{"id":"4f66c977-a058-409d-bc1c-eebf40318a77","arxiv_id":"1908.09182","paper_version":4,"verdict":"REJECT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"For the Heisenberg group Brownian motion, the ratio of tube probabilities around horizontal paths phi and psi converges to exp(-1/2||phi||^2 + 1/2||psi||^2), giving Onsager-Machlup functional L(p,v) = -1/2||v||^2.","lead":"This paper derives the Onsager-Machlup functional for the hypoelliptic Brownian motion on the Heisenberg group, obtaining a kinetic energy formula for the most probable horizontal path. It is the first such result in a sub-Riemannian setting, but the proof as written contains gaps in the small-ball asymptotics.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 4.7 cannot support the uψ/z interchange: it relates radius ε to radius sqrt(3ε^2+εCγ), so the claimed equality of tube probabilities is not established.","rationale":"The reader's weakest_assumption identifies the same load-bearing gap: the asymptotic interchangeability of the tube probabilities is not justified, specifically because Lemma 4.7 and (3.15) only supply inclusions at different radii. My stress-test sharpens this: the radius mismatch is not a small correction but scales as sqrt(C_γ ε) versus ε, so no standard regular-variation property of Gaussian small-ball probabilities could bridge it. The conditional-expectation identity in Proposition 4.9 relies on the same missing tube concentration. The claimed formula may be correct, but the submitted proof does not establish it, so the REJECT verdict is appropriate. I found no independent evidence (machine-checked proofs, numerical verification, or a parameter-free derivation) that would override the gap.","tokens_in":17022,"tokens_out":14939,"duration_ms":158898,"concrete_test":"Test Lemma 4.7 directly in the simplest nontrivial case: take ψ=0 and φ(t)=(at,0,0), so γ=φ, C_γ=a, u_ψ=g_t, and z is given by z_horizontal=W, z_3(t)=1/2∫_0^t ω(W,dW) - a∫_0^t W_2(s)ds. Lemma 4.7 asserts that lim_{ε→0} P(sup_{[0,1]} ρ(z_t,e)<ε) / P(sup_{[0,1]} ρ(g_t,e)<ε) = 1 for every fixed a. Derive (or simulate via rare-event methods at small ε) this ratio. If it is not 1, or if a-dependent corrections appear at leading order, the radius-mismatch argument in Lemma 4.7 collapses and Proposition 4.9 is invalidated.","verdict_should_be":"REJECT","load_bearing_attack":"The central reduction (4.8) and Proposition 4.9 require replacing P(sup ρ(u_ψ,e)<ε) by P(sup ρ(z,e)<ε). Lemma 4.7 only proves inclusions at mismatched radii: A_ε ⊆ {sup ρ(z,e)<r(ε)} and B_ε ⊆ {sup ρ(u_ψ,e)<r(ε)}, where r(ε)=sqrt(3ε^2+εC_γ) ~ sqrt(C_γ ε). Since r(ε)/ε → ∞, these inclusions are far too weak to imply equality of the limits. For small-ball probabilities of the expected form P(ε) ~ ε^a exp(-b/ε^2), P(r(ε))/P(ε) is exponentially large in 1/ε, so the asserted equality does not follow without an additional regular-variation property that is neither stated nor proved. The same defect appears in (3.15): equivalence of d_cc and ρ only gives inclusions at radii ε/c and cε, and the displayed equality of limits requires scale-regularity of the small-ball probability that is absent. The subsequent conditional-expectation step in Proposition 4.9, which assumes that ∫⟨c_γ,d(W-φ)⟩ conditioned on {sup ρ(u_φ,e)<ε} concentrates at 0, is another unproved tube estimate of the same kind. Thus Proposition 4.9, and with it Theorem 3.13, is not proven in this version.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims to determine the Onsager-Machlup functional for the hypoelliptic Brownian motion on the Heisenberg group. For horizontal finite-energy curves φ,ψ starting at the identity, the main result (Theorem 3.13) states that the ratio of Carnot-Carathéodory tube probabilities converges to exp(-1/2||φ||^2_{H(H)} + 1/2||ψ||^2_{H(H)}). The proof proceeds in two stages: first, by left-invariance and the assumed asymptotic equivalence of the Carnot-Carathéodory and homogeneous distances (Eq. (3.15)), the problem is reduced to tube probabilities for the processes u^φ_t = φ(t)^{-1}g_t around the identity. Second, a Girsanov transformation and an auxiliary process z_t are used to compare u^φ and u^ψ, leading via Lemma 4.8 to the claimed exponential ratio. The paper also introduces the notion of finite-energy horizontal curves and the Cameron-Martin space H(H).","tokens_in":17319,"tokens_out":11050,"duration_ms":110692,"significance":"If the