{"id":"29f034ce-8455-414a-9b96-e5a94e0b9d86","arxiv_id":"2412.04015","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"The scaled density fluctuation field of a Glauber-Kawasaki lattice gas near a stationary flat interface converges to e(ϑ)√̟B_t in dimension 1 and to e(ϑ)Z_t(θ), with Z solving the additive-noise one-dimensional stochastic heat equation, in dimension 2.","lead":"This paper proves that, for a lattice gas whose particles both hop and are created and annihilated, the random density fluctuations near a flat interface converge after scaling to a Brownian motion in one dimension and to a stochastic heat equation in two dimensions. The fluctuation mode is exactly the shape of the interface transition layer, a form of interface rigidity now derived from the particle dynamics rather than assumed.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma B.3 is likely false: the periodic two-kink profile makes ∂ϑρ_K a two-bump zero eigenfunction of A_K, so e^{t K A_K} does not converge in L²(√KT) to the one-bump projection e⟨F,e⟩, and the e-shape of the limit is unsupported.","rationale":"We focused on the central claim Theorem 2.4 and traced where the shape e enters. The only place the rank-one projection is justified is Lemma B.3, whose proof imports Proposition B.4 and Lemma B.7 from the companion paper [14] and Carr–Pego [5]. The reader flagged this as the weakest assumption; we agree, and we identify a concrete mechanism that makes Lemma B.3 fail as stated. The profile ρ_K is not the single standing wave φ on R; it is the periodic two-layer solution on √KT. The linearized operator A_K therefore has a periodic zero eigenfunction ∂ϑρ_K with two well-separated bumps. Even after the fast K-scaled evolution, e^{t K A_K} F converges, for t up to fixed T, to the projection onto this two-bump eigenfunction, up to the antisymmetric near-zero mode which survives because its eigenvalue is exponentially small in √K and remains o(1) after multiplication by Kt. Hence the L²(√KT) distance to the one-bump e_K is O(1). This is not a minor gap: Lemma 7.6 bounds the discrepancy between the martingale driven by the true semigroup and the one driven by e_K S_{t−s}⟨H⟩ by exactly this L² norm. If the norm does not vanish, the covariance of the limit is not (2.29); the Gaussian limit would have extra contributions from the far interface, and the factorization X_t = e(ϑ)B_t or e(ϑ)Z_t would break. The theorem might be salvageable by working with local seminorms adapted to compact ϑ-supports, since the far interface is invisible to the test functions, but that is not what the paper proves. This is a central, load-bearing correctness concern, not a matter of style. Unless the proposed numerical test overturns it, the verdict should be REJECT.","tokens_in":54265,"tokens_out":25627,"duration_ms":258612,"concrete_test":"Discretize the periodic operator K A_K on √KT for K = 10^2, 10^3, 10^4 with a spectral scheme resolving both interfaces (mesh size ≪ K^{−1/2}), compute ρ_K numerically from (2.8), and evaluate I_K = ∫_0^1 ‖∂ϑ e^{t K A_K} F − ⟨F,e⟩∂ϑe_K‖²_{L²(√KT)} dt for F(ϑ) = e^{−ϑ²}. If I_K does not tend to 0 and remains O(1), Lemma B.3 is false and the projection step in Lemma 7.6 fails. An even simpler preliminary check: compute ‖∂ϑρ_K/‖∂ϑρ_K‖_{L²(√KT)} − e_K‖_{L²(√KT)}; if this distance is approximately √(2 − √2) > 0 for large K, the fixed points of the semigroup are incompatible with Lemma B.3.","verdict_should_be":"REJECT","load_bearing_attack":"Lemma B.3 asserts that for fixed T and F ∈ C_c^∞(R), the time integral of ‖∂ϑP^K_t F − ⟨F,e⟩∂ϑe_K‖²_{L²(√KT)} tends to 0. The operator A_K = ∂ϑ² − V''(ρ_K) is periodic on √KT, where ρ_K has two transition layers: the decreasing kink near ϑ = 0 and the increasing antikink near ϑ ≈ h2√K, see (2.7)–(2.10). Hence A_K has the periodic zero eigenfunction ∂ϑρ_K. Its L²-normalized version g_K = ∂ϑρ_K/‖∂ϑρ_K‖ has two separated bumps, g_K ≈ (e_1 + e_2)/√2, with e_2 the translate of e at the second interface. The one-bump e_K defined in Section 7 is not close to g_K in L²(√KT): ‖g_K − e_K‖² ≈ 2 − √2 > 0. The near-zero antisymmetric mode (relative translation of the two kinks) has eigenvalue O(K e^{−c√K}) (Carr–Pego [5]), so on the fixed interval [0,T] this mode neither relaxes nor decays: K e^{−c√K} T → 0. Thus e^{t K A_K} F retains an O(1) component at the second interface, and the L²(√KT)-error in Lemma B.3 does not vanish. Lemma 7.6 relies exactly on this lemma to replace T^K_{t−s} H by e_K S_{t−s}⟨H⟩; if the replacement error is O(1), the covariance computation leading to (2.29) and the factorization X_t = e(ϑ)√̟B_t or e(ϑ)Z_t(θ) has no basis. The result is imported from the single-kink setting of [14, Prop. B.4] and is not valid for the periodic two-layer operator.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a Glauber+Kawasaki particle system on the discrete torus, with Glauber rates scaled by K=K_N and Kawasaki rates by N^2, started from a product measure whose density is a stationary two-layer profile for the reaction-diffusion equation. It proves a relative-entropy bound (Theorem 2.2 and Corollary 2.3), a Boltzmann-Gibbs principle (Theorem 6.1), and then shows that the rescaled density fluctuation field X^N_t converges, for K_N satisfying explicit growth conditions in d=1 or 2, to a Gaussian field. In d=1 the limit is e(ϑ)√̟B_t, and in d=2 it is e(ϑ)Z_t(θ), where e is the normalized derivative of the standing wave φ, B_t is Brownian motion, and Z_t solves the stochastic heat equation on the one-dimensional torus driven by space-time white noise. Tightness is proved for the time-integrated field. The main technical route is: entropy production estimates, two-block replacement and Boltzmann-Gibbs homogenization, approximation of the density field by a Dynkin martingale, and a Sturm-Liouville reduction that projects the field onto the one-bump mode e.","tokens_in":54780,"tokens_out":19170,"duration_ms":196729,"significance":"If the proof is correct, this is a substantial result: it derives, directly from the microscopic Glauber+Kawasaki dynamics, the linear fluctuation of a stationary flat interface and shows that the fluctuation retains the shape of the standing-wave profile. The covariance formula (2.29) is derived from the martingale quadratic variation rather than imposed, and the growth conditions on K_N are stated explicitly. The paper is carefully structured and contains substantial technical work in the entropy production and Boltzmann-Gibbs sections. Its main vulnerability is the spectral reduction to the one-bump mode e, which is imported from the companion paper [14] and is exactly the step that produces the e(ϑ) factorization in the final limit.","major_comments":[{"comment":"There is an inconsistency in the definition of P^K_t, and as written Lemma B.3 is false. In Section 7, P^K_t is defined as the semigroup of K A_K and Lemma 7.6 uses it through T^K_{t-s}F = P^K_{t-s}F S_{t-s}G. In Appendix B, however, Lemma B.3 states that P^K_t is the semigroup of A_K, without the factor K. Taken literally, the lemma cannot hold: for a generic F in C_c^∞(R), the integrand at t=0 is ||∂ϑF − ⟨F,e⟩∂ϑe_K||²_{L²(√KT)}, which is generally bounded away from zero, and the unaccelerated semigroup e^{tA_K} does not erase this contribution over a fixed interval [0,T]. Since Lemma 7.6 is the only step that replaces the semigroup T^K_{t-s} by e_K times the heat semigroup, the version with the factor K is the one needed. The authors should restate Lemma B.3 with the semigroup actually used, prove it, or give a precise statement in [14] that covers exactly that version.","section":"Appendix B, Lemma B.3; Section 7, Lemma 7.6"},{"comment":"The reduction to the one-bump mode e is not justified in the periodic two-layer geometry. The operator A_K = ∂ϑ² − V''(ρ_K) on the torus √KT has the exact periodic zero eigenfunction ∂ϑρ_K, which has two separated transition-layer bumps, and, by the Carr–Pego metastability analysis, a near-zero antisymmetric mode with exponentially small eigenvalue. Lemma B.3 and its supporting import from [14] concern a one-layer/one-kink setting and do not by themselves show that the contribution