{"id":"5f21db46-f140-4f11-a11a-12f2953a06c9","arxiv_id":"2501.16499","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For the 1D Schrödinger map equation with zero Neumann data, non-trivial statistically stationary solutions exist, constructed by vanishing-viscosity limits of invariant measures of the stochastic Landau-Lifshitz-Gilbert equation.","lead":"Researchers prove that the deterministic one-dimensional Schrödinger map equation, a geometric evolution equation for a vector field on the sphere, has statistically stationary random solutions that are not just constant states. They build them as limits of randomly forced Landau-Lifshitz-Gilbert equations, and also obtain stationary solutions for the stochastic version and for the vortex-filament, binormal curvature flow equation.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The uniform ν-bounds and all non-triviality statements rest on the imported identity E‖z^ν×∂²_xz^ν‖²=‖∂_xh‖², but the regularity hypotheses for that identity are never stated; the paper's only LLG well-posedness result (Prop 3.1) assumes h∈W^{2,∞}, while Theorems 1.1 and 1.2 assume only h∈W^{1,∞}.","rationale":"The paper's overall strategy is coherent and the conservation-law argument is genuinely novel; I do not see an internal contradiction that would justify rejection. The most load-bearing condition, however, is the imported identity of Proposition 3.3, because without it there are no uniform H² bounds, no tightness, and no limit Z. The reader's weakest_assumption flagged the same identity, together with the all-time-event Itô step. I focus on the regularity of h because the all-time-event issue in Corollary 3.11 only affects the side statement Corollary 1.4, whereas a failure of the identity under W^{1,∞} would undermine Theorems 1.1 and 1.2 themselves. The lower-bound constants in Section 6 appear consistent after dividing the displayed inequality by ‖∂_xh‖², so I do not treat that as load-bearing. The concrete check I propose settles the matter directly: re-derive the identity at the stated regularity or explicitly restrict/approximate h.","tokens_in":32323,"tokens_out":20581,"duration_ms":184766,"concrete_test":"Independently derive Proposition 3.3's identity for an invariant measure of (1.2) under h∈W^{1,∞}, without invoking [21]. Specifically, write the Itô formula for ‖∂_xz^ν_t‖²_{L²} and check whether the stochastic integral 2∫(∂_xz^ν, √ν∂_x(hz^ν)×dW) can be defined as an H¹-valued martingale when only ∂_xh∈L^∞. If the derivation requires ∂²_xh (equivalently h∈W^{2,∞}) at any point, then Theorem 1.1 needs h∈W^{2,∞} or an explicit mollification of h with uniform bounds in ν; if the derivation closes with h∈W^{1,∞}, the concern is resolved.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central construction is a Kuksin-type limit of stationary LLG solutions z^ν. Every uniform bound used for tightness—Lemmas 3.5, 3.7, 3.8—and the non-triviality arguments of Sections 5 and 6 feed off Proposition 3.3's identity E[‖z^ν×∂²_xz^ν‖²_{L²}] = ‖∂_xh‖²_{L²}. The paper cites this identity to [21] without stating the hypotheses under which it is proved. The only well-posedness result actually stated, Proposition 3.1, requires h∈W^{2,∞}; Proposition 3.2 makes the pattern explicit, requiring h∈W^{k+1,∞} for H^k-valued solutions. The main theorems, however, assume only h∈W^{1,∞}. This is not merely a cosmetic mismatch: for an H¹-valued solution, the martingale term ∫∂_x(hz)×dW is an H¹-valued stochastic integral only if the integrand has one derivative, which in turn involves ∂²_xh. If [21]'s invariant-measure identity indeed requires W^{2,∞}, then the approximating stationary LLG solutions used in Section 4 are not available at the assumed regularity, and Theorem 1.1 must either be restricted to h∈W^{2,∞} or supplemented by an explicit smoothing/approximation argument that preserves ‖∂_xh‖_{L²}. The same regularity issue propagates to (5.1) and (6.7), since these are obtained by passing the corresponding LLG-level identities to the limit. If the identity is in fact valid for h∈W^{1,∞}, the concern is vacuous; the point is that the paper never supplies or cites the required check.