{"id":"cd29d5b1-969b-486a-ac2e-ee1e50032074","arxiv_id":"2502.04587","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The Wiener chaos order of the solution to the stochastic Schrödinger and heat equations is asymptotically Gaussian around a mean proportional to time, so noise sensitivity sets in at perturbation scale 1/t.","lead":"Random noise disturbs the solution of the Schrödinger and heat equations in layers, and this paper finds how many layers are active at time t: about a constant times t, with Gaussian fluctuations. It also shows that the solution becomes sensitive to perturbing the noise once the perturbation acts on a time scale of order 1/t.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 3's identity (3.3) for spacetime white noise is asserted via an unshown limiting argument; it is the single step on which the SHE half of Theorem 1.1 rests, though a direct chaos-expansion proof should close it.","rationale":"The paper's central claim is Theorem 1.1: a CLT for the Fourier spectrum N_t. In the Schrödinger case the argument is a direct saddle-point computation with R∈S; the strict definiteness of ∇²R(0) flagged by the reader is automatic for a nondegenerate Schwartz covariance and in any case only affects the normalizing constant, not the ratio. The genuinely load-bearing step is in Section 3: (3.3) is the sole mechanism by which the OU semigroup is converted into a change of temperature β→βe^{-s}. Without it, (3.5) does not follow and the SHE half of Theorem 1.1 has no proof. The written text defers the white-noise limit to a single sentence. This is a real gap in exposition, but not a sign that the theorem is false: the chaos expansion of the mild solution gives the nth chaos coefficient exactly as β^n times a β-independent kernel, so the identity is analytically exact. A revision should insert this argument. Since the reader's CONDITIONAL verdict already asks for justification of this passage, my read does not move the verdict.","tokens_in":6770,"tokens_out":24253,"duration_ms":248774,"concrete_test":"Verify (3.3) directly from the Wiener chaos expansion of the mild SHE. Iterate Z_β = 1 + β∫ p_{t-s}(·-y) Z_β(s,y) ξ(ds,dy) to obtain Z_β = Σ_{n≥0} β^n I_n(k_n) with kernels k_n independent of β; since P_s I_n = e^{-ns} I_n, conclude P_s Z_β = Σ (βe^{-s})^n I_n(k_n) = Z_{βe^{-s}}. If this holds, the approximation passage in Section 3 is not a correctness risk and the paper only needs to add the argument; if the kernel is found to depend on β through the Wick renormalization, then (3.5) and the stated CLT for N_t would need revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The SHE part of Theorem 1.1 depends on the exact identity P_s Z_β(t,0) = Z_{βe^{-s}}(t,0) (Eq. 3.3). This is the only bridge between the OU semigroup and the explicit second-moment formula f(β,t), and hence the only route to the Laplace transform (3.5) and the CLT. The paper verifies (3.3) for spatially smooth noise by Feynman-Kac and then states 'An approximation leads to the same conclusion' without presenting the limit. This matters because for white noise the Feynman-Kac formula involves the divergent constant R(0)t, so the limiting argument must control how the Wick renormalization and the Gaussian average over the independent copy of the noise interact; it is not a purely cosmetic step. A direct proof is available: the mild equation gives the nth chaos of Z_β as β^n I_n(k_n) with k_n independent of β, and P_s multiplies the nth chaos by e^{-ns}, so (3.3) is exact. Thus the concern is about a missing justification in the written proof, not about the truth of the theorem.