result were established, it would be a notable first example of an Onsager-Machlup functional in a hypoelliptic setting, with the natural candidate L(p,v)=-1/2||v||^2_{H_p}. The use of the group structure and Girsanov transformation is a sensible strategy, and the proposed functional is consistent with the Euclidean and Riemannian cases. However, the proof as written has decisive gaps at exactly the points where small-ball asymptotics need to be controlled: the equivalence of tube probabilities under different distances and the comparison of the auxiliary processes u^ψ and z are not justified by the estimates provided. Because these gaps affect the central claim, the manuscript is not ready for publication in its present form.","major_comments":[{"comment":"The equality of limits of tube probabilities under d_cc and ρ does not follow from the bi-Lipschitz equivalence (3.12). Equivalent metrics only give inclusions of the form {d_cc<ε}⊆{ρ<c^{-1}ε} and {ρ<ε}⊆{d_cc<Cε}, which determine the small-ball limit only up to a scale factor in the radius. The equality in (3.15) requires a further regularity property of the small-ball probability under radius scaling, and no such property is stated or proved. Since (3.15) is used in (4.3) and (4.8) as the first reduction step, this gap is load-bearing for Theorem 3.13.","section":"Section 3.2, Eq. (3.15)"},{"comment":"The proof of Lemma 4.7 establishes only one-sided inclusions at mismatched radii: if ρ(u^ψ,e)<ε, then ρ(z,e)<r(ε), and conversely, where r(ε)=sqrt(3ε^2+εC_γ). Since r(ε)/ε→∞ as ε→0, these inclusions do not imply that the limits (4.15) and (4.16) exist together or are equal. For small-ball probabilities of the expected polynomial order P(ε)≍ε^a, the ratio P(r(ε))/P(ε)≍(r(ε)/ε)^a diverges, so the asserted equality would generally fail. The lemma needs a genuine small-ball asymptotic comparison under radius scaling, which is not provided; the current argument cannot support the substitution of P(ρ(z,e)<ε) for P(ρ(u^ψ,e)<ε) in Proposition 4.9.","section":"Lemma 4.7"},{"comment":"The step after the identity for ∫⟨cγ,dW⟩ is not justified. The proof claims that the conditional exponential moments of ∫⟨cγ(s),dW(s)⟩ given the tube A_ε = {sup ρ(u^φ,e)<ε} converge to exp(c⟨γ,φ⟩), which is equivalent to asserting that the stochastic integral ∫⟨cγ(s),dW_s-φ'(s)ds⟩ is asymptotically negligible on A_ε. No estimate for this term on the tube is given; this is a non-trivial tube estimate of the same nature as Lemma 4.7, and it is exactly the kind of small-ball control the paper lacks. Consequently, the application of Lemma 4.8 is not supported, and Proposition 4.9, which is the core of Theorem 3.13, is not established.","section":"Proposition 4.9"}],"minor_comments":[{"comment":"The third coordinate of γ is defined with an integral from 0 to 1, which appears to be a typo for an integral from 0 to t. As written, γ has zero third component derivative and is not a horizontal curve in general; please correct the formula, since the proof of Proposition 4.9 relies on γ∈H(H).","section":"Eq. (4.11)"},{"comment":"In the final line of the proof, 'sup_{t∈[0,1]} ρ(u^ψ_t(t),e)' should read 'sup_{t∈[0,1]} ρ(u^ψ_t,e)'.","section":"Lemma 4.7, proof"},{"comment":"The citation for [7] contains 'XXX' in place of the arXiv identifier or page range and should be completed.","section":"References"},{"comment":"In the chain of equalities, the stochastic integral in the second line is written with subscript u^φ_t while the integration variable is s; this should be u^φ_s for consistency.","section":"Proposition 4.9, display"}],"recommendation":"reject","confidential_remarks":"The paper addresses an interesting and timely problem, and the proposed Onsager-Machlup functional is the natural one, but the proof as written has central gaps in small-ball asymptotic comparisons. In particular, Lemma 4.7 and Eq. (3.15) do not justify the interchanges on which Proposition 4.9 rests, and the conditional expectation estimate in Proposition 4.9 is asserted without proof. These are not merely local presentation issues; they invalidate the main argument. Should the authors supply the missing small-ball estimates, the result may be salvageable, but in the current form the manuscript does not meet the standard for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear —,\n\nFirst thing to know: this is the first genuine attempt at an Onsager-Machlup functional for a hypoelliptic diffusion, and the formula it claims — L(p,v) = −½‖v‖² for horizontal v, with no curvature correction — is exactly what the elliptic analogy suggests for the Heisenberg group. The formula may well be true. The machinery, however, does not prove it in this version.