{"id":"6d4c7f2a-dfe7-4661-89a2-49b9eb4491a3","arxiv_id":"2504.17094","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Small-noise fluctuations and large deviations of generalized Dean-Kawasaki SPDEs are characterized on C^2 bounded domains with Dirichlet boundary conditions.","lead":"This paper proves central limit and large deviation results for a class of conservative stochastic PDEs with small noise on bounded domains with Dirichlet boundary conditions. It extends known torus results to settings where particles can enter or leave through the boundary.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 6's L∞ estimate rests on an unproved Moser-iteration step (52); since Theorem 7's singular CLT relies entirely on it, this is the load-bearing soft spot.","rationale":"The reader's conditional verdict is appropriate. The Assumption 2 limitation is real and acknowledged, but it is a scope restriction: the paper states the CLT under that assumption and remarks that it is necessary. The more load-bearing internal risk is the proof of Theorem 6, which the reader also flagged as delegated. Theorem 6 is the exact bridge that converts the regularized CLT (Theorem 5) into the singular CLT in probability (Theorem 7), and its proof contains an asserted Moser-iteration step, equation (52), that is not derived and begins with an undefined L^0 norm. Because the entire singular CLT depends on this estimate, the paper should either include the full derivation or clearly mark the result as conditional on the verification of (52). This does not change the overall verdict from conditional, but it identifies a sharper point to settle than the acknowledged Assumption 2 restriction.","tokens_in":68847,"tokens_out":10333,"duration_ms":107651,"concrete_test":"Re-derive equation (52) from (51) in full detail: write the log-convexity inequality, the Sobolev embedding, and Hölder exponents explicitly, and track every constant through the Moser iteration p_{k+1} = ((d+2)/d)p_k + 2, replacing the p0 = 0 step by a well-defined L^1 or L^2 base case. Then verify that the constant product in (53) remains finite for d = 1, 2, 3 with these explicit constants. If the exponent in (52) or the summability of the constants fails, Theorem 6 is unproved and Theorem 7's singular CLT does not follow.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The headline CLT in probability (Theorem 7) is the paper's main result, and its transition from the regularized CLT (Theorem 5) to the singular equation is carried entirely by Theorem 6: one needs P(ρ^{ε,K} < M/2) → 0 to replace the singular σ by a globally Lipschitz σ̃. The proof of Theorem 6, however, jumps from the p-th moment bound (51) to the 'improved estimate' (52) by invoking log-convexity, Sobolev embedding and Hölder 'as in Theorem 3.9 of [DFG20]'. Neither the derivation of (52) nor the associated constants is shown. The subsequent Moser iteration starts at p0 = 0 and uses (52) with a p−2 = 0 'norm', which is not defined in the paper, so the base case of the iteration is only formal. If (52) fails, or if the constants in the product bound (53) diverge in dimension d ≥ 2, the asserted bound E‖(ρ^{ε,K} − M)_−‖_{L∞(U×[0,T])} ≤ c(ε(‖FK3‖ + ‖∇·FK2‖))^γ is not established, and the final probability bound in Theorem 7 loses its last term. This is an internal proof concern, distinct from the acknowledged restriction of Assumption 2 to constant boundary data.