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the Fourier spectrum N_t of the spatial Fourier mode at the origin of two linear SPDEs with multiplicative Gaussian noise: the stochastic Schrödinger equation in Stratonovich form and the 1+1-dimensional stochastic heat equation. The Fourier spectrum is defined as the probability distribution on Wiener chaos orders determined by the normalized second-moment contributions. Theorem 1.1 claims a central limit theorem: (N_t - μt)/(σ√t) converges in law to a standard normal, with explicit constants in the proofs (μ=σ²=R(0) for the Schrödinger equation; μ=β⁴/2, σ²=β⁴ for the stochastic heat equation). The proof for the Schrödinger case uses exact chaos coefficients and Laplace's method after a change of variables, while the heat-equation case combines the Chen–Dalang second-moment formula with an Ornstein–Uhlenbeck semigroup identity. The paper derives as a consequence that the onset of chaos occurs at perturbation strength s∼1/t.","tokens_in":6968,"tokens_out":14834,"duration_ms":141758,"significance":"The result gives a sharp, quantitative description of noise-induced chaos for two canonical SPDEs: the second moment of the solution is asymptotically dominated by Wiener chaos of order proportional to t, and the Fourier spectrum is approximately Poisson with mean of order t. The proofs are explicit and use no fitted parameters: the Schrödinger computation is a direct saddle-point calculation from exact chaos coefficients, and the heat-equation part reduces to known exact second-moment asymptotics. The derived 1/t decorrelation threshold is a clear, falsifiable prediction. The main shortcoming is a missing justification for the white-noise identity (3.3), but the statement is true and a short chaos-expansion proof would close the gap; after that revision the paper would be a clean and self-contained note.","major_comments":[{"comment":"The identity P_s Z_β(t,0)=Z_{βe^{-s}}(t,0) for spacetime white noise is asserted after a smooth-noise Feynman–Kac verification, with the sentence 'An approximation leads to the same conclusion' and no limiting argument. This identity is the only bridge between the Ornstein–Uhlenbeck semigroup and the explicit second-moment formula f(β,t), so the exact Laplace transform (3.5) and the SHE half of Theorem 1.1 rest on it. The omitted limit is nontrivial because for white noise the Feynman–Kac expression involves the divergent constant R(0)t and one must track how the Wick renormalization interacts with the Gaussian average over the independent noise copy. The claim is nevertheless true: for the mild solution, the n-th Wiener chaos of Z_β(t,0) is β^n times a β-independent chaos coefficient, and the OU semigroup P_s multiplies that chaos by e^{-ns}. I recommend that the authors add this direct chaos-expansion verification or a complete approximation argument before acceptance.","section":"Section 3, Eq. (3.3)"}],"minor_comments":[{"comment":"The theorem states only that 'there exist σ, μ>0', but the proofs determine them explicitly (μ=σ²=R(0) for the Schrödinger case; μ=β⁴/2, σ²=β⁴ for the SHE case). Stating these values in the theorem would make the result more precise and would make the correlation corollary in §1.3 immediate.","section":"Theorem 1.1"},{"comment":"The sentence 'which we assume is strictly negative definite' introduces a hypothesis on ∇²R(0) that is not listed in Theorem 1.1 or in the case-1 assumptions. Under the stated Schwartz-class and positive-definiteness assumptions, this strictness is in fact automatic for a nondegenerate covariance because the continuous nonnegative spectral density cannot be supported on a hyperplane; however, the paper should either justify this or move the condition into the theorem statement to avoid the appearance of an unstated assumption.","section":"Section 2, near Eq. (2.5)"},{"comment":"The phrase 'where F(⋅) is arbitrary square integrable function' should read 'an arbitrary square-integrable function'.","section":"Section 3, Eq. (3.2)"},{"comment":"The display 'E exp{iθ Nt−R(0)t√t }' contains a parenthesis typo and should be written as 'E exp{iθ (N_t−R(0)t)/√t}'.","section":"Section 2, characteristic-function display"},{"comment":"The interchange of the Brownian expectation and the Gaussian expectation over the independent copy of the noise is not explicitly justified; a brief appeal to stochastic Fubini would make the smooth-noise argument more transparent.","section":"Section 