from the second interface cancels or is negligible on the fixed time horizon [0,T]. If the intended semigroup is e^{tK A_K}, then the near-zero mode does not relax on that time scale because K e^{−c√K} T → 0, so the problem may be repairable; but this two-layer spectral reduction is exactly what must be proved. Without it, the factorization X_t = e(ϑ)√̟B_t (or e(ϑ)Z_t(θ)) is unsupported. Please provide a self-contained proof of the two-layer version of Lemma B.3 or an explicit reference in [14] that covers the periodic two-kink operator.","section":"Appendix B, Lemma B.3 and [14, Proposition B.4]"}],"minor_comments":[{"comment":"The text says the field is divided by N^{d/2}K^{1/4}, but the definition (2.23) uses the factor (N^d K)^{−1/2} = N^{−d/2}K^{−1/2}; the martingale computations in (7.7) confirm the latter. Please correct the introductory sentence.","section":"Section 1, paragraph before (2.23)"},{"comment":"The proof says the time integral converges to 0 as N → ∞, but the limit in the lemma is K ↑ ∞ with N fixed as a lattice parameter. Please correct the variable.","section":"Appendix B, proof of Lemma B.3"},{"comment":"The notation P^K_t is used with two different meanings: the semigroup of K A_K in Section 7 and the semigroup of A_K in Appendix B. Please unify the notation so that the statement used in Lemma 7.6 is unambiguous.","section":"Section 7 and Appendix B"},{"comment":"The paper proves finite-dimensional convergence for X^N_t and tightness only for the time-integrated process ∫_0^t X^N_s ds. This is stated, but it would be helpful to make the scope explicit in the abstract or introduction so that readers do not infer process-level tightness of X^N_t itself.","section":"Theorems 2.4 and 2.5"}],"recommendation":"major_revision","confidential_remarks":"The central spectral input (Proposition B.4 and Lemma B.7 of [14]) comes from a companion preprint that shares a co-author with the present paper. That reliance is legitimate if the companion result is correct and is stated precisely, but it makes verification difficult. I recommend that the editor ask the authors to either prove the needed two-layer spectral lemma in this paper or provide a precise theorem number in [14] with the periodic two-kink hypotheses. If the corrected Lemma B.3 is supplied, the paper appears to be a strong contribution to the field; if the lemma is not corrected, the e(ϑ) factorization of the limit is not established."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper is worth knowing about, and the first half is genuinely good. The entropy production bound for inhomogeneous product measures (Theorem 2.2) and the Boltzmann-Gibbs principle (Theorem 6.1) are new and look carefully done; the two-block replacement arguments with the flows and the inhomogeneity-dependent constants are substantial. The martingale identification in Lemma 7.2 is also clean, and the paper is honest about limitations: tightness only for the time-integrated field, the reference measure nu^N_0 rather than the unknown stationary one, and the growth restriction K_N ≤ delta0 sqrt(log N).\n\nThe trouble is concentrated exactly where the reader suspected: the Sturm-Liouville projection onto e. Lemma B.3 is likely false. The profile rho_K on sqrt(K)T has two transition layers, so the operator A_K has a zero eigenfunction dϑ rho_K with two separated bumps of opposite sign. The normalized eigenfunction g_K is approximately (e_1 - e_2)/sqrt(2), not the single-bump e_K. The antisymmetric mode (relative translation of the two kinks) has eigenvalue O(K e^{-c sqrt(K)}), so on the fixed time interval [0,T] it neither decays nor relaxes; e^{t K A_K} F converges to g_K <F,g_K>, which has an O(1) component at the second interface. Thus the L^2(sqrt(K)T) error in Lemma B.3 does not vanish, and Lemma 7.6's replacement of T^K_{t-s}H by e_K S_{t-s}<H> is not justified. The covariance (2.29) and the factorization X_t = e(vartheta) sqrt(varrho) B_t / e(vartheta) Z_t(theta) rest on that replacement. If the projection is onto the two-bump mode, the first-interface observable picks up independent noise from the second interface, which changes the covariance constant (by a factor of 2, on the back of an envelope) and breaks the claim that the limit is exactly e times a single tangential noise.