\n\nThe novel part is real: the authors avoid Riemannian geometry entirely, using left-invariance and the explicit group structure to reduce the tube problem to two-dimensional Brownian motion plus a conditional exponential moment. The definition of the Cameron-Martin space of finite-energy horizontal curves is clean, and the Maurer-Cartan computations in Section 4.1 are correct and useful. If the paper is revised, these pieces will survive.\n\nThe soft spot is where the proof changes distance and then changes process. Equation (3.15) asserts that the Carnot-Carathéodory tube and the rho-tube have the same limit. From the equivalence c·rho ≤ d_cc ≤ C·rho you only get inclusions at radii ε/c and Cε. For that to yield equality of limits you need the small-ball probability to be regular in ε — i.e., P(aε)/P(ε) → 1 — and for a hypoelliptic diffusion with decay like exp(−K/ε²) that is false. The same defect appears in Lemma 4.7: the inclusions land at radius sqrt(3ε² + εCγ) ~ sqrt(ε), not ε, so the claimed equality of limits in (4.15) and (4.16) is not a consequence of the lemma. The conditional-expectation estimate in Proposition 4.9 inherits the problem. This is a load-bearing gap, not a cosmetic one.\n\nI don't see circularity or any attempt to fit the answer; the paper is honest and the computations are straightforward. The result is plausible and the topic is important. The right response is not to throw the paper out but to send it to a referee who knows small-ball estimates for hypoelliptic processes, with the clear expectation that the tube-comparison estimates need real work.","headline":"First genuine attempt at an Onsager-Machlup functional for a hypoelliptic diffusion, with a plausible formula, but the proof relies on unsupported small-ball comparisons across equivalent metrics and matched radii — deserves a serious referee, not desk rejection.","tokens_in":17801,"tokens_out":3007,"would_cite":false,"duration_ms":29777,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["58J65","60J60","35R03","53C17"],"pacs":[],"model":"deepseek-v4-flash","headline":"Brownian motion on the Heisenberg group has an Onsager-Machlup functional equal to minus half the squared horizontal energy, so small-tube probabilities around horizontal curves are governed by the same energy law as in Euclidean space.","keywords":["Onsager-Machlup functional","Heisenberg group","hypoelliptic Brownian motion","Carnot-Carathéodory distance","Cameron-Martin-Girsanov theorem","sub-Riemannian geometry","tube probability asymptotics","stochastic exponential"],"falsifier":"Compute the small-ball probabilities $P(\\sup_{t\\in[0,1]} \\rho(u^\\psi_t,e)<\\varepsilon)$ and $P(\\sup_{t\\in[0,1]} d_{cc}(g_t,\\psi(t))<\\varepsilon)$ to the same order in $\\varepsilon$ for one explicit horizontal path $\\psi$, such as a line segment. If their ratio does not tend to 1 as $\\varepsilon\\to 0$, or if the limit changes when $\\varepsilon$ is replaced by the mismatched radius appearing in the inclusion argument, then the asymptotic interchange behind Lemma 4.7 fails and the claimed ratio in Proposition 4.9 is not established.","tokens_in":16798,"feed_emoji":"🎲","tokens_out":10650,"duration_ms":97373,"temperature":0.7,"pith_summary":"This paper proves that Brownian motion on the Heisenberg group has an Onsager-Machlup functional of the same quadratic form as Euclidean Brownian motion. For any two horizontal finite-energy curves $\\varphi$ and $\\psi$, the ratio of the probabilities that the hypoelliptic Brownian path stays inside an $\\varepsilon$-tube around $\\varphi$ versus around $\\psi$ converges, as $\\varepsilon$ tends to zero, to $\\exp(-\\frac{1}{2}\\|\\varphi\\|^2_{\\mathcal{H}(H)} + \\frac{1}{2}\\|\\psi\\|^2_{\\mathcal{H}(H)})$. This is the first Onsager-Machlup result for a hypoelliptic diffusion, and it matters because the Heisenberg group is the simplest nontrivial sub-Riemannian space, where the Riemannian comparison and curvature techniques used in the classical results are unavailable. The proof replaces those tools with the group's left-invariance and a Cameron-Martin-Girsanov shift of the two-dimensional Brownian motion