{"id":"dcc0e184-f5f6-4796-8f8d-fd668a121840","arxiv_id":"1908.08850","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Tightness is proved for reversible gradient dynamics of critical strip-wetting models, and a new continuous wetting measure with a local-time tilt converges to reflecting Brownian motion as the strip shrinks, leaving a Bessel SPDE as the conjectured limit.","lead":"This paper proves that the random dynamics of certain discrete wetting models are tight, meaning they cannot escape to infinity, and it constructs a new continuous wetting measure that converges to a reflecting Brownian motion. It is a step toward showing that the long-time fluctuations of wetting interfaces obey one universal stochastic equation, the Bessel SPDE.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.1's tightness proof needs the upper frame bound for the hat functions x_k^N; the Cauchy-Schwarz estimate as written is too weak, though the bound is true and repairable.","rationale":"The reader's weakest assumption correctly identifies the missing frame-type estimate in the proof of Theorem 2.1. This is the most load-bearing concern because the entire tightness argument for the rescaled reversible dynamics rests on the uniform martingale quadratic variation bound; without it, the BDG estimate and the subsequent H^{-1} tightness criterion fail. The needed estimate is true for the piecewise-linear hat basis, so the concern is a proof gap rather than a counterexample. I also inspected the proof of Theorem 4.1. The Girsanov step is supported by a bounded-bracket martingale, the SDE (4.5) has pathwise uniqueness by the monotonicity of the drift, and the comparison/tightness argument is plausible; the singular drift at 0 is a minor technical point since solutions are positive for positive times. The conjectures in Sections 3 and 4 are clearly labeled and do not affect the validity of the main theorems. Therefore the reader's conditional verdict is appropriate and no adjustment is needed.","tokens_in":16610,"tokens_out":18651,"duration_ms":195715,"concrete_test":"Re-derive the inequality between (2.2) and the BDG bound using the pointwise support bound 0 ≤ x_k^N ≤ N^{-1/2} 1_{[(k-1)/N,(k+1)/N]} and the fact that these intervals overlap at most twice; this gives Σ_{k=1}^N |<h,x_k^N>|^2 ≤ 4 N^{-2} ||h||^2 and makes the quadratic variation in (2.2) bounded uniformly in N. If this derivation cannot be carried out, the tightness proof of Theorem 2.1 lacks its key estimate.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 2.1 derives from (2.2) the quadratic variation <M^i>_t = 2N^2 Σ_{k=1}^N |<h,x_k^N>|^2 t. The text bounds this by 'Cauchy-Schwarz followed by the latter bound', referring to ||x_k^N||^2 ≤ 2N^{-2}. Read literally, this gives Σ |<h,x_k^N>|^2 ≤ N ||h||^2 ||x_k^N||^2 ≤ 2N^{-1}||h||^2, so the factor 2N^2 leaves an N-dependence, and the uniform BDG estimate fails. What is actually needed is the upper frame bound Σ |<h,x_k^N>|^2 ≤ C N^{-2}||h||^2. This bound is true: the hat functions form a Bessel sequence with constant C/N^2, and one can obtain it directly from the pointwise support bound 0≤x_k^N≤N^{-1/2}1_{I_k} with I_k=[(k-1)/N,(k+1)/N] and the at-most-two-fold overlap of the I_k, yielding Σ |<h,x_k^N>|^2 ≤ 4N^{-2}||h||^2. But the proof as written does not give this argument. If the estimate failed, the martingale quadratic variation in (2.2) would be unbounded in N, the BDG bound would carry a factor √N, and the H^{-1}(0,1) tightness conclusion would not follow. This is a genuine but repairable gap: the theorem appears true, but its proof currently relies on an unstated estimate.