\n\nThis is a load-bearing flaw, not a cosmetic one. The remedy is not obvious because the second layer is a genuine feature of the standing wave on the torus; one would need to either track both layers in the statement, or prove a spectral gap uniform in K despite the exponentially small antisymmetric eigenvalue. I do not see how the current argument can be patched without changing the result.\n\nShould a serious editor send it to referees? Yes. The entropy and BG machinery is substantial and likely reusable, the authors are clearly in control of the probabilistic parts, and the spectral issue is subtle enough to need expert judgment. But the main theorem as stated is not established. I would not cite the fluctuation result in its current form.\n\nRecommendation: engage with the probability estimates, but referee a corrected version with the two-layer spectrum addressed.","headline":"Genuinely new entropy machinery, but the spectral lemma that produces the e-shape is likely false on the periodic two-kink torus, so the main fluctuation theorem is not established as stated.","tokens_in":55275,"tokens_out":12911,"would_cite":false,"duration_ms":119045,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","82C22","60H15","35K57"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that the space-time mass fluctuation field of Glauber-Kawasaki particle dynamics near a stationary flat interface converges, in dimensions one and two, to a Gaussian field whose normal-direction profile is the normalized…","keywords":["Glauber-Kawasaki dynamics","mass fluctuation field","interface fluctuations","standing wave","Sturm-Liouville semigroup","stochastic heat equation","Boltzmann-Gibbs principle","nonequilibrium fluctuations"],"falsifier":"Take $d=1$, choose a smooth compactly supported $F$ with $\\int_{\\mathbb{R}}F(\\vartheta)e(\\vartheta)\\,d\\vartheta=0$, and simulate the particle system on a large torus with $K_N\\asymp\\delta_0\\sqrt{\\log N}$; the theorem predicts $\\operatorname{Var}_{\\nu^N}(X_t^N(F))\\to0$ for each $t$, and in $d=2$ the covariance must factor as $e(\\vartheta)e(\\vartheta')$ times the heat-kernel covariance of $Z_t(\\theta)$. A simulation that sees a nonzero variance for $F\\perp e$, or a non-factorized two-point correlation, would falsify the rank-one projection at the core of the argument.","tokens_in":53957,"feed_emoji":"📈","tokens_out":13008,"duration_ms":117684,"temperature":0.7,"pith_summary":"The paper establishes the scaling limit of the space-time mass fluctuation field of a Glauber+Kawasaki particle system around a stationary flat interface, in dimensions one and two. It shows that, when the Glauber rate $K_N$ diverges no faster than $\\sqrt{\\log N}$ and the normal coordinate is rescaled by $\\sqrt{K_N}$, the fluctuation field converges to a Gaussian field. The limit is not an arbitrary Gaussian noise: it is the normalized derivative $e$ of the standing wave $\\varphi$ multiplying, in $d=1$, a Brownian motion, and in $d=2$, the solution of a one-dimensional stochastic heat equation with additive white noise. The result matters because it is a rigorous microscopic-to-macroscopic derivation of linear interface fluctuations, showing that the interface keeps the shape of the transition layer.","feed_headline":"Interface fluctuations reduce to one shape times noise","feed_subtitle":"In d=1 a Brownian motion; in d=2 a stochastic heat equation, with the interface profile fixed as a spatial factor.","key_machinery":"The load-bearing object is the Sturm-Liouville operator $A_K=\\partial_\\vartheta^2-V''(\\rho_K)$ acting on $\\sqrt{K}\\,\\mathbb{T}$, for which $\\partial_\\vartheta\\rho_K$ is the ground state. The argument shows that the semigroup $e^{tK A_K}$ converges to the rank-one projection $F\\mapsto \\langle F,e\\rangle e$, where $e$ is the normalized derivative of the standing wave $\\varphi$. This projection is what turns the microscopic stochastic differential of the fluctuation field into a martingale whose only remaining spatial dependence is tangential: in $d=1$ there is no tangential space, so the limiting martingale is a Brownian motion; in $d=2$ the tangential variable carries the stochastic heat equation. The convergence is fed by a relative-entropy bound for the inhomogeneous product measure $\\nu^N$ and a Boltzmann-Gibbs principle that homogenizes the microscopic Glauber rates, with the required spectral-gap and semigroup estimates supplied by the Sturm-Liouville analysis of the companion paper.","core_discovery":"On the paper's own terms, the central discovery is Theorem 2.4: under $\\nu^N$ and the growth conditions (2.17), (2.18), the random vector $(X^N_{t_1}(F_1),\\dots,X^N_{t_p}(F_p))$ converges in distribution to a centered Gaussian vector whose covariance is given by (2.29). Equivalently, for $d=1$ the limit field is $X_t(\\vartheta)=e(\\vartheta)\\sqrt{\\varpi}B_t$ with $B$ a standard Brownian motion, and for $d=2$ it is $X_t(\\vartheta,\\theta)=e(\\vartheta)Z_t(\\theta)$, where $Z$ solves $\\partial_t Z_t=\\Delta_\\theta Z_t+\\sqrt{\\varpi}\\xi$ with $Z_0=0$ and $\\xi$ space-time white noise on $\\mathbb{R}_+\\times\\mathbb{T}$. Here $e=\\partial_\\vartheta\\varphi/\\|\\partial_\\vartheta\\varphi\\|_{L^2(\\mathbb{R})}$ is the normalized derivative of the decreasing standing wave $\\varphi$ of $\\partial_\\vartheta^2\\varphi-V'(\\varphi)=0$, and $\\varpi$ is the positive constant built from the static compressibility and the Glauber rate. The appearance of $e(\\vartheta)$ as a deterministic factor means the fluctuation field inherits the shape of the transition layer $\\varphi$ in the direction normal to the interface.","pith_inferences":["If the $K_N$-growth restriction were pushed to polynomial growth, the same homogenization machinery could plausibly produce nonlinear SPDE limits (with a genuine reaction-diffusion nonlinearity) rather than the linear Gaussian field; the paper notes this possibility but does not carry it out.","The one-mode factorization suggests that on these macroscopic time scales the interface profile is effectively fixed, consistent with the exponential metastability time scale of interface motion; on time scales short compared with that, no interface displacement is seen.","A numerical test of the factorization is feasible: measure the two-point covariance of the fluctuation field in $d=2$ and check that it separates as $e(\\vartheta)e(\\vartheta')$ times a heat-kernel covariance in $\\theta-\\theta'$; the paper proves this only after taking $N\\to\\infty$, so a finite-size check would be an independent extension."],"forward_implications":["In $d=1$, the mass fluctuation at the flat interface is statistically a single Brownian motion whose spatial profile is frozen as $e(\\vartheta)$; no nontrivial SPDE emerges in the normal direction.","In $d=2$, the normal profile is again frozen as $e(\\vartheta)$, while the tangential coordinate evolves by the linear stochastic heat equation $\\partial_t Z_t=\\Delta_\\theta Z_t+\\sqrt{\\varpi}\\xi$.","Fluctuations have largest amplitude at the interface and decay away from it, because $e$ decays exponentially; the stable phases far from the interface do not fluctuate in the limit.","The convergence holds for Glauber rates $K_N$ up to order $\\sqrt{\\log N}$; within this regime the limit is Gaussian, and nonlinear terms in the fluctuation SPDE would require a faster growth of $K_N$."],"supporting_citations":[{"why":"Companion paper supplying the Sturm-Liouville semigroup estimates (Lemma B.7, Proposition B.4) and the formal SPDE expansion that Theorem 2.4 makes rigorous.","marker":"[14]"},{"why":"Metastable patterns analysis for $u_t=\\epsilon^2u_{xx}-f(u)$; source of the standing-wave and ground-state spectral structure used in Appendix B.","marker":"[5]"},{"why":"Establishes motion by mean curvature from Glauber-Kawasaki dynamics, the hydrodynamic limit that the fluctuation theorem refines.","marker":"[15]"},{"why":"Reaction-diffusion