisfied. Let E be the event where concentration occurs: E := {ω : W u 1 ( ˆPR(j,k,n,h)...

  50. [59]

    Proof of Lemma G.2.Again, the proof amounts to an application of Lemma G.4

    □ G.3. Proof of Lemma G.2.Again, the proof amounts to an application of Lemma G.4. More specifically, we apply that lemma to a ball of radius somewhat greater thanℓ: Proof of Lemma G.2.Let δ := 2κℓand recall the definition ofr′ δ from Appendix G.1. Taking c1 sufficiently small...

  51. [60]

    Then, by(3.28) from Corollary 3.12, if c1 is sufficiently small andc2 is sufficiently large, P(Xt = x0 + Wt) ≥ 1 − c′/2

    It hence follows thatx0 is at distance≥ (c2 − 1)ℓ − 2 √ t ≥ (c2−3) √ tfrom all barriers due to the triangle inequality. Then, by(3.28) from Corollary 3.12, if c1 is sufficiently small andc2 is sufficiently large, P(Xt = x0 + Wt) ≥ 1 − c′/2. (I.4) In other words, we haveXt = x0...

  52. [61]

    Now, by Lemma 5.3 with η = q/4, it holds withc3 := (3/4)q that P M (p, ⃗ v)/N(p, ⃗ v) < c3 and N (p, ⃗ v) ≥ n0 ≤ 32q−2 exp(−q2n0/8)

    : x′ 0 ∈ R+(p, ⃗ v)} ≥q (I.14) provided that c1 is taken sufficiently small andc2 sufficiently large. Now, by Lemma 5.3 with η = q/4, it holds withc3 := (3/4)q that P M (p, ⃗ v)/N(p, ⃗ v) < c3 and N (p, ⃗ v) ≥ n0 ≤ 32q−2 exp(−q2n0/8). (I.15) This proves (5.7). As for (5.8), le...

  53. [62]

    Recall that c3 = (3/4)q and fix some η < q/4

    : x′ 0 ∈ R+(p, ⃗ v)} (I.17) ≥ 1 − q/2. Recall that c3 = (3/4)q and fix some η < q/4. Then, we have that1 − q/2 − η > 1 − c3. Hence, the combination of Lemma 5.3 with (I.16) and (I.17) yields (5.8). □ Appendix J. Proof of Lemma 5.5 We rely on the tools from Section 3.2 with δ :...

  54. [1981]

    doi:10.1214/aop/1176994472

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.