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies scaling limits of gradient (Langevin) dynamics associated with critical wetting models. In the discrete setting, for the strip-pinned wetting model of [DO19], the authors prove that the diffusively rescaled reversible processes (Y^N) are tight in C([0,T],H^{-1}(0,1)) (Theorem 2.1), using the Lyons–Zheng decomposition and an estimate of the martingale quadratic variation. In the continuum setting, they introduce the tilted Bessel measure P^{1,η}_a(dX) = (X_1∧η)/X_1 · a/(a∧η) exp(L^η_1/(2η)) P^3_a(dX), identify it as the law of the SDE dX_t = 1_{X_t≤η}/X_t dt + dB_t, and prove weak convergence to the law P^1_a of a one-dimensional Bessel process started from a as η→0 (Theorem 4.1). They also derive an integration-by-parts formula for the discrete measures, conjecture that the limit of (Y^N) is the reversible gradient dynamics of reflected Brownian motion, and formulate a Bessel-SPDE conjecture for a further continuous approximation.","tokens_in":16931,"tokens_out":19244,"duration_ms":188422,"significance":"If the proofs are completed, the paper makes two useful contributions. The continuum result of Theorem 4.1 is particularly clean: the density is an explicit exponential martingale (4.6), the family P^{1,η}_a monotonically interpolates between P^3_a and P^1_a, and the proof works on any finite time interval while correctly identifying the failure of the infinite-horizon analogue. The paper is also honest about its limitations: Conjectures 3.4 and 4.5 are clearly labeled as open, as are the convergence of the discrete integration-by-parts formula and the strong Feller property. Even if the conjectures remain unresolved, the explicit static approximation of reflected Brownian motion by a local-time-tilted Bessel/meander law is a valuable and potentially reusable result.","major_comments":[{"comment":"The claimed uniform bound on the sharp bracket is not obtained by the argument as written. Cauchy–Schwarz together with ‖x_k^N‖² ≤ 2N^{-2} gives Σ_{k=1}^N |⟨h,x_k^N⟩|² ≤ 2N^{-1}‖h‖², and hence the prefactor 2N² in (2.2) yields a bound of order N, not a constant. The argument needs the frame-type estimate Σ_{k=1}^N |⟨h,x_k^N⟩|² ≤ C N^{-2}‖h‖², which is true (it follows from x_k^N ≤ N^{-1/2}1_{I_k}, with I_k=[(k-1)/N,(k+1)/N], and the uniform bound on the overlap of the intervals I_k), but it is neither stated nor proved. Since this estimate is exactly what removes the N-dependence from the BDG bound and thus supports the H^{-1}(0,1) tightness conclusion, it should be added.","section":"Section 2, proof of Theorem 2.1, after Eq. (2.2)"},{"comment":"The convergence of the SDE for Z^η is not fully justified. The text has only pointwise a.s. convergence Z^η→Z and the a.s. bound Z≥Z^0>0 for a.e. t. To let η→0 in (4.7) one must prove convergence of the stochastic integrals, e.g. by E∫_0^1(√Z^η_s−√Z_s)² ds→0, using the domination 0≤(√Z^η_s−√Z_s)²≤Z^η_s≤Z^∞_s and the integrability of Z^∞, and one must justify the vanishing of the occupation drift 2∫_0^t 1_{Z^η_s≤η²}ds by dominated convergence. These are repairable but should be written out. In addition, the sentence 'the only possible subsequential weak limit... is given by the law of X' is used before the continuity of the pointwise limit X has been established; the argument should be reordered so that the SDE identification also yields continuity of X.","section":"Section 4.2, proof of Theorem 4.1, passage from Eq. (4.7) to the limit"}],"minor_comments":[{"comment":"The phrase 'reflecting Brownian bridge started from 0' appears to be a typo; elsewhere in the paper P^1_0 denotes the law of reflecting Brownian motion (one-dimensional Bessel process) started from 0, not a bridge.","section":"Proof of Theorem 2.1"},{"comment":"The assertion that pathwise uniqueness follows from monotonicity of x ↦ 1_{x≤η}/x is not a standard criterion as stated, because the coefficient is singular at 0 and discontinuous at η; please replace it by a reference or proof (e.g., via the Bessel SDE and Yamada–Watanabe).","section":"Section 4.2, Eq. (4.5)"},{"comment":"The Cauchy–Schwarz argument as written yields only L¹-boundedness of exp(X_1/η−M^ǫ_1), which does not by itself imply uniform integrability; the conclusion follows if one notes that both factors are bounded in L^p for every p<∞, but this should be stated explicitly.","section":"Proposition 