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the generalized Dean--Kawasaki SPDE (1) on a bounded C^2 domain with Dirichlet boundary data and truncated, spatially correlated noise. It proves quantitative law-of-large-numbers estimates for a regularized equation (Propositions 4 and 5), a central limit theorem in H^{-s} for the regularized equation (Theorem 5), an L^\\infty estimate and a CLT in probability for the singular equation (Theorems 6 and 7) under a joint \\varepsilon-K scaling, and a large deviations principle on L^1(U\\times[0,T]) with a rate function matching the zero-range-process rate function (Theorem 8). The CLT is proved only under Assumption 2, where the initial condition and the Dirichlet boundary data coincide with the same random positive constant M, so the hydrodynamic limit is the constant M. The LDP section considers general H^1(\\partial U) boundary data via the weak approach of Budhiraja--Dupuis--Maroulas, relying on skeleton-equation well-posedness imported from Fehrman--Gess and the author's prior work. The paper identifies the explicit joint scaling \\varepsilon K^{d+2}\\to 0 for eigenfunction noise in Remark 17.","tokens_in":69151,"tokens_out":13907,"duration_ms":129839,"significance":"If the proof gaps are repaired, the results would be a meaningful extension of Dean--Kawasaki fluctuation theory from the torus to bounded domains with Dirichlet boundary conditions, covering fast-diffusion and porous-medium nonlinearities \\Phi(\\xi)=\\xi^m for all m>0, including the critical square-root diffusion, and matching the zero-range-process rate function. Strengths of the manuscript include the detailed estimate chains in Theorem 5 and Propositions 4--5, the explicit joint scaling in Remark 17, and the careful discussion in Section 4.2 of why uniformity over boundary data fails. However, the singular CLT depends on an unproved L^\\infty bound, and the canonical eigenfunction noise does not satisfy Assumption 1 as stated, so the paper's advertised scope is not currently established.","major_comments":[{"comment":"Assumption 1(5) is incompatible with Assumption 1(3) for any nontrivial noise. Since f_k vanishes on \\partial U, the sum F_1=\\sum_k f_k^2 vanishes on \\partial U; a continuous function on U that is constant and vanishes on \\partial U must be identically zero, forcing f_k\\equiv 0 for every k. Thus the \"probabilistically stationary\" condition cannot be met by nonzero Dirichlet-compatible noise. In particular, Example 1 does not satisfy Assumption 1(5): for U=(0,\\pi) with e_k=(2/\\pi)^{1/2}\\sin(kx), F_1^K(x) is a nontrivial trigonometric sum, not a constant. The theorems in Sections 3 and 4 all assume Assumption 1, so the main example and the critical square-root case are not covered as stated. Please remove or reformulate point 5, or prove that the results hold without it and state exactly where point 5 is used.","section":"Section 2, Assumption 1, points 3 and 5"},{"comment":"The proof of the L^\\infty bound for the singular equation is incomplete. Equation (52) is asserted to follow from (51) \"as in Theorem 3.9 of [DFG20]\" without showing the log-convexity/Sobolev/H\\\"older argument, and this is the only step that upgrades L^p control to L^\\infty. The Moser iteration then starts at p_0=0 and applies (52) with p=q_1=p_0+2=2, whose right-hand side is the undefined L^0-norm of (\\rho^{\\varepsilon,K}-M)_-; moreover, the recursion p_k=((2+d)/d)p_{k-1}+2 does not match the exponent produced by (52), which is ((2+d)/d)(p_{k-1}+2)=p_k+4/d. Consequently the asserted bound E\\|(\\rho^{\\varepsilon,K}-M)_-\\|_{L^\\infty}\\le c(\\varepsilon(\\|F_3^K\\|+\\|\\nabla\\cdot F_2^K\\|))^\\gamma is not established. Since Theorem 7 uses exactly this bound to control P(S^c)+P(\\tilde S^c), the central limit theorem for the singular equation rests on a gap.","section":"Section 3.2, Theorem 6, equations (51)--(53)"},{"comment":"Theorem 9 states well-posedness of the skeleton equation, equivalence of weak and kinetic solutions, and an L^1-contraction for general H^1(\\partial U) boundary data, but the proof is a one-sentence assertion that the torus result of [FG23] is adapted with arguments from [Pop25]. No details are given for the Dirichlet boundary conditions or for the general coefficients, and the statement says \"under some assumptions\" without listing them. Proposition 8, Theorem 10, and Proposition 15 all rely on Theorem 9, so the large deviations principle is conditional on an