{"id":"19f98238-d68b-4745-b6b8-fec880088ad4","arxiv_id":"1908.01387","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Conditioned Brownian motion confined to a shrinking tube around a submanifold converges weakly to a Feynman-Kac-type Wiener measure on the submanifold, now proved for general Riemannian manifolds.","lead":"This paper proves that Brownian motion in a curved space, forced to stay inside a very thin tube around a lower-dimensional surface, converges as a random path to a known motion on the surface as the tube shrinks. It completes earlier work that had only shown convergence at individual times, by adding the missing compactness step for general Riemannian manifolds.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The key sub-Gaussian estimate Proposition 2 is not self-contained: it imports uniform spectral, coercivity, and Sobolev estimates from the unpublished companion paper [7]; any failure in those imports breaks the tightness proof.","rationale":"The reader's verdict CONDITIONAL is accurate. I followed the proof chain: Theorem 2 follows from Proposition 3 applied to the projected processes; Proposition 3 requires the uniform sub-Gaussian estimate Corollary 2; Corollary 2 follows from Proposition 2; Proposition 2 is proved in Section 4 by combining a logarithmic Sobolev inequality with Davies' method. The logarithmic Sobolev inequality (Proposition 9 and Theorem 3) depends on ultracontractivity (Proposition 7), which rests on [7, Cor 4] and [7, Cor 10]. The ground-state estimates (Corollaries 4 and 5) also rest on [7, Thm 1 and Props 2 and 3]. The paper is therefore not self-contained in exactly the way the reader describes. I checked for internal errors independent of [7]: there are normalization slips, for example the limit ⟨φε,e^{-(1−t)H^0_ε/2}σ^{-1}_*√ρ⟩ is claimed to be 1 but is actually ∫φ0 unless φ0 is normalized per fibre, and the Hardy-inequality proof in Proposition 6 drops a nonnegative horizontal-derivative term, but both are absorbable and do not change the conclusion. The overstatement of Proposition 2 for all t>0 is also not load-bearing because the tightness argument only needs t−s≤1. Thus the main risk remains the unverified external basis. If [7] is made available and its statements check out, the argument appears coherent; hence CONDITIONAL rather than REJECT.","tokens_in":20562,"tokens_out":37589,"duration_ms":379886,"concrete_test":"Obtain the companion manuscript [7] and verify Theorem 1, Proposition 2, Corollary 4, and Corollary 10, in particular the uniform Sobolev bound ||u||_{H^{2n}} ≤ D_n(||u||_2 + ||(H^0_ε+α)^n u||_2) with D_n independent of ε and the spectral gap λ_ε−λ_0/ε^2→0. As a spot check, run the simplest nontrivial model case (e.g. L=S^1 embedded in R^3 or R^4) and test numerically whether the ultracontractivity constant in Proposition 7 remains bounded as ε→0 for small t; if the constant diverges, the sub-Gaussian bound and the tightness argument fail.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 2 is the only bridge from the semigroup convergence in [7] to tightness: it feeds Corollary 2, which feeds Proposition 3 and Theorem 2. Its proof in Section 4 is built on Proposition 7 (ultracontractivity), and Proposition 7 literally invokes [7, Cor 4] (spectral gap) and [7, Cor 10] (uniform Sobolev estimates). Proposition 5 (spectral gap and L^2 ground-state convergence) is quoted from [7, Props 2 and 3]; Corollary 4 then uses [7, Thm 1] to upgrade to uniform convergence in all derivatives, and Corollary 5 (the uniform boundary decay cδ ≤ φε ≤ Cδ) uses Corollary 4. The Rosen lemma and the Hardy inequality use [7, Prop 2] and [7, Prop 6] (Sasaki metric). All of these are nontrivial uniform-in-ε statements, not reproduced here, and [7] is a submitted but unpublished manuscript. If, for example, the constants D_n in [7, Cor 10] depend on ε rather than being ε-independent, the L2-L∞ bound in Proposition 7 acquires an ε-dependent constant, the sub-Gaussian estimate in Corollary 2 is no longer uniform, and the conclusion of Theorem 2 does not follow. This is a load-bearing external dependence, not a cosmetic citation; the central claim is conditional on [7].","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims weak convergence of Brownian motion on a Riemannian manifold M conditioned to remain in a small tubular neighbourhood L(epsilon) of a closed submanifold L up to a finite time T, as the tube radius epsilon tends to zero. The