hydrodynamic limit (Theorem 2.1), which fixes the deterministic profile $u(t,v)$.","marker":"[7]"},{"why":"Entropy production, Boltzmann-Gibbs principle, and martingale tightness scheme, adapted here to inhomogeneous product measures.","marker":"[18]"},{"why":"Provides the flow and concentration estimates used in the two-block replacement bounds of Theorem 4.1.","marker":"[19]"},{"why":"Martingale problem and quadratic variation tools used to identify covariances of the limit field.","marker":"[24]"}],"fun_headline_variants":["Interface fluctuations reduce to a standing-wave shape times noise","Interface fluctuation field: fixed profile times Brownian or SHE","Shape from standing wave multiplies the Gaussian fluctuation field","Interface fluctuation limit: one deterministic profile times noise"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything hinges on the linearized reaction-diffusion operator near the interface losing all but one effective mode fast enough: the semigroup $e^{tK A_K}$ must converge to the projection onto $e$ at a rate uniform enough for the martingale estimates; if this one-mode collapse is too slow or non-uniform, the Gaussian limit would not have the shape $e$.","fun_headline_variants_meta":{"raw":{"variants":["Interface fluctuations reduce to a standing-wave shape times noise","Interface fluctuation field: fixed profile times Brownian or SHE","Shape from standing wave multiplies the Gaussian fluctuation field","Interface fluctuation limit: one deterministic profile times noise"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000828,"raw_usage":{"total_tokens":3798,"prompt_tokens":1306,"completion_tokens":2492,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":922,"completion_tokens_details":{"reasoning_tokens":2429}},"tokens_in":922,"tokens_out":2492,"duration_ms":20214,"temperature":1.0,"reasoning_tokens":2429,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T21:53:04.709876+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $d=1$, choose a smooth compactly supported $F$ with $\\int_{\\mathbb{R}}F(\\vartheta)e(\\vartheta)\\,d\\vartheta=0$, and simulate the particle system on a large torus with $K_N\\asymp\\delta_0\\sqrt{\\log N}$; the theorem predicts $\\operatorname{Var}_{\\nu^N}(X_t^N(F))\\to0$ for each $t$, and in $d=2$ the covariance must factor as $e(\\vartheta)e(\\vartheta')$ times the heat-kernel covariance of $Z_t(\\theta)$. A simulation that sees a nonzero variance for $F\\perp e$, or a non-factorized two-point correlation, would falsify the rank-one projection at the core of the argument.","supporting_citations":[{"cited_title":"Stochastic PDE approach to fluctuating interfaces","cited_arxiv_id":"2412.00708","evidence_quote":"Companion paper supplying the Sturm-Liouville semigroup estimates (Lemma B.7, Proposition B.4) and the formal SPDE expansion that Theorem 2.4 makes rigorous."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Metastable patterns analysis for $u_t=\\epsilon^2u_{xx}-f(u)$; source of the standing-wave and ground-state spectral structure used in Appendix B."},{"cited_title":"Funaki and K","cited_arxiv_id":null,"evidence_quote":"Establishes motion by mean curvature from Glauber-Kawasaki dynamics, the hydrodynamic limit that the fluctuation theorem refines."},{"cited_title":"De Masi, P","cited_arxiv_id":null,"evidence_quote":"Reaction-diffusion hydrodynamic limit (Theorem 2.1), which fixes the deterministic profile $u(t,v)$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Entropy production, Boltzmann-Gibbs principle, and martingale tightness scheme, adapted here to inhomogeneous product measures."},{"cited_title":"Non-equilibrium fluctuations for a reaction-diffusion model via relative entropy","cited_arxiv_id":"1810.03418","evidence_quote":"Provides the flow and concentration estimates used in the two-block replacement bounds of Theorem 4.1."},{"cited_title":"Kipnis, C","cited_arxiv_id":null,"evidence_quote":"Martingale problem and quadratic variation tools used to identify covariances of the limit field."}],"review_version":1}