unverified adaptation. Please provide a complete proof or a precise reduction to the cited results that verifies every hypothesis, and clarify how Proposition 15's appeal to Theorem 21 of [FG23] is adapted to the bounded-domain setting.","section":"Section 4.1, Theorem 9"},{"comment":"The CLT is proved only under Assumption 2, where the initial condition and the Dirichlet boundary data coincide with the same random positive constant M. This restriction is acknowledged in Remarks 4 and 15, but the abstract presents the CLT as applying to general Dirichlet boundary conditions. Please state explicitly in the abstract and introduction that the CLT is for constant coinciding initial and boundary data, and that variable boundary data are treated only in the LDP.","section":"Section 3, Assumption 2, and abstract"}],"minor_comments":[{"comment":"The phrase \"Let v^{\\varepsilon,K} be a weak solution to (3)\" should be \"Let v^{\\varepsilon,K} be defined by (3)\", since (3) defines v^{\\varepsilon,K} in terms of the kinetic solution \\rho^{\\varepsilon,K} and the hydrodynamic limit \\bar\\rho.","section":"Theorem 7, statement"},{"comment":"The notation n_p conflicts with the regularization index n: in (52) and (53) n_p is used both as an exponent exponent and as the reciprocal p^{-1}. Please disambiguate these quantities.","section":"Section 3.2, equations (52)--(53)"},{"comment":"The text repeatedly writes \"Mozer iteration\"; this should be \"Moser iteration\".","section":"Section 3.2, Theorem 6"},{"comment":"The double-sum estimates in the uniqueness proof contain repeated or mismatched summation indices; the sums over j and k should be written separately with the weighting by \\lambda_j^{-s} displayed consistently.","section":"Proposition 6, uniqueness proof, equation (34) and following"}],"recommendation":"major_revision","confidential_remarks":"The two blocking issues are the incompatibility of Assumption 1(5) with nonzero Dirichlet-compatible noise and the missing Moser-iteration step in Theorem 6. Both are likely local fixes, but until they are resolved the singular CLT is not proven. The manuscript also leans heavily on [Pop25] and [FG23] for Theorem 9; the adaptations should be spelled out rather than asserted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main thing you should know: this is a real extension, not a repackaging. The torus CLT (Clini-Fehrman) and torus LDP (Fehrman-Gess) are moved to C^2 bounded domains with Dirichlet boundary data, boundary terms included, full range m in (0,infinity) for Phi(xi)=xi^m, and the rate function matches the zero-range process on the bounded domain. Theorems 7 and 8 are new statements, and the LDP half is largely written out.\n\nCredit where due: the energy estimates (Propositions 4 and 5) are detailed, the joint scaling in Remark 17 is explicit (epsilon K^{d+2} -> 0 for the eigenfunction noise), and the paper is upfront about Assumption 2 -- the CLT only treats constant initial/boundary data M, and Remark 15 admits the L^4 and L^{2beta+4} bounds in the CLT proof would fail otherwise. The abstract's Dirichlet-generality is really delivered by the LDP section, not the CLT, which is a mismatch worth knowing. No circularity: nothing is fitted, and the reliance on prior work (Pop25, DFG20, FG23) is legitimate.\n\nNow the soft spot. The stress-test note lands. Theorem 6 is the load-bearing step for the singular CLT, and the paper itself says the 'improved estimate' (52) follows 'as in Theorem 3.9 of [DFG20]' -- that is a sketch. The Moser iteration starts at p_0 = 0, which means applying (52) with p = 2 leaves a factor ||(rho-M)_-||_{L^{p-2}} with p-2 = -2, an undefined 'norm'; the base case is formal. The paper does bound the constant product in (53), so I would guess the result is true and the gap is fixable, but as written the CLT in probability rests on an unfinished proof, and the bound E||(rho^{epsilon,K}-M)_-||_{L^infinity} <= c(epsilon(...))