claimed limit is a Wiener measure on L with explicit density exp(-1/2 \\int_0^T U(omega(s)) ds). The proof strategy is: (i) reduce tightness of the conditioned processes to tightness of their projections onto L (Proposition 1); (ii) establish a uniform sub-Gaussian estimate for the Dirichlet heat kernel of the rescaled and eigenvalue-shifted semigroup (Proposition 2 and Corollary 2); (iii) derive moment estimates for the projected processes (Theorem 2). The kernel estimate is obtained from logarithmic Sobolev inequalities, a Hardy inequality, and a Rosen lemma, using several uniform spectral and Sobolev facts imported from the companion paper [7].","tokens_in":20850,"tokens_out":12982,"duration_ms":118682,"significance":"The limit measure identification is attractive and extends prior Euclidean work [9] to general Riemannian submanifolds. The paper has useful structural contributions: Proposition 1 cleanly reduces tightness of tube-valued processes to tightness of their projections, and the Davies-type semigroup route is appropriate. The main result, if fully supported, would be a substantive contribution to the theory of conditioned Brownian motion and small-noise diffusions in tubes. However, the central estimate is not self-contained; its validity is conditional on nontrivial uniform-in-epsilon results in the unpublished companion paper [7], and the proof of the key estimate contains a time-uniformity gap.","major_comments":[{"comment":"The semigroup bound (22) is derived only for t ≤ 1, but Proposition 2 and Corollary 2 are stated for all t > 0. The final 'absorption' of exp(t/2(λ0/ε^2 − λε)) into the constant is not justified for all t > 0: Proposition 5 gives only convergence as ε → 0 and does not control the sign or provide a bound independent of t for fixed ε. In addition, the factor in (23) is written with λε − λ0/ε^2 while the absorption step uses λ0/ε^2 − λε; these must be reconciled. The statements should be restricted to bounded time intervals, and the applications in Theorem 2 should be adjusted accordingly; the proof as written supports the needed t-s ∈ (0,1) bound but not the full t > 0 claim.","section":"§4.5, Proposition 2 and Eqs. (22)–(24)"},{"comment":"The proof of Proposition 2 is heavily dependent on the companion manuscript [7]: Theorem 1, Propositions 2 and 3, Corollaries 4 and 10, and Proposition 6 are used to obtain the spectral gap, equi-coercivity, uniform Sobolev estimates, ground-state convergence, the Sasaki-metric identity, and the form of the quadratic form. These are nontrivial uniform-in-ε results and are not proved or stated in this paper. Since [7] is submitted but not publicly available, the central claim of the paper cannot be independently checked as it stands. The authors should include full statements and proofs of the imported results, or provide a publicly accessible version of [7].","section":"§1.1.c, §§4.1–4.5"},{"comment":"Theorem 2 is proved for a time horizon equal to 1: the proof uses Qε(s,W;t,W′) with 1−t and normalization Z(x,ε)=∫ K_1^ε(x,V)Σ^{-1}1(V), and all times lie in [0,1]. The theorem and Corollary 3, however, are stated for an arbitrary finite T>0. No reduction from T to 1 is described. Since Brownian motion on a Riemannian manifold does not scale by a simple time change without changing the metric, this gap needs to be addressed explicitly, or the results should be stated only for T=1.","section":"§3, Theorem 2 and Corollary 3"},{"comment":"The paper explicitly says 'We are not going into the details of this proof' after deriving the differential of ||f_s||_{p(s)}^{p(s)}. This is the step that turns the logarithmic Sobolev inequalities of Theorem 3 into the L^2-L∞ bound (22). Because the theorem is central, the details cannot simply be omitted; at minimum the hypotheses of [3, Theorem 2.2.7] should be verified explicitly for the present ε-dependent operators, including the integrability of Γ(p).","section":"§4.5, between Theorem 3 and Theorem 4"}],"minor_comments":[{"comment":"The same symbol h is used for a function on L and for its pullback to L(1); this is a common abuse but should be flagged, since both appear in the same estimate.","section":"§3, proof of Corollary 2"},{"comment":"The statement of Proposition 2 should record explicitly that the constants C and k are independent of ε, t, and h; currently this is implicit, while the proof produces estimates only for ε < ε0.","section":"§4, Proposition 2 statement"},{"comment":"The display after the transformation to the 1-tube would benefit from a step-by-step derivation; the action of Σ_ε on Qε and on the distance function is introduced without explanation, and the reader must reconstruct the cancellation.","section":"§3, Theorem 2"},{"comment":"The notation λ0/ε^2 is ambiguous: it presumably means λ0 ε^{-2}. Please clarify the parenthesization to avoid the literal reading (λ0)/ε^2.","section":"§4.1, Proposition 5"},{"comment":"Reference [7] should be assigned an arXiv identifier or otherwise be made publicly accessible, since it carries a large part of the technical burden of the paper.