4.4"},{"comment":"The claim that 1/(2η) is the smallest coefficient α for which exp(αL^η_∞) is not integrable is not proved; if kept, it should be accompanied by a reference or a short derivation.","section":"Remark 4.3"}],"recommendation":"major_revision","confidential_remarks":"For the editor: I see no disclosure or novelty problem. The overlap with [EAZ19] is acknowledged, and the conjectures citing it are clearly labeled. The main issue is rigor in the two proofs identified above; both are repairable without changing the scope of the paper. The manuscript fits the journal's range in mathematics and probability."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague — two things stand out. Theorem 2.1 answers the tightness conjecture from [DO19] for reversible gradient dynamics of shrinking-strip wetting models, and Theorem 4.1 introduces a genuinely new continuous wetting measure — a Brownian meander tilted by local time near zero — and proves it converges to reflecting Brownian motion. The explicit density, the BES(3)-to-reflecting-BM interpolation, and the fact that the conjectures are clearly labeled as conjectures are all real positives. The paper is a serious step toward the conjectured Bessel SPDE limit.\n\nIt deserves a serious referee. The Girsanov/Itô–Tanaka computation behind Theorem 4.1 is clean, and the convergence argument is essentially sound. The conjectures in Sections 3 and 4.3 are honest about what is open, and the self-citation to [EAZ19] is not a problem because the relevant statements are explicitly conjectural.\n\nThere are two soft spots, both repairable. In the proof of Theorem 2.1, the quadratic variation bound after Eq. (2.2) does not follow from Cauchy–Schwarz plus ‖x_k^N‖² ≤ 2N^{-2}. That route gives Σ |⟨h,x_k^N⟩|² ≤ 2N^{-1}‖h‖², so the factor 2N² leaves a divergent N-dependence. What is needed is the Bessel-sequence bound Σ |⟨h,x_k^N⟩|² ≤ C N^{-2}‖h‖², which is true for these hat functions because their supports overlap at most twice, but it is not stated. Without that estimate the BDG bound and the H^{-1} tightness conclusion do not go through as written. This is a genuine gap, but local and easily fixed.\n\nSecond, SDE (4.5) is used for a = 0 with a singular drift 1_{X≤η}/X, and the paper does not prove pathwise uniqueness or strong existence there. This is probably minor — the convergence P^{1,η}_0 → P^1_0 can likely be obtained from the a > 0 case via the Imhof relation or a direct Girsanov argument — but the proof as written leans on an unstated assumption.\n\nBottom line: the main results look true, the paper is well situated in the literature, and the gaps are genuine but local. I would send it to peer review and ask for a revised proof of the frame estimate and a justification of the a = 0 SDE step. For people working on wetting models, gradient interface dynamics, or Bessel SPDEs, this is worth reading.","headline":"A useful, honest paper: it proves the tightness conjecture for shrinking-strip wetting dynamics and gives a genuinely new continuous wetting measure converging to reflecting Brownian motion, but the proof of the tightness theorem has a repairable gap and the a=0 SDE step is under-justified.","tokens_in":17515,"tokens_out":3785,"would_cite":true,"duration_ms":39495,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","60H15","60J55"],"pacs":[],"model":"deepseek-v4-flash","headline":"Rescaled wetting dynamics are tight, and a new tilted Bessel path measure converges to reflecting Brownian motion.","keywords":["wetting models","shrinking strip","gradient dynamics","scaling limits","reflecting Brownian motion","Brownian meander","Bessel SPDEs","local times"],"falsifier":"Compute, for a sine mode $e_1$ and growing $N$, the quantity $N^2\\sum_{k=1}^N \\langle e_1,x_k^N\\rangle^2$. If this grows without bound, the uniform quadratic-variation estimate underpinning Theorem 2.1 fails and the tightness claim has no support. A second check: simulate $X_t^\\eta = a + \\int_0^t 1_{\\{X_s\\le \\eta\\}}X_s^{-1}\\,ds + B_t$ for decreasing $\\eta$ and compare the empirical law at time 1 to the Bessel-1 density; Theorem 4.1 asserts the distance goes to zero.","tokens_in":16375,"feed_emoji":"🌊","tokens_out":10707,"duration_ms":95870,"temperature":0.7,"pith_summary":"This paper proves two approximation results for critical wetting models, interfaces pulled toward a wall by a thin strip of pinning. It shows that the rescaled reversible gradient dynamics of the discrete shrinking-strip model are tight in $C([0,T],H^{-1}(0,1))$, settling a conjecture from earlier work. It also constructs a continuous wetting measure $P^{1,\\eta}_a$, given as a 3-dimensional Bessel law tilted by an explicit local-time factor, and proves that as $\\eta\\to0$ it converges weakly to the law of reflecting Brownian motion on $[0,1]$. A sympathetic reader would care because these results make a concrete bridge between discrete interface models and a continuum SPDE universality class, and they give a computable pathwise approximation of reflecting Brownian motion.","feed_headline":"A continuous wetting model converges to reflecting Brownian motion","feed_subtitle":"The same paper proves discrete wetting dynamics are tight, supporting a Bessel-SPDE limit.","key_machinery":"The load-bearing device is an explicit Radon–Nikodym density with a local-time tilt: against the 3-dimensional Bessel law, $P^{1,\\eta}_a$ weights paths by $\\frac{X_1\\wedge\\eta}{X_1}\\frac{a}{a\\wedge\\eta}\\exp(L^\\eta_1/(2\\eta))$. Itô–Tanaka rewrites this density as the Girsanov factor of a martingale, identifying it with the SDE whose drift is $1_{\\{X\\le\\eta\\}}/X$; comparison of squared Bessel processes then lets the proof pass $\\eta\\to0$. On the discrete side, the machinery is the Lyons–Zheng decomposition, which expresses time-reversal-stable martingale increments with quadratic variation fixed by the gradient form; the whole tightness proof reduces to bounding $N^2\\sum_{k=1}^N \\langle h,x_k^N\\rangle^2$ uniformly in $N$.","core_discovery":"The paper's central claim is that the continuous wetting measure $P^{1,\\eta}_a$, defined on path space by the explicit density $\\frac{X_1\\wedge\\eta}{X_1}\\frac{a}{a\\wedge\\eta}\\exp\\bigl(\\frac{1}{2\\eta}L^\\eta_1\\bigr)$ with respect to the law of a 3-dimensional Bessel process started at $a$, is the law of the unique strong solution of the truncated-drift SDE $X_t = a + \\int_0^t \\frac{1_{\\{X_s\\le \\eta\\}}}{X_s}\\,ds + B_t$, and that as $\\eta\\to0$ these laws converge weakly to $P^1_a$, the law of a reflecting Brownian motion on $[0,1]$. It also claims that the rescaled reversible dynamics of the discrete shrinking-strip wetting model form a tight family in $C([0,T],H^{-1}(0,1))$, so the sequence has subsequential limiting dynamics. In both cases the stated result is a genuine approximation theorem: the continuous family $P^{1,\\eta}_a$ interpolates monotonically between the Bessel-3 law and the Bessel-1 law, and the discrete tightness was previously an open conjecture.","pith_inferences":["A direct route to the open conjectures is to prove that the rescaled discrete integration-by-parts boundary term converges to the Bessel-SPDE boundary term at generator level; the paper identifies this as the missing step.","The monotone interpolation $P^1_a \\preceq P^{1,\\eta}_a \\preceq P^3_a$ suggests a broader universality: any pinning shape with the same effective critical parameter should produce the same $\\eta\\to0$ law, making the specific choice of $\\phi_a$ irrelevant for the scaling limit.","Because the continuous wetting measure is an explicit tilt of a 3-dimensional Bessel law, it can serve as an importance-sampling proposal for reflecting Brownian motion, and the comparison argument gives a checkable way to measure how the bias disappears as $\\eta\\to0$."],"forward_implications":["The rescaled discrete dynamics have subsequential weak limits in $C([0,T],H^{-1}(0,1))$, so any future proof of the Bessel-SPDE limit only needs identification of the limit, not