{"id":"82b3f8aa-3e35-45b9-a2d7-2571dc01b494","arxiv_id":"2412.14740","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Semipermeable barrier locations can be recovered from discrete samples of reflected Brownian motion, with T^{-2/3} error for fixed sampling intervals and exp(-c√T) error in a high-frequency regime.","lead":"The paper gives explicit algorithms and proofs for recovering one-dimensional semipermeable barriers in a plane from discrete samples of a reflected Brownian motion. The achievable error depends sharply on sampling frequency, with polynomial rates for fixed-frequency samples and an exponential rate when samples are frequent.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified.","rationale":"The reader's weakest_assumption correctly identifies the mixing/exploration condition as the most fragile premise in Theorem 2.3. I agree that if πmin is very small or tmix is very large, the observation-time bound becomes impractical, but this is an explicitly quantified modeling limitation rather than an internal inconsistency. The partial recovery theorem (Theorem 2.5) sidesteps mixing entirely, which supports the paper's overall structure. My own scrutiny of the proofs, including the constant alignment in Appendix C, the concentration exponent in Lemma 4.10, the coupling error in Corollary 3.12, and the covering-time argument in Lemma 6.1, did not reveal a load-bearing gap. The paper is transparent about its conjectures and limitations, and the claims are stated with appropriate qualifiers. Therefore the reader's ACCEPT verdict should stand unchanged.","tokens_in":66695,"tokens_out":36577,"duration_ms":308899,"concrete_test":"Run the synthetic simulation described in Figure 1 with a known semipermeable barrier and a stationary start, then apply Algorithms 1 and 2 for increasing observation periods T, keeping t fixed and small enough to satisfy the theorem's conditions. On a log-log plot of the observed Hausdorff error versus T, verify that the slope approaches -2/3 and that the required T to reach a given ε is consistent with (2.8), including the tmix/πmin prefactor. If the empirical rate is significantly slower, the claim in Theorem 2.3 would need revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After reviewing the theorems and their proofs, the central claims appear internally consistent. The recovery rates follow from explicit concentration (Lemma 4.10), coupling (Corollary 3.12), and covering-time (Lemma 6.1) arguments, with the exploration cost honestly encoded through tmix/πmin in (2.8). The weakest premise is indeed the mixing/stationarity requirement of Theorem 2.3: if πmin is tiny or tmix is huge, (2.8) demands enormous observation times. However, this is a transparent modeling condition, not a hidden gap; Theorem 2.5 (partial recovery) does not require mixing, and Theorem 2.7 concerns the continuous trace rather than the discrete estimator. I located no circular step or omitted proof. Appendix C's iteration from E0 = √2(ε³/κ)^{1/4} down to ε is valid for small ε because E0/ℓ = O((εκ)^{1/4}) → 0, and the constants can be chosen so that the first refinement step lies within the range of Proposition 4.2. The paper also explicitly flags its conjectures (optimality of 2/3, exponential rate for m>0) as expectations rather than theorems.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":66838,"tokens_out":24202,"duration_ms":190348,"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.","major_comments":[],"minor_comments":[{"comment":"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.","section":"Appendix F (proof of Proposition 4.1)"},{"comment":"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.","section":"Appendix G.4"},{"comment":"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.","section":"Section 2.2 / Theorem 2.3"},{"comment":"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.","section":"Lemma 4.10"},{"comment":"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.","section":"Sections 2.1 and 2.3.3"},{"comment":"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.","section":"Appendix D (proof of Lemma 4.8)"}],"recommendation":"minor_revision","confidential_remarks":"I have no concerns about attribution or scope; the related-work discussion is accurate and the references to Lejay, Matthews, Chen-Friedman, and Paulin are standard for the cited facts. The manuscript is a strong fit for a probability journal. In my view the remaining issues are presentation-level and can be resolved without further verification."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this if you care about inference from stochastic process samples. It is the first mathematical treatment of recovering semipermeable barrier locations from discrete observations of a reflected Brownian motion in a planar domain, and it delivers: explicit algorithms, regime-dependent rates, and proofs that go beyond hand-waving. The planar coupling argument is the technical centerpiece; it reduces the problem to one-dimensional reflected Brownian motion and makes the constants depend on interpretable quantities like curvature, permeability, mixing time, and minimal stationary density.\n\nWhat is actually new: prior work by Cholaquidis et al. estimated the domain boundary for classical reflected Brownian motion with polynomial rates. This paper handles semipermeable barriers and shows that the sampling rate changes the qualitative picture—polynomial in the fixed-frequency regime, exponential in the high-frequency regime for the outer barrier. The distinction between complete and partial recovery is sensible and practically relevant.