^gamma is not established. That is the main thing a referee should demand.\n\nMinor: Proposition 6's uniqueness proof has a duplicated summation index (the double sum over j and j,k) -- cosmetic. Theorem 9 delegates skeleton well-posedness to 'adapting' FG23/Pop25; acceptable, but thin.\n\nBottom line: for Dean-Kawasaki / fluctuating hydrodynamics people, this is the bounded-domain reference for CLT and LDP behavior. It deserves a serious referee. My recommendation: send it to review, and instruct the referee to enforce a complete proof of Theorem 6 -- or, if it is genuinely a re-run of DFG20, a precise theorem statement with all constants tracked.","headline":"Genuine bounded-domain extension of the torus CLT/LDP results for Dean-Kawasaki SPDEs; the singular CLT rests on a sketched Moser-iteration estimate (Theorem 6) that needs a completed proof before the paper is fully verified.","tokens_in":69673,"tokens_out":4691,"would_cite":true,"duration_ms":42404,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60F05","60F10","60H15","82B21","82B31","60K35","35Q70"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves a central limit theorem and a large-deviations rate function for generalized Dean–Kawasaki SPDEs on bounded domains with Dirichlet boundary data.","keywords":["Central limit theorem","Large deviations principle","Dean–Kawasaki equation","Dirichlet boundary conditions","Stochastic kinetic solutions","Interacting particle systems","Zero range process"],"falsifier":"A direct numerical experiment would settle the scaling claim: take the Dirichlet-eigenfunction noise of Example 1 and keep $\\varepsilon K^{d+2}=\\delta>0$ fixed as $\\varepsilon\\to0$; Theorem 7's error bound has prefactors $\\|F_3^K\\|_{L^\\infty}\\sim K^{d+2}$, so the right-hand side should not vanish and $v^{\\varepsilon,K}$ should fail to converge to the linearized SPDE if the estimates are sharp. A complementary analytic check is homogeneous Dirichlet data $\\Phi(\\rho)|_{\\partial U}=0$ with fast diffusion $\\Phi(\\xi)=\\xi^m$, $m<1$, where deterministic finite-time extinction is known; then Theorem 6's positivity estimate is false, so Theorem 7 cannot cover that boundary data.","tokens_in":68632,"feed_emoji":"📈","tokens_out":13682,"duration_ms":109046,"temperature":0.7,"pith_summary":"Generalized Dean–Kawasaki SPDEs describe how the density of an interacting particle system fluctuates when driven by small multiplicative noise. The paper proves two fluctuation results for these equations on arbitrary $C^2$ bounded domains with Dirichlet boundary conditions: under the joint scaling $\\varepsilon K^{d+2}\\to 0$, the rescaled fluctuation $v^{\\varepsilon,K}=\\varepsilon^{-1/2}(\\rho^{\\varepsilon,K}-\\bar\\rho)$ converges in probability to the solution of the linearized SPDE (a central limit theorem), and the solutions satisfy a large deviations principle with the same variational rate function that governs the zero-range process. This matters because the boundary condition is what makes a particle model describe absorption or injection of particles at the boundary, turning fluctuation results on the torus into results for realistic bounded regions. The technical price is a set of new $L^p$ energy estimates and an $L^\\infty$ positivity estimate that keep the singular square-root diffusion away from its zero set.","feed_headline":"Dean–Kawasaki noise: CLT and rate function proved on bounded domains","feed_subtitle":"Matches zero-range particle rate, with Dirichlet boundaries as absorption or injection","key_machinery":"The load-bearing mechanism is the theory of stochastic kinetic solutions: the equation is renormalized away from the zero set of the density, where the square-root diffusion coefficient is singular, and the Dirichlet condition is encoded through $\\Phi(\\rho^{\\varepsilon,K})|_{\\partial U}=\\bar f$. On top of that solution theory, the CLT rests on three devices: Assumption 2, which fixes the initial and