that drives the hypoelliptic process.","feed_headline":"Heisenberg Brownian tubes follow the Euclidean energy law","feed_subtitle":"Around horizontal curves, the tube probability ratio tends to exp(-Δ energy / 2), the flat-space Onsager-Machlup form.","key_machinery":"The argument runs on three pieces working together. First, because the Carnot-Carathéodory distance is left-invariant and equivalent to the homogeneous distance $\\rho$, the tube probability around a curve $\\varphi$ is rewritten as the probability that the left-translated process $u^\\varphi_t=\\varphi(t)^{-1}g_t$ stays within $\\varepsilon$ of the identity. Second, for horizontal $\\varphi$ the left Maurer-Cartan form of $u^\\varphi$ is explicit: its horizontal part is the shifted Brownian motion $W_t-\\varphi(t)$, and its vertical part is a stochastic-area integral plus a drift built from $\\varphi'$. Third, the proof applies the Cameron-Martin-Girsanov theorem to the driving two-dimensional Brownian motion, implemented by the stochastic exponential $E^\\gamma_1$ for a horizontal curve $\\gamma$ whose Maurer-Cartan form is $c_\\gamma=c_\\varphi-c_\\psi$, so that the law of $u^\\varphi$ under a tilted measure matches the law of an auxiliary process $z$ that stays close to $u^\\psi$. A conditioning lemma (Lemma 4.8) turns conditional exponential moments into the exponential of inner products, and the energy difference $-\\frac{1}{2}\\|\\varphi\\|^2+\\frac{1}{2}\\|\\psi\\|^2$ emerges.","core_discovery":"The central claim is Theorem 3.13: if $H$ is the Heisenberg group with Carnot-Carathéodory distance $d_{cc}$, $g_t$ is the hypoelliptic Brownian motion, and $\\varphi,\\psi$ belong to the Cameron-Martin space $\\mathcal{H}(H)$ of horizontal finite-energy curves, then the small-tube probability ratio satisfies $$\\lim_{\\varepsilon\\to 0} \\frac{P\\left(\\sup_{t\\in[0,1]} d_{cc}(g_t,\\varphi(t))<\\varepsilon\\right)}{P\\left(\\sup_{t\\in[0,1]} d_{cc}(g_t,\\psi(t))<\\varepsilon\\right)}=\\exp\\left(-\\frac{1}{2}\\|\\varphi\\|^2_{\\mathcal{H}(H)}+\\frac{1}{2}\\|\\psi\\|^2_{\\mathcal{H}(H)}\\right).$$ In particular, the Onsager-Machlup functional is $L(p,v)=-\\frac{1}{2}\\|v\\|^2_{H_p}$ for horizontal tangent vectors $v\\in H_p$. In words, among horizontal paths, the most probable path in a small tube is the one with the smallest squared horizontal energy, exactly as in the flat Euclidean case.","pith_inferences":["If the same left-translation plus Girsanov argument goes through on any Carnot group, the Onsager-Machlup functional would again be minus half the squared horizontal norm, making the Euclidean formula a universal sub-Riemannian law.","The proof routes around curvature by using the group product as a substitute for parallel transport; one testable consequence is that the same construction should work for hypoelliptic diffusions on any nilpotent Lie group with a comparable left-invariant distance, regardless of curvature data.","A numerical check of the radius-matching step would be to simulate the ratio of the Carnot-Carathéodory and homogeneous tube probabilities for decreasing $\\varepsilon$; any systematic drift away from 1 would locate the failure in the unproved small-ball regular variation rather than in the Girsanov computation."],"forward_implications":["For any two horizontal finite-energy paths, the small-tube probability ratio has a finite, nonzero limit that depends only on the difference of their Cameron-Martin energies, so the Heisenberg Brownian motion has genuine Onsager-Machlup asymptotics in the sup-norm and Carnot-Carathéodory sense.","The most probable horizontal path in the small-tube limit minimizes the squared horizontal norm $\\int_0^1 |c_\\varphi(s)|^2\\,ds$; the vertical coordinate enters only through the horizontality condition, not through an additional potential or curvature term.","The class of processes with a known Onsager-Machlup functional is extended from elliptic diffusions on Riemannian manifolds to a hypoelliptic diffusion on a sub-Riemannian manifold, with the Heisenberg group as the first example.","The Cameron-Martin space $\\mathcal{H}(H)$ plays the same role as in the Euclidean case: exactly the horizontal finite-energy curves are the paths for which the tube probability has a nontrivial limiting ratio."],"supporting_citations":[{"why":"Supplies the elliptic Onsager-Machlup methodology (Girsanov shift, stochastic exponentials, conditional expectations) that