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper's central theorem depends almost entirely on [7]; I would urge the editor to obtain [7] before making a final decision, or to require the authors to include the relevant results in the present paper. The stress-test concern about external dependence is real, and it is compounded by the independent time-uniformity gap in §4.5 and the T=1 versus general T issue in Theorem 2. There is no evidence of internal circularity, but the manuscript as it stands is not verifiable by a reader without access to [7]."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a real step beyond [9] and [7]. It proves tightness of Brownian motion conditioned to stay in a thin tube around a general Riemannian submanifold, and identifies the limit as Wiener measure on the submanifold with a geometric potential. The Euclidean version was known; the marginal convergence was in [7]; the tightness was open for general embeddings. That makes Theorem 2 and Corollary 3 genuinely new.\n\nWhat I like: the strategy is coherent. You prove a sub-Gaussian Dirichlet heat kernel estimate (Prop 2) by Davies' log-Sobolev machinery, then tightness follows by a standard moment criterion (Prop 3). The reduction of tightness to the projected processes (Prop 1) is clean and saves real work. The paper is honest about following Davies and about what it borrows.\n\nThe soft spot is the dependence on [7]. The list is long and load-bearing: Theorem 1 (semigroup convergence), equi-coercivity and Gamma-convergence (Prop 2), the spectral gap (Cor 4), the uniform Sobolev estimates (Cor 10), and the Sasaki metric expression (Prop 6) all come from the companion paper, which is only 'submitted'. Proposition 7's ultracontractivity literally uses [7, Cor 10]; if those constants are not epsilon-independent, the L2-Linf bound develops an epsilon-dependence, the sub-Gaussian estimate fails to be uniform, and tightness does not follow. The stress-test note is right: this is not a cosmetic citation. I don't see a way to verify the central claim from this manuscript alone.\n\nSome smaller things: Proposition 2 states t>0 and epsilon>0, but the proof of the main bound (22) is explicitly for t≤1. The tightness argument only needs t-s≤1, so the fix is to state the proposition for that range or add a short argument for t>1. The absorption of exp(t/2(lambda_0/epsilon^2 - lambda_epsilon)) into a constant also deserves one sentence, since it uses Proposition 5 and smallness of epsilon. There are a few typos in constants (notably in Corollary 2's proof).\n\nBottom line: the mathematics looks plausible and the conclusion likely holds if [7] is correct. But as submitted, the paper cannot be refereed on its own merits. I would send it to peer review, with the requirement that the author either include the needed results from [7] or supply a publicly available version of [7], and that the t≤1 point be fixed. I would not cite the result until the companion is available.","headline":"New tightness result for conditioned Brownian motion in tubes around general Riemannian submanifolds; the proof is plausible but leans so heavily on an unpublished companion paper that it cannot be evaluated on its own.","tokens_in":21399,"tokens_out":3686,"would_cite":false,"duration_ms":38259,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60B10","35K08","58J65","28C20"],"pacs":[],"model":"deepseek-v4-flash","headline":"Brownian motion confined to a shrinking tube around a submanifold converges weakly to a tilted Wiener measure on the submanifold.","keywords":["conditional Brownian motion","tubular neighbourhoods","tightness","heat kernel estimate","sub-Gaussian estimate","logarithmic Sobolev inequality","Dirichlet semigroup","Wiener measure"],"falsifier":"Compute the Dirichlet spectrum of the Laplacian on a thin tube