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs statistically stationary solutions to the one-dimensional deterministic Schrödinger map equation with null Neumann boundary conditions. The method is Kuksin's fluctuation-dissipation limit: for each viscosity ν one takes a stationary solution z^ν of the stochastic Landau-Lifshitz-Gilbert equation forced by multiplicative noise of amplitude √ν h, proves ν-uniform bounds using the identity E‖z^ν×∂²_xz^ν‖²_{L²}=‖∂_xh‖²_{L²}, and then passes to the limit via Skorokhod-Jakubowski compactness. The limit Z is shown to be a stationary process whose trajectories solve the SME and satisfy several conservation laws; the paper further claims non-triviality of Z in space, time, and randomness when ∂_xh≠0, a quantitative lower bound on the set of non-trivially evolving trajectories, existence of stationary martingale solutions to the stochastic SME, and existence of statistically stationary solutions to the binormal curvature flow obtained by integrating Z in space.","tokens_in":32731,"tokens_out":20805,"duration_ms":192576,"significance":"If the proof can be completed, these are novel results: they appear to provide the first construction of non-trivial statistically stationary solutions for the one-dimensional SME, without recourse to the Hasimoto transform and directly in the real-valued formulation. The paper also contains a number of genuinely useful ideas: the geometric multiplicative noise that preserves S², the use of the space-average conservation law to prove genuine randomness, and the transfer to the binormal curvature flow. The main limitation is that several load-bearing steps, in particular the imported identity (3.2) and the all-time nondegeneracy argument, are not stated with sufficient hypotheses or are not fully justified. The central compactness strategy is coherent and likely repairable, but the claims as written outrun the evidence provided.","major_comments":[{"comment":"The central estimate (3.2) is imported from [21] without stating the regularity assumptions under which it is proved. The only well-posedness results stated in the paper, Proposition 3.1 and Proposition 3.2, require h∈W^{2,∞} and h∈W^{k+1,∞} respectively, whereas Theorems 1.1, 1.2 and 1.5 assume only h∈W^{1,∞}. Since the existence of the stationary LLG solutions, the uniform bounds in Lemmas 3.5, 3.7 and 3.8, and the non-triviality arguments in Sections 5 and 6 all feed on identity (3.2), the paper must either state a precise W^{1,∞} version of Proposition 3.3 with a reference or proof, or restrict the main theorems to h∈W^{2,∞}; alternatively, a smoothing/approximation argument would have to be supplied that preserves the exact value ‖∂_xh‖²_{L²}.","section":"Section 3, Proposition 3.3; Theorems 1.1, 1.2, 1.5"},{"comment":"The proof of Corollary 3.11 is not valid as written. The event Γ is defined using all times t≥0, so the indicator 1_Γ is not adapted to the filtration at time t and cannot be inserted as a fixed factor inside the Itô computation performed 'following Lemma 3.10'. Additionally, the conclusion P(Γ)=0 only implies that almost surely there exists some (possibly random) time at which ‖∂_xz_t‖>0; it does not imply the stated event {ω : ‖∂_xz_t(ω)‖²>0 for every t≥0} has probability one. Corollary 1.4 and the corresponding all-time nondegeneracy statements in the introduction are therefore unsupported and need either a corrected argument or a weakened formulation.","section":"Section 3.2, Corollary 3.11 and Corollary 1.4"},{"comment":"The proof assumes that if t↦Z_t is constant in time, then the identity (5.1) can be treated exactly as in Theorem 1.2(a). However, the argument in Theorem 1.2(a) uses ∂_xZ=0 to