boundary data to the same positive constant $M$ and makes the hydrodynamic limit the constant $M$; the auxiliary potentials $\\Theta_{\\Phi,p}$, defined by $\\Theta_{\\Phi,p}'(\\xi)=(\\xi-\\bar\\rho)^{(p-2)/2}\\Phi'(\\xi)^{1/2}$, which convert the diffusion into gradients whose $L^2$ norm is controlled by the energy estimates; and a Moser-iteration $L^\\infty$ estimate showing the solution stays above $M/2$ with high probability, so the singular coefficient can be replaced by a bounded one. For the large deviations principle, the weak approach of [BDM08] represents the noise through a controlled SPDE; the skeleton limit of the controlled equation is a parabolic-hyperbolic PDE whose solution map is compact in $L^1(U\\times[0,T])$, and the rate function is the minimal $L^2$ cost of a control driving that skeleton equation.","core_discovery":"The paper's central claim is twofold. Theorem 7 states that for stochastic kinetic solutions of (1) under Assumptions 1, 2, 3 and 6, the process $v^{\\varepsilon,K}=\\varepsilon^{-1/2}(\\rho^{\\varepsilon,K}-\\bar\\rho)$ converges in probability in $L^2([0,T];H^{-s}(U))$ for every $s>d/2$ to the unique strong solution of the linearized SPDE $\\partial_t v=\\Delta(\\Phi'(\\bar\\rho)v)-\\nabla\\cdot(\\sigma(\\bar\\rho)\\dot\\xi+\\nu'(\\bar\\rho)v)$ with zero initial and boundary data, whenever the eigenfunction noise is truncated so that $\\varepsilon K^{d+2}\\to 0$. Theorem 8 states that for boundary data $\\bar f\\in H^1(\\partial U)$ and initial data with finite $\\Phi$-entropy, the solutions satisfy a large deviations principle on $L^1(U\\times[0,T])$ with rate function $$I_{\\bar f,\\rho_0}(\\rho)=\\frac12\\inf_{g\\in $L^{2}$(U\\times[0,T];\\mathbb R^d)}\\left\\{\\|g\\|^2_{$L^{2}$(U\\times[0,T];\\mathbb R^d)} : \\partial_t\\rho=\\$\\Delta$\\Phi(\\rho)-\\nabla\\cdot(\\$\\sigma$(\\rho)g+\\nu(\\rho)),\\ \\Phi(\\rho)|_{\\partial U}=\\bar f,\\ \\rho(\\cdot,0)=\\rho_0\\right\\}.$$ This rate function is exactly the one established for the zero-range process in earlier work, so the SPDE reproduces the particle system both in first-order fluctuations and in the exponential cost of rare events.","pith_inferences":["One can test sharpness of the scaling: if $\\varepsilon K^{d+2}\\to c>0$, the error prefactors in Theorem 7 should fail to vanish, so $v^{\\varepsilon,K}$ should stop converging; this would show the scaling is necessary rather than merely sufficient.","The CLT is restricted to random constant initial and boundary data, and the paper's Remark 15 says only $L^2$ bounds are available without this; a non-constant boundary profile with a nontrivial stationary state likely needs a genuinely different fluctuation theory with boundary layers.","The successful rate-function match with zero-range processes suggests the same variational rate function might describe other boundary-driven particle models such as exclusion processes with reservoirs, but establishing that would require a separate particle-side large-deviations argument not attempted here."],"forward_implications":["On any $C^2$ bounded domain, typical fluctuations of the density are asymptotically Gaussian and governed by the linearized SPDE (4), so SPDE simulations can replace expensive particle simulations in the central-limit regime.","Because the rate function matches the zero-range-process rate, probabilities of rare density profiles can be computed from the variational problem $I_{\\bar f,\\rho_0}$ instead of by simulating huge particle systems.","Dirichlet boundaries correspond to absorption or injection of particles, so the results cover heat reservoirs, traffic inflow and outflow, wealth entering or leaving a system, and population immigration or emigration on bounded regions.","The joint scaling $\\varepsilon K^{d+2}\\to 0$ gives a concrete prescription: for noise amplitude $\\varepsilon$, the number $K$ of eigenmodes must grow faster than $\\varepsilon^{-1/(d+2)}$ for the limit theorems to