around a circle in $\\mathbb{R}^3$, for example a solid torus of minor radius $\\varepsilon$; if $\\lambda_\\varepsilon-\\lambda_0/\\varepsilon^2$ does not tend to $0$, or if the heat kernel at two points with the same projection violates the bound $\\exp(-d_L^2/(4Bt))$ with $d_L$ the geodesic distance on the circle, the central claim collapses.","tokens_in":20240,"feed_emoji":"🌀","tokens_out":5657,"duration_ms":54706,"temperature":0.7,"pith_summary":"Brownian motion on a Riemannian manifold, conditioned to stay inside a tube of radius $\\varepsilon$ around a closed submanifold $L$, is shown to have a well-defined limit as $\\varepsilon\\to 0$: the family of conditional path measures is tight, and the limit is a Wiener measure on $L$ tilted by a geometric potential $U$. The paper establishes the missing tightness half of a convergence programme whose marginal convergence came from a companion paper. If correct, the result gives a precise sense in which tube confinement effectively freezes the normal directions and leaves only motion along $L$, with an explicitly identified change of measure. The proof runs through a uniform sub-Gaussian bound on the Dirichlet heat kernel of the tube.","feed_headline":"Tube-confined Brownian motion converges to a weighted Wiener measure","feed_subtitle":"The limit is Brownian motion on the submanifold with a geometric potential, explicitly identified.","key_machinery":"The load-bearing object is the sub-Gaussian kernel estimate for the rescaled Dirichlet semigroup: $K^\\varepsilon_t(W,W') \\le C t^{-(m+3)/2}\\,\\varphi_\\varepsilon(W)\\,\\varphi_\\varepsilon(W')\\,\\exp\\bigl(-d_L(\\pi(W),\\pi(W'))^2/(4Bt)\\bigr)$. Here $\\varphi_\\varepsilon$ is the normalized positive ground state of the tube Laplacian after subtracting the leading eigenvalue $\\lambda_0/\\varepsilon^2$, and $\\pi$ projects the tube onto $L$. The estimate is derived through a ground-state transform, logarithmic Sobolev inequalities, the Rosen lemma, a Hardy inequality in the Sasaki metric, and weighted $L^p$ estimates from heat-kernel theory; it converts spectral information into the Gaussian-in-distance control needed for tightness.","core_discovery":"The paper's central claim is Theorem 2: for a fixed starting point on $L$, the processes $Y^\\varepsilon$ obtained by conditioning Brownian motion on $M$ to remain in $L(\\varepsilon)$ up to time $T$ form a tight family; hence, together with convergence of one-dimensional marginals, the conditional measures $\\mu^x_\\varepsilon$ converge weakly to $\\mu^x_0$, the Wiener measure on $L$ weighted by $\\exp\\bigl(-\\tfrac12\\int_0^T U(\\omega(s))\\,ds\\bigr)$, normalized. The potential $U$ is built from the geometry of the embedding: $U=\\rho^{1/2}\\Delta \\rho^{-1/2}$, where $\\rho$ is the density of the induced Riemannian volume relative to the Sasaki volume on the tube. A sympathetic reading: the paper claims that the only information surviving the thin-tube limit is tangential Brownian motion on $L$ plus a deterministic geometric weight, and it supplies a proof of the path-level convergence.","pith_inferences":["The explicit form of $U$ suggests a testable numerical prediction: simulate Brownian motion conditioned to a thin tube and compare the empirical path distribution with the tilted Wiener measure; a discrepancy in the potential term would localize a failure in the companion semigroup convergence.","The same sub-Gaussian machinery might extend to tubes around non-compact or higher-codimension submanifolds, provided the spectral-gap and equi-coercivity inputs can be established there.","If the companion paper's spectral-gap result fails in some geometry, the tightness conclusion could still hold while the identified limit measure would need correction; the two claims are not logically identical.","The result suggests that the effective long-time behaviour of confined Brownian motion is governed by the submanifold Laplacian plus potential, which may be relevant to diffusion in narrow channels and polymer confinement models."],"forward_implications":["If the central claim holds, the conditional Brownian path measures converge weakly, not just in finite-dimensional distributions, to the explicit tilted Wiener measure on $L$.","The limit measure is determined by the geometric potential $U$, so the shape of the embedding governs the limiting distribution through the density $\\rho$.","Tightness of the projected processes on $L$ is equivalent