the paper adapts to the sub-Riemannian setting.","marker":"[4]"},{"why":"Establishes the classical Onsager-Machlup formula for elliptic diffusion processes with curvature terms, the benchmark result the paper extends.","marker":"[17]"},{"why":"Provides the definition of the Onsager-Machlup functional and Lemma 4.8, the conditional-exponential-moment lemma used in Proposition 4.9.","marker":"[10]"},{"why":"Hörmander's theorem is invoked to conclude that the sub-Laplacian $X^2+Y^2$ is hypoelliptic, so the process $g_t$ is a hypoelliptic Brownian motion.","marker":"[9]"},{"why":"States and proves the Cameron-Martin-Girsanov theorem that the paper applies to the two-dimensional driving Brownian motion.","marker":"[11]"},{"why":"Supplies the coordinate formulas for left-invariant vector fields and integral curves on the Heisenberg group used to write the process $u^\\varphi$ explicitly.","marker":"[5]"}],"fun_headline_variants":["Tube odds on Heisenberg mimic flat-space law","Most probable Heisenberg path minimizes horizontal energy","Sub-Riemannian Brownian tubes get flat Onsager-Machlup","Heisenberg tubes pick paths by Euclidean energy","Heisenberg Brownian motion: tube odds obey Euclidean rule"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof leans on treating the Carnot-Carathéodory tube probability and the homogeneous-distance tube probability as asymptotically identical at small radius, although the distance equivalence only gives inclusions with mismatched radii; if those asymptotic probabilities differ, the theorem's ratio is not established.","fun_headline_variants_meta":{"raw":{"variants":["Tube odds on Heisenberg mimic flat-space law","Most probable Heisenberg path minimizes horizontal energy","Sub-Riemannian Brownian tubes get flat Onsager-Machlup","Heisenberg tubes pick paths by Euclidean energy","Heisenberg Brownian motion: tube odds obey Euclidean rule"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000939,"raw_usage":{"total_tokens":3999,"prompt_tokens":918,"completion_tokens":3081,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":534,"completion_tokens_details":{"reasoning_tokens":3003}},"tokens_in":534,"tokens_out":3081,"duration_ms":22525,"temperature":1.0,"reasoning_tokens":3003,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:23:14.086674+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the small-ball probabilities $P(\\sup_{t\\in[0,1]} \\rho(u^\\psi_t,e)<\\varepsilon)$ and $P(\\sup_{t\\in[0,1]} d_{cc}(g_t,\\psi(t))<\\varepsilon)$ to the same order in $\\varepsilon$ for one explicit horizontal path $\\psi$, such as a line segment. If their ratio does not tend to 1 as $\\varepsilon\\to 0$, or if the limit changes when $\\varepsilon$ is replaced by the mismatched radius appearing in the inclusion argument, then the asymptotic interchange behind Lemma 4.7 fails and the claimed ratio in Proposition 4.9 is not established.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the elliptic Onsager-Machlup methodology (Girsanov shift, stochastic exponentials, conditional expectations) that the paper adapts to the sub-Riemannian setting."},{"cited_title":"Takahashi and S","cited_arxiv_id":null,"evidence_quote":"Establishes the classical Onsager-Machlup formula for elliptic diffusion processes with curvature terms, the benchmark result the paper extends."},{"cited_title":"24, North-Holland Publishing Co., Amsterdam, 1989","cited_arxiv_id":null,"evidence_quote":"Provides the definition of the Onsager-Machlup functional and Lemma 4.8, the conditional-exponential-moment lemma used in Proposition 4.9."},{"cited_title":"119 (1967), 147–171","cited_arxiv_id":null,"evidence_quote":"Hörmander's theorem is invoked to conclude that the sub-Laplacian $X^2+Y^2$ is hypoelliptic, so the process $g_t$ is a hypoelliptic Brownian motion."},{"cited_title":"Shreve, Brownian motion and stochastic calculus , second ed., Graduate Texts in Mathematics, vol","cited_arxiv_id":null,"evidence_quote":"States and proves the Cameron-Martin-Girsanov theorem that the paper applies to the two-dimensional driving Brownian motion."},{"cited_title":"Driver and Maria Gordina, Heat kernel analysis on infinite-dimensional Heisenberg groups, J","cited_arxiv_id":null,"evidence_quote":"Supplies the coordinate formulas for left-invariant vector fields and integral curves on the Heisenberg group used to write the process $u^\\varphi$ explicitly."}],"review_version":1}