conclude Z_t=⟨Z_t⟩; here the hypothesis only gives Z_t×∂²_xZ_t=0. To make the reduction, the paper needs the (true but unproved) fact that every null-Neumann steady state of the SME is constant in space. This classification should be stated and proved explicitly, since it is the bridge from time-constancy of the trajectory to ∂_xh=0.","section":"Section 5.2, proof of Theorem 1.2(b)"}],"minor_comments":[{"comment":"The numerical example h(x)=0.1 cos(x) on D=[0,2π] is inconsistent: for this function one has ⟨h²⟩=0.005, ⟨|h|⟩²≈0.00405 and ‖∂_xh‖²_{L²}=0.005, not the values 1/2, 4/π² and 1/2 used in the displayed inequality. The claimed bound P(Γ^C)>0.2298 must be recomputed or the example changed to h=cos x.","section":"Section 6"},{"comment":"The statement '∂_xX, ∂²_xX≠0 if and only if ∂_xh≠0' is not accurate because ∂_xX=Z is never zero, as Z is S²-valued; the intended assertion is about ∂²_xX (or about ∂_xZ and ∂²_xZ at the level of Z).","section":"Section 8, Theorem 1.6(c)(i)"},{"comment":"The transform f(v)=∫_0^x v(y)dy presumes that 0 belongs to the interval D; for a general bounded interval D one should fix a base point x_0∈D and use f(v)(x)=∫_{x_0}^x v(y)dy.","section":"Section 8, definition of the transform f"},{"comment":"The claim that steady states of the SME with null Neumann boundary conditions coincide with constant maps is used in the proof of Theorem 1.2(b) but not proved; a short proof or a precise citation should be added.","section":"Section 1, Remark 1.3 and Section 5.2"},{"comment":"The passage from the LLG-level identity (3.4) to the corresponding bound for the limit Z needs the same uniform-integrability justification as in Section 5; the text states the bound but does not display the required estimates for E‖∂_xZ_t‖⁴.","section":"Section 6"}],"recommendation":"major_revision","confidential_remarks":"The main novelty depends on an imported identity from [21], which shares an author with this manuscript. This is not improper in itself, but Proposition 3.3 is quoted without hypotheses, making the regularity gap impossible to adjudicate. If the W^{1,∞} version of the invariant-measure identity is available, the central theorem may well be sound; otherwise the assumptions must be strengthened. The all-time nondegeneracy claims and the Section 6 numerical example also need correction. I do not recommend rejection, because the core tightness/compactness strategy appears coherent and the identified gaps are local and likely repairable within the scope of the manuscript."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Clear take: this is the first construction of non-trivial statistically stationary solutions for the 1D Schrödinger map equation, the stochastic SME, and the binormal curvature flow. The approach is a Kuksin-type vanishing-noise limit of the stochastic LLG equation, and the novelty is the adaptation to a sphere-constrained equation with multiplicative Stratonovich noise. The new conservation law (Lemma 3.10) used to prove qualitative non-triviality is a real contribution. The paper is honest about what it cannot do (e.g., the Hashimoto transform to NLS does not preserve stationarity). The citation to [21] is heavy but it is a prior standalone result, not circular. \n\nMain soft spots, in my view:\n\n1. Regularity mismatch: Theorems 1.1, 1.2, and 1.5 assume h∈W^{1,∞}, but the only LLG well-posedness result stated (Prop 3.1) requires h∈W^{2,∞}. The load-bearing identity E‖z^ν×∂²_x z^ν‖²=‖∂_x h‖² is imported from [21] without stating its hypotheses. If that identity indeed needs W^{2,∞}, the approximating stationary solutions used in Section 4 are not justified at the stated regularity. This is not fatal to the overall strategy, but it needs either a restriction to W^{2,∞} or an explicit smoothing argument that preserves ‖∂_x h‖_{L²}.\n\n2. The 1_Γ Itô step in Corollary 3.11: Γ is defined over all t≥0, so 1_Γ is not obviously F_0-measurable, and using it inside an Itô computation needs justification. The result may survive, but as written this is a gap.