\n\nThe soft spots are real but not fatal. The main guarantee, Theorem 2.3, requires the chain to mix and the stationary measure to have a positive lower density (tmix/πmin in the bound). When the process is trapped by a low-permeability barrier, the observation time needed becomes astronomical. That is a transparent modeling condition, not a hidden gap, but it limits the direct applicability to real movement data. Theorem 2.7 handles only the outer impermeable barrier; the exponential claim for semipermeable barriers remains a conjecture, as the authors say. The optimality of the T^{-2/3} rate is also an expectation by analogy with i.i.d. boundary estimation, not a proved lower bound. The appendices contain many implicit constants and the proofs are long; a referee should check at least the concentration lemma and the coupling estimates. I did not find a circular step or a fitting-to-data move. The reindeer case study is illustrative and uses ad hoc parameter choices, but it is presented that way.\n\nWho is this for: probabilists and statisticians working in set estimation or statistical inference for diffusions. It deserves a serious referee pass. I would send it to review and expect heavy but productive work on the presentation of constants and the mixing assumptions.\n\nRecommendation: accept, after a careful referee round. The central claims appear well supported and the limitations are honestly flagged.","headline":"First real treatment of semipermeable barrier recovery from sampled Brownian motion; solid proofs, heavy mixing assumptions, deserves refereeing.","tokens_in":67386,"tokens_out":2574,"would_cite":true,"duration_ms":22958,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60J65","62M05","62G05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Discrete tracks of a reflected Brownian motion reveal the hidden barriers that shape them.","keywords":["semipermeable barriers","reflected Brownian motion","snapping out Brownian motion","set estimation","barrier recovery","transition kernel discontinuity","Hausdorff distance"],"falsifier":"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).","tokens_in":66446,"feed_emoji":"🧭","tokens_out":1464,"duration_ms":13468,"temperature":0.7,"pith_summary":"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.","feed_headline":"Discrete Brownian tracks reveal hidden barriers","feed_subtitle":"New algorithms locate semipermeable obstacles from movement data, with sharp recovery rates by sampling regime.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Closest prior work: estimates a domain and its boundary from reflected Brownian motion samples; the paper extends this to semipermeable barriers.","marker":"[17]"},{"why":"Provides the minimax rate for boundary estimation from i.i.d. samples that the paper uses as a benchmark for the expected optimality of the $T^{-2/3}$ rate.","marker":"[1]"},{"why":"Introduces snapping out Brownian motion, the process that this paper studies with semipermeable barriers.","marker":"[30]"},{"why":"Provides the Markov chain concentration inequality used to control the empirical transition kernel estimates in Lemma 4.10.","marker":"[44]"},{"why":"Supplies the covering-time estimates used in the proof of Theorem 2.7 for the exponential-rate result.","marker":"[41]"},{"why":"Gives the narrow escape asymptotics used in Lemma 6.1 to bound the time to cover the boundary with small arcs.","marker":"[15]"},{"why":"Supplies the reindeer movement dataset used for the real-life case study in Section 7.","marker":"[39]"}],"fun_headline_variants":["Sampling rate governs barrier recovery rate","Exponential gains in barrier recovery from frequent data","Hidden semipermeable walls from Brownian samples","Fast barrier recovery with high-frequency Brownian data","Discontinuities in transition kernel expose barriers"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Sampling rate governs barrier recovery rate","Exponential gains in barrier recovery from frequent data","Hidden semipermeable walls from Brownian samples","Fast barrier recovery with high-frequency Brownian data","Discontinuities in transition kernel expose barriers"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000618,"raw_usage":{"total_tokens":2855,"prompt_tokens":919,"completion_tokens":1936,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":535,"completion_tokens_details":{"reasoning_tokens":1868}},"tokens_in":535,"tokens_out":1936,"duration_ms":11069,"temperature":1.0,"reasoning_tokens":1868,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T11:56:47.684722+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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).","supporting_citations":[{"cited_title":"Cholaquidis, R","cited_arxiv_id":null,"evidence_quote":"Closest prior work: estimates a domain and its boundary from reflected Brownian motion samples; the paper extends this to semipermeable barriers."},{"cited_title":"Aamari, C","cited_arxiv_id":null,"evidence_quote":"Provides the minimax rate for boundary estimation from i.i.d. samples that the paper uses as a benchmark for the expected optimality of the $T^{-2/3}$ rate."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces snapping out Brownian motion, the process that this paper studies with semipermeable barriers."},{"cited_title":"Loe, B.B","cited_arxiv_id":null,"evidence_quote":"Supplies the reindeer movement dataset used for the real-life case study in Section 7."}],"review_version":1}