3, proof of (3.3) for smooth noise"}],"recommendation":"major_revision","confidential_remarks":"This is a short, clearly written note with a single technically load-bearing gap: the white-noise identity (3.3) is asserted without proof. The gap is easily repairable via the direct chaos-expansion argument, and the result itself appears correct. I recommend major revision rather than rejection; after the authors supply the missing argument, the paper should be suitable for publication in a probability journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a short, honest calculation paper that proves a CLT for the Wiener chaos order (the 'Fourier spectrum') of the stochastic heat and Schrödinger solutions, and from that reads off the 1/t noise-sensitivity threshold. The result is not wrong; the main soft spot is a missing limiting argument in the heat equation section.\n\nWhat is actually new: the CLT for the distribution of N_t, the chaos order under the second-moment measure, is not in the cited literature. The paper correctly identifies that the asymptotic Poisson/Gaussian behavior of N_t is what drives noise sensitivity, and it computes the characteristic function explicitly. The Schrödinger side is clean: the generating function is exact, and Laplace's method plus the strict negative definiteness of the Hessian of R at 0 gives the CLT. The SHE side is also essentially correct, combining the known Chen–Dalang second moment with the Ornstein–Uhlenbeck scaling identity.\n\nThe soft spots are real but not fatal. Equation (3.3), the identity P_s Z_beta(t,0) = Z_{beta e^{-s}}(t,0) for spacetime white noise, is verified for smooth noise and then asserted to follow by approximation. That is the single bridge to the Laplace transform and the CLT, and the limiting argument is not cosmetic: for white noise the Feynman–Kac expression contains a divergent constant, and one has to control how the Wick renormalization interacts with the OU average. As the stress-test note says, a direct chaos expansion proof closes the gap easily (the nth chaos of Z_beta carries beta^n and the OU semigroup multiplies by e^{-ns}), so the theorem is true but the written proof leaves a step to the reader. The second issue is smaller: the strict negative definiteness of the Hessian is assumed mid-proof in Section 2 but not stated in Theorem 1.1. Both are easy fixes for a revision.\n\nThe citation pattern is fine. The self-citation [5] supplies the Fourier-domain equation, which is appropriate. There are no fitted parameters and no circularity. The paper does not overclaim; it explicitly frames itself as a benchmark for nonlinear problems.\n\nWho this is for: anyone working on noise sensitivity for SPDEs, Wiener chaos decompositions, or exactly solvable models. It is a modest but genuinely useful benchmark, not a breakthrough. I would bring it to a reading group and I would cite it if I worked in this area.\n\nRecommendation: send to peer review. A serious referee will ask for the missing limiting argument in (3.3) and for the Hessian condition to be stated, but the core argument holds up and the result deserves to be in the literature.","headline":"A correct and useful short paper: the chaos-order CLT for two linear SPDEs, with one asserted approximation that needs a referee's attention.","tokens_in":7502,"tokens_out":1890,"would_cite":true,"duration_ms":20095,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H15","60F05","60H07","35R60"],"pacs":[],"model":"deepseek-v4-flash","headline":"For the stochastic heat and Schrödinger equations, the Wiener chaos spectrum of the solution is asymptotically Gaussian, placing the onset of noise chaos at perturbation scale $1/t$.","keywords":["chaos","noise sensitivity","Fourier spectrum","stochastic heat equation","stochastic Schrödinger equation","Wiener chaos","central limit theorem"],"falsifier":"Approximate spacetime white noise by smooth mollified noise and compute the ratio $E[Z_\\beta(t,0) P_s Z_\\beta(t,0)]/E[Z_\\beta(t,0)^2]$ as the mollification