\n\n3. Minor: the illustrative lower bound in Section 6 claims P(Γ^C)>0.2298 for h=0.1 cos(x), but the displayed computation uses ⟨h²⟩=1/2, which would correspond to α=1, not α=0.1. The constant does not match the example. This is an error in an illustrative calculation, not in the main existence results.\n\nThe central existence theorem (Theorem 1.1 a,b) may survive these issues, but the uniform bounds rely on the identity, so the regularity point is the one to fix. I would send this to serious refereeing. The right reader is a probabilist or PDE analyst working on stationary measures for dispersive or geometric equations. The paper deserves a revision and a knowledgeable referee, not a desk reject.","headline":"Genuinely new Kuksin-type construction for a geometric PDE with a repairable regularity gap and a minor numerical typo; worth serious refereeing.","tokens_in":33293,"tokens_out":5128,"would_cite":true,"duration_ms":44957,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60G10","60H15","60L90","60H30"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every $h\\in W^{1,\\infty}(D;\\mathbb{R})$, there is a stationary stochastic process whose trajectories are strong global solutions of the deterministic 1D Schrödinger map equation; it is genuinely random and non-trivial in space and…","keywords":["Statistical solutions","Schrödinger map equation","Stochastic Schrödinger map equation","Landau-Lifschitz-Gilbert equation","Binormal curvature flow","Vortex filament equation","Stationary stochastic process","Invariant measure"],"falsifier":"Run a structure-preserving discretization of the stochastic LLG with $h(x)=0.1\\cos(x)$ on $D=[0,2\\pi]$, wait until the law is stationary, and estimate both $\\mathbb{E}\\|\\mathrm{z}^{\\nu}\\times\\partial_x^2\\mathrm{z}^{\\nu}\\|_{L^2}^2$ and $\\mathbb{P}(\\|\\partial_x \\mathrm{z}^{\\nu}_t\\|_{L^2}>0\\ \\forall t\\ge 0)$; Corollary 1.4 and Proposition 3.3 predict the probability is $1$ and the identity equals $0.01\\pi\\approx0.0314$, so a reproducible violation of either prediction would settle the central claim.","tokens_in":32043,"feed_emoji":"🌀","tokens_out":11383,"duration_ms":96386,"temperature":0.7,"pith_summary":"This paper asks whether the deterministic one-dimensional Schrödinger map equation—the Landau–Lifschitz equation for a curve moving on the unit sphere—admits statistically stationary solutions beyond trivial constant states. The authors answer yes: for any real function $h$ with one weak derivative and bounded gradient, they build a stationary random process $Z$ whose trajectories solve the deterministic equation, with uniform bounds on the first and second spatial derivatives in terms of $\\|\\partial_x h\\|_{L^2}$. If $\\partial_x h\\neq 0$, the process is genuinely random and its dynamics are non-trivial in both space and time, and a concrete example gives a lower bound of more than $22.98\\%$ on the probability of seeing non-constant trajectories. The same limiting argument produces stationary solutions for the stochastic Schrödinger map equation and, after integrating in space, for the binormal curvature flow (vortex filament equation). Up to the authors' knowledge, this is the first construction of non-trivial statistically stationary solutions for the 1D SME.","feed_headline":"Noise limit yields stationary Schrödinger-map flows","feed_subtitle":"For the 1D Schrödinger map equation, non-trivial statistically stationary solutions exist, and at least 23% of trajectories move.","key_machinery":"The construction follows Kuksin's fluctuation–dissipation scheme. One starts from the stochastic Landau–Lifschitz–Gilbert equation $\\mathrm{u}^\\nu_t = \\mathrm{u}^\\nu_0 + \\int_0^t (\\mathrm{u}^\\nu\\times\\partial_x^2\\mathrm{u}^\\nu - \\nu\\,\\mathrm{u}^\\nu\\times(\\mathrm{u}^\\nu\\times\\partial_x^2\\mathrm{u}^\\nu))\\,dr + \\sqrt{\\nu}\\int_0^t h\\,\\mathrm{u}^\\nu\\times\\circ dW_r$, which preserves the sphere and has an invariant measure for each $\\nu\\in(0,1]$. The load-bearing identity, imported from earlier work, is that every stationary solution $\\mathrm{z}^\\nu$ satisfies $\\mathbb{E}\\|\\mathrm{z}^\\nu\\times\\partial_x^2\\mathrm{z}^\\nu\\|_{L^2}^2=\\|\\partial_x h\\|_{L^2}^2$; combined with the geometric identity $\\|\\partial_x^2\\mathrm{z}\\|_{L^2}^2=\\|\\mathrm{z}\\times\\partial_x^2\\mathrm{z}\\|_{L^2}^2+\\|\\partial_x\\mathrm{z}\\|_{L^4}^4$, this yields uniform-in-$\\nu$ bounds in $H^1\\cap H^2$ and Hölder time regularity. Tightness in $L^2_{\\mathrm{loc}}([0,\\infty);H^1)\\cap C_w([0,\\infty);H^1)$ and the Skorokhod–Jakubowski theorem produce a limit $Z$ on a new probability space; the $\\nu$-scaled noise and dissipative terms vanish in the limit, leaving the deterministic SME. The extra conservation law $\\langle Z_t\\rangle=\\langle Z_0\\rangle$, which is not conserved by the stochastic LLG, is what lets the authors prove genuine randomness and non-trivial space–time dynamics, with Lemma 3.10 supplying the limiting identity used to rule out spatially trivial solutions.","core_discovery":"The central discovery is Theorem 1.1: for every $h\\in W^{1,\\infty}(D;\\mathbb{R})$, there exists a probability space and a stationary stochastic process $Z$ such that $\\hat{\\mathbb{P}}$-a.s. each trajectory $Z(\\hat\\omega)$ is a strong global solution to the deterministic SME $z_t=z_0+\\int_0^t z_r\\times\\partial_x^2 z_r\\,dr$ with $|z_t|=1$ and null Neumann boundary conditions, lying in $L^\\infty([0,\\infty);H^1)\\cap L^2_{\\mathrm{loc}}([0,\\infty);H^2)\\cap C([0,\\infty);H^1)$. The stationary law satisfies $\\hat{\\mathbb{E}}\\|Z_t\\times\\partial_x^2Z_t\\|_{L^2}^2\\le \\|\\partial_x h\\|_{L^2}^2$ and $\\hat{\\mathbb{E}}\\|\\partial_x^2Z_t\\|_{L^2}^2+\\hat{\\mathbb{E}}\\|\\partial_x Z_t\\|_{L^4}^4+\\hat{\\mathbb{E}}\\|\\partial_x Z_t\\|_{L^2}^p\\lesssim \\|\\partial_x h\\|_{L^2}^2+1$, and the quantities $\\|\\partial_x Z_t\\|_{L^2}^2$ and $\\|Z_t-Q\\|_{L^2}^2$ are conserved along every trajectory. Theorem 1.2 sharpens this: for $h\\neq 0$, $\\partial_x Z$ and $\\partial_x^2 Z$ are non-zero on a set of positive probability exactly when $\\partial_x h\\neq 0$; then $t\\mapsto Z_t$ is not constant, and $Z$ cannot be written as a countable mixture of deterministic solutions or as a spatially piecewise-constant random field. The paper presents this as the first existence result for non-trivial statistically stationary solutions to the 1D SME.","pith_inferences":["Editorial extension: the identity $\\mathbb{E}\\|\\mathrm{z}^{\\nu}\\times\\partial_x^2\\mathrm{z}^{\\nu}\\|_{L^2}^2=\\|\\partial_x h\\|_{L^2}^2$ suggests a general principle—the spatial gradient of the noise coefficient fixes the mean curvature energy of any stationary state—that one could test on other sphere-valued geometric flows where the same algebraic identity may fail.","Editorial extension: the spatial-mean conservation law $\\langle Z_t\\rangle=\\langle Z_0\\rangle$, used here to prove genuine randomness, may be the right diagnostic for stationarity in other geometric evolution equations whose stochastic approximations share the pointwise sphere constraint.","Editorial extension: the paper leaves the Hashimoto transform and the cubic NLS link unresolved; a promising follow-up is to design the stochastic approximation directly in curvature–torsion variables, so that the time shift needed for the non-local equation is built into the noise and stationarity is preserved."],"forward_implications":["For every $h\\in W^{1,\\infty}(D;\\mathbb{R})$, the deterministic SME admits a stationary stochastic process of strong global solutions whose law is time-invariant, with the explicit second-derivative bounds of Theorem 1.1(b).","If $\\partial_x h\\neq 0$, the stationary solution is genuinely random: it is not constant in $\\hat\\omega$