hold.","The large deviations principle is uniform over compact sets of initial data, while the paper proves a structural obstruction, boundary normal-derivative terms, preventing uniformity over boundary data."],"supporting_citations":[{"why":"Supplies well-posedness and uniqueness of stochastic kinetic solutions of (1) on bounded domains with Dirichlet boundary data, the backbone solution theory for every theorem.","marker":"[Pop25]"},{"why":"Proves the central limit theorem for generalized Dean–Kawasaki SPDEs on the torus, the result that is extended here to C^2 domains and to the full range of fast-diffusion coefficients.","marker":"[CF23]"},{"why":"Provides the kinetic solution theory, the regularization of the square-root coefficient, and the L-infinity and tightness estimates adapted to the bounded-domain setting.","marker":"[FG24b]"},{"why":"Establishes the torus large deviations principle and skeleton-equation well-posedness whose rate function is matched in Theorem 8.","marker":"[FG23]"},{"why":"Supplies the variational representation and weak approach to infinite-dimensional large deviations that underpin Assumption 4 and Theorem 8.","marker":"[BDM08]"},{"why":"Gives the template CLT and LDP for conservative SPDEs, including the Moser-iteration L-infinity argument reused for the singular square-root diffusion.","marker":"[DFG20]"},{"why":"Identifies the zero-range-process large deviations rate function with which the SPDE rate function is compared and matched.","marker":"[BKL95]"}],"fun_headline_variants":["Dean–Kawasaki SPDE: CLT and LDP on bounded domains with Dirichlet BC","Small noise limits for Dean–Kawasaki on bounded domains with absorption","Zero-range matching: CLT and LDP for conservative SPDEs with Dirichlet data","Noise fluctuations and rare events for Dean–Kawasaki with Dirichlet boundaries","CLT and large deviations for SPDEs with absorbing boundaries"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire CLT proof depends on Assumption 2, that the initial density and the Dirichlet boundary value are one and the same random positive constant $M$; if the boundary value differs from the initial data, the hydrodynamic limit is no longer constant, and the paper says only $L^2$ estimates are available, so the $L^4$ and $L^{2\\beta+4}$ bounds at the heart of the CLT cannot be obtained.","fun_headline_variants_meta":{"raw":{"variants":["Dean–Kawasaki SPDE: CLT and LDP on bounded domains with Dirichlet BC","Small noise limits for Dean–Kawasaki on bounded domains with absorption","Zero-range matching: CLT and LDP for conservative SPDEs with Dirichlet data","Noise fluctuations and rare events for Dean–Kawasaki with Dirichlet boundaries","CLT and large deviations for SPDEs with absorbing boundaries"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000366,"raw_usage":{"total_tokens":1982,"prompt_tokens":972,"completion_tokens":1010,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":588,"completion_tokens_details":{"reasoning_tokens":904}},"tokens_in":588,"tokens_out":1010,"duration_ms":7477,"temperature":1.0,"reasoning_tokens":904,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T10:49:32.629277+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct numerical experiment would settle the scaling claim: take the Dirichlet-eigenfunction noise of Example 1 and keep $\\varepsilon K^{d+2}=\\delta>0$ fixed as $\\varepsilon\\to0$; Theorem 7's error bound has prefactors $\\|F_3^K\\|_{L^\\infty}\\sim K^{d+2}$, so the right-hand side should not vanish and $v^{\\varepsilon,K}$ should fail to converge to the linearized SPDE if the estimates are sharp. A complementary analytic check is homogeneous Dirichlet data $\\Phi(\\rho)|_{\\partial U}=0$ with fast diffusion $\\Phi(\\xi)=\\xi^m$, $m<1$, where deterministic finite-time extinction is known; then Theorem 6's positivity estimate is false, so Theorem 7 cannot cover that boundary data.","supporting_citations":[],"review_version":1}