width goes to zero; if the limit is not $f(\\beta e^{-s/2},t)/f(\\beta,t)$, where $f(\\beta,t)=E[Z_\\beta(t,0)^2]$, the load-bearing identity fails and the central limit theorem for $N_t$ does not follow.","tokens_in":6548,"feed_emoji":"🎲","tokens_out":13075,"duration_ms":112879,"temperature":0.7,"pith_summary":"This paper shows that for two canonical equations with multiplicative Gaussian noise—the Itô–Schrödinger equation with smooth spatial covariance and the one-dimensional stochastic heat equation driven by spacetime white noise—the distribution of the Wiener chaos order contributing to the solution's second moment becomes Gaussian after centering and rescaling by $\\sqrt{t}$. Concretely, the Fourier spectrum $N_t$ satisfies $(N_t - \\mu t)/(\\sigma \\sqrt{t}) \\Rightarrow N(0,1)$, with $\\mu = \\sigma^2 = R(0)$ in the Schrödinger case and $\\mu = \\beta^4/2$, $\\sigma^2 = \\beta^4$ in the heat case with inverse temperature $\\beta$. Because $N_t$ behaves like a Poisson random variable of order $t$, the correlation between the solution and the solution driven by a perturbed noise of strength $s$ decays like $e^{-sN_t}$, so perturbations far larger than $1/t$ destroy correlation while those far smaller leave it intact. The paper's contribution is to make the onset of chaos quantitative in two models where every coefficient of the Wiener chaos expansion is explicit.","feed_headline":"Noise sensitivity threshold for two stochastic PDEs is exactly 1/t","feed_subtitle":"Fourier spectra turn Gaussian, so a perturbation of size 1/t marks the onset of total decorrelation.","key_machinery":"The central object is the Fourier spectrum $N_t$, the probability distribution on non-negative integers that assigns to each $n$ the fraction of the solution's second moment carried by the $n$-th Wiener chaos. For the Schrödinger case the load-bearing computation is the exact coefficient $c^2_{t,n} = (t^n/n!) e^{-R(0)t} \\int R(y)^n \\Phi_0(y)\\,dy$, which makes the characteristic function of $N_t$ a ratio of integrals that can be evaluated by Laplace asymptotics. For the stochastic heat equation the load-bearing mechanism is the identity $P_s Z_\\beta(t,0) = Z_{\\beta e^{-s}}(t,0)$ for the Ornstein–Uhlenbeck semigroup—the semigroup that damps the $n$-th chaos by $e^{-ns}$—which converts the generating function $E e^{-sN_t}$ into a ratio of explicit second moments. The normalization comes from the exact formula $E[Z_\\beta(t,0)^2] = 2e^{\\beta^4 t/4} \\int_{-\\infty}^{\\beta^2 \\sqrt{t/2}} (1/\\sqrt{2\\pi}) e^{-y^2/2}\\,dy$.","core_discovery":"The paper establishes a central limit theorem for the Fourier spectrum $N_t$, defined by $P(N_t=n)=c^2_{t,n}/E X_t^2$, where $c^2_{t,n}$ is the contribution of the $n$-th Wiener chaos to the second moment of the solution. For $X_t = \\hat{\\phi}(t,0)$ of the Itô–Schrödinger equation, the exact formula $c^2_{t,n} = (t^n/n!) e^{-R(0)t} \\int R(y)^n \\Phi_0(y)\\,dy$ turns the characteristic function of $N_t$ into a ratio of exponential integrals, and a Taylor expansion around $y=0$ yields $(N_t - R(0)t)/\\sqrt{R(0)t} \\Rightarrow N(0,1)$. For $X_t = Z(t,0)$ of the stochastic heat equation, the paper proves the Ornstein–Uhlenbeck semigroup identity $P_s Z_\\beta(t,0) = Z_{\\beta e^{-s}}(t,0)$, giving the exact Laplace transform $E e^{-sN_t} = E Z_{\\beta e^{-s/2}}(t,0)^2 / E Z_\\beta(t,0)^2$; analytic continuation of this formula gives $(N_t - \\beta^4 t/2)/(\\beta^2 \\sqrt{t}) \\Rightarrow N(0,1)$. In both cases $N_t$ is asymptotically equivalent to a Poisson variable with intensity of order $t$, so the solution's second moment is dominated by Wiener chaos of order $t$.","pith_inferences":["The identity $P_s Z_\\beta = Z_{\\beta e^{-s}}$ is likely to hold for other Gaussian noises and initial data, in which case the same Poisson-type CLT should hold in any strong-disorder regime with exponentially growing second moment; the paper conjectures similar behavior but does not prove it.","The geometric-Brownian analogy suggests viewing $N_t$ as a count of branching or