to tightness of the full tube processes, so checking one-dimensional distance moments suffices.","The sub-Gaussian estimate gives a uniform modulus of continuity for the family, implying compact containment and hence the standard tightness criterion applies.","For embeddings in Euclidean space the result recovers the previously known surface limit as a special case."],"supporting_citations":[{"why":"Supplies the semigroup convergence, equi-coercivity, spectral gap $\\lambda_\\varepsilon-\\lambda_0/\\varepsilon^2\\to 0$, uniform Sobolev estimates, and Sasaki-metric formula on which Section 4 rests.","marker":"[7]"},{"why":"Gives the logarithmic Sobolev and sub-Gaussian heat-kernel methodology used to turn spectral bounds into kernel estimates.","marker":"[3]"},{"why":"Provides the tightness theorem used to reduce tightness of the full process family to tightness of its projections onto $L$.","marker":"[4]"},{"why":"Established the Euclidean embedding case, which the present paper extends to general Riemannian manifolds.","marker":"[9]"},{"why":"Supplies the moment-based tightness criterion verified via the sub-Gaussian estimate.","marker":"[6]"},{"why":"Gives the gradient characterisation of geodesic distance used to pass from basic functions to $d_L$ in the kernel bound.","marker":"[2]"}],"fun_headline_variants":["Thin-tube Brownian motion converges to weighted Wiener measure","Tube-confined Brownian paths: tightness and limit identified","Conditioned Brownian motion on tubes has explicit weak limit","Sub-Gaussian heat kernel estimate yields tube path convergence"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes, without proof here, the companion paper's convergence results: semigroup convergence to the projected Laplacian, the spectral gap $\\lambda_\\varepsilon-\\lambda_0/\\varepsilon^2\\to 0$, equi-coercivity, uniform Sobolev estimates, and the Sasaki-metric formula; if any of those fail, the sub-Gaussian estimate and hence tightness do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Thin-tube Brownian motion converges to weighted Wiener measure","Tube-confined Brownian paths: tightness and limit identified","Conditioned Brownian motion on tubes has explicit weak limit","Sub-Gaussian heat kernel estimate yields tube path convergence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000142,"raw_usage":{"total_tokens":1131,"prompt_tokens":873,"completion_tokens":258,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":489,"completion_tokens_details":{"reasoning_tokens":190}},"tokens_in":489,"tokens_out":258,"duration_ms":3888,"temperature":1.0,"reasoning_tokens":190,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:17:08.368811+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the Dirichlet spectrum of the Laplacian on a thin tube around a circle in $\\mathbb{R}^3$, for example a solid torus of minor radius $\\varepsilon$; if $\\lambda_\\varepsilon-\\lambda_0/\\varepsilon^2$ does not tend to $0$, or if the heat kernel at two points with the same projection violates the bound $\\exp(-d_L^2/(4Bt))$ with $d_L$ the geodesic distance on the circle, the central claim collapses.","supporting_citations":[{"cited_title":"A convergence result for dir ichlet semi- groups on tubular neighbourhoods with an application to sto chastic processes","cited_arxiv_id":null,"evidence_quote":"Supplies the semigroup convergence, equi-coercivity, spectral gap $\\lambda_\\varepsilon-\\lambda_0/\\varepsilon^2\\to 0$, uniform Sobolev estimates, and Sasaki-metric formula on which Section 4 rests."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the logarithmic Sobolev and sub-Gaussian heat-kernel methodology used to turn spectral bounds into kernel estimates."},{"cited_title":"Ethier and Thomas G","cited_arxiv_id":null,"evidence_quote":"Provides the tightness theorem used to reduce tightness of the full process family to tightness of its projections onto $L$."},{"cited_title":"Sidorova, O","cited_arxiv_id":null,"evidence_quote":"Established the Euclidean embedding case, which the present paper extends to general Riemannian manifolds."},{"cited_title":"Foundations of modern probability","cited_arxiv_id":null,"evidence_quote":"Supplies the moment-based tightness criterion verified via the sub-Gaussian estimate."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the gradient characterisation of geodesic distance used to pass from basic functions to $d_L$ in the kernel bound."}],"review_version":1}