precompactness.","The continuous family $P^{1,\\eta}_a$ gives an explicit monotone interpolation $P^1_a \\preceq P^{1,\\eta}_a \\preceq P^{1,\\eta'}_a \\preceq P^3_a$ for $\\eta \\le \\eta'$, so reflecting Brownian motion is the $\\eta\\to0$ endpoint of a continuous family of Bessel-type laws.","The mollified continuous wetting measures have a well-defined reversible dynamics described by an SPDE with reflection at 0 and an attractive term near level $\\eta$; sending $\\eta\\to0$ formally recovers the conjectured Bessel-1 dynamics.","The static convergence of $P^{1,\\eta}_a$ works on any finite time horizon, giving a finite-time continuous approximation of reflecting Brownian motion that is not available as an absolutely continuous tilt on the infinite horizon."],"supporting_citations":[{"why":"Defines the shrinking-strip wetting model whose rescaled dynamics are proved tight, and supplies the static convergence to the reflected Brownian bridge used at the end of Theorem 2.1.","marker":"[DO19]"},{"why":"Proves the critical wetting scaling limits that set the target laws (Brownian meander and reflecting Brownian motion) for the continuous approximation.","marker":"[DGZ05]"},{"why":"Provides the Lyons–Zheng decomposition and Revuz-correspondence machinery that turn the gradient-form structure into explicit quadratic variations.","marker":"[FOT10]"},{"why":"Supplies Itô–Tanaka, Girsanov, comparison, and tightness criteria used to identify $P^{1,\\eta}_a$ as a strong solution and to pass to the $\\eta\\to0$ limit.","marker":"[RY13]"},{"why":"Gives the tightness criterion in $C([0,T],H^{-1})$ that closes Theorem 2.1 once the moment bounds are obtained.","marker":"[EK86]"},{"why":"Introduces the Bessel SPDE family whose $\\delta=1$ member is the conjectured limit of the wetting dynamics, and supplies the integration-by-parts formula that the discrete formula is compared with.","marker":"[EAZ19]"},{"why":"Shows the gradient Dirichlet forms are closable, so the mollified continuous wetting measure defines a genuine reversible Markov process.","marker":"[Zam02]"}],"fun_headline_variants":["Wetting dynamics converge to reflecting Brownian motion","Reflecting Brownian motion from wetting model limits","Tight wetting dynamics suggest Bessel SPDE limit","Continuous wetting converges, discrete dynamics tight","Bessel SPDE emerges from wetting model dynamics"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of the discrete tightness result depends on a sharp collective estimate for the tent-shaped coordinate functions—the sum of squared overlaps with any fixed height profile must be of order $N^{-2}$—and the text's displayed Cauchy–Schwarz bound does not by itself give that estimate.","fun_headline_variants_meta":{"raw":{"variants":["Wetting dynamics converge to reflecting Brownian motion","Reflecting Brownian motion from wetting model limits","Tight wetting dynamics suggest Bessel SPDE limit","Continuous wetting converges, discrete dynamics tight","Bessel SPDE emerges from wetting model dynamics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000269,"raw_usage":{"total_tokens":1634,"prompt_tokens":967,"completion_tokens":667,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":583,"completion_tokens_details":{"reasoning_tokens":593}},"tokens_in":583,"tokens_out":667,"duration_ms":6508,"temperature":1.0,"reasoning_tokens":593,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:30:01.957716+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for a sine mode $e_1$ and growing $N$, the quantity $N^2\\sum_{k=1}^N \\langle e_1,x_k^N\\rangle^2$. If this grows without bound, the uniform quadratic-variation estimate underpinning Theorem 2.1 fails and the tightness claim has no support. A second check: simulate $X_t^\\eta = a + \\int_0^t 1_{\\{X_s\\le \\eta\\}}X_s^{-1}\\,ds + B_t$ for decreasing $\\eta$ and compare the empirical law at time 1 to the Bessel-1 density; Theorem 4.1 asserts the distance goes to zero.","supporting_citations":[],"review_version":1}