collision events in a hidden particle picture, a viewpoint that could link chaos spectra of SPDEs with overlap statistics in last-passage percolation.","One could test the paper's closing conjecture that the free energy $\\log Z(t,0)$ has Fourier spectrum of order $t^{1/3}$ by numerically estimating its low-order chaos coefficients, providing evidence for or against a $t^{1/3}$ chaos time scale."],"forward_implications":["If the noise is perturbed with strength $s\\sim t^{-\\alpha}$, the correlation $\\mathrm{Cor}[X_t(0),X_t(s)]$ tends to $1$ for $\\alpha>1$ and to $0$ for $\\alpha<1$; the onset of chaos is exactly at $s\\sim t^{-1}$.","The dominant Wiener chaos order grows linearly in $t$, so the second moment of the solution is concentrated on chaos of order comparable to $t$, not on low-order chaos.","Since $N_t$ becomes a Poisson variable of intensity $R(0)t$ or $\\beta^4 t/2$, the Gaussian limit is the usual Poisson-to-normal transition for a large intensity parameter.","In the Schrödinger picture, $N_t$ can be read as the number of scatterings of an underlying compound Poisson process, tying the chaos spectrum to the kinetic equation for the second moment."],"supporting_citations":[{"why":"Supplies the exact second moment formula $E[Z_\\beta(t,0)^2]$ used as the normalization in the stochastic-heat Laplace transform.","marker":"[2]"},{"why":"Gives the Fourier-domain Itô formulation of the random Schrödinger equation from which the explicit chaos coefficients are derived.","marker":"[5]"},{"why":"Provides the Wiener chaos expansion and the Ornstein–Uhlenbeck semigroup action $P_s I_n = e^{-ns} I_n$ on which both proofs rest.","marker":"[6]"}],"fun_headline_variants":["1/t: exact chaos threshold for stochastic heat and Schrodinger","Stochastic PDEs: at 1/t, Fourier spectra become Gaussian","1/t noise triggers Gaussian chaos in stochastic heat and Schrodinger"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The stochastic heat equation proof assumes, with the limiting argument left out, that applying the noise-smoothing semigroup to the solution with inverse temperature $\\beta$ gives exactly the solution with inverse temperature $\\beta e^{-s}$ even for spacetime white noise.","fun_headline_variants_meta":{"raw":{"variants":["1/t: exact chaos threshold for stochastic heat and Schrodinger","Stochastic PDEs: at 1/t, Fourier spectra become Gaussian","1/t noise triggers Gaussian chaos in stochastic heat and Schrodinger"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000925,"raw_usage":{"total_tokens":3948,"prompt_tokens":914,"completion_tokens":3034,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":530,"completion_tokens_details":{"reasoning_tokens":2976}},"tokens_in":530,"tokens_out":3034,"duration_ms":24371,"temperature":1.0,"reasoning_tokens":2976,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T22:13:52.437615+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Approximate spacetime white noise by smooth mollified noise and compute the ratio $E[Z_\\beta(t,0) P_s Z_\\beta(t,0)]/E[Z_\\beta(t,0)^2]$ as the mollification width goes to zero; if the limit is not $f(\\beta e^{-s/2},t)/f(\\beta,t)$, where $f(\\beta,t)=E[Z_\\beta(t,0)^2]$, the load-bearing identity fails and the central limit theorem for $N_t$ does not follow.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the exact second moment formula $E[Z_\\beta(t,0)^2]$ used as the normalization in the stochastic-heat Laplace transform."},{"cited_title":"Gaussian fluctuations from random Schr¨ odinger equation, Communications in Partial Differential Equations 46.2 (2021): 201-232","cited_arxiv_id":null,"evidence_quote":"Gives the Fourier-domain Itô formulation of the random Schrödinger equation from which the explicit chaos coefficients are derived."},{"cited_title":"Malliavin calculus and related topics , (2006)","cited_arxiv_id":null,"evidence_quote":"Provides the Wiener chaos expansion and the Ornstein–Uhlenbeck semigroup action $P_s I_n = e^{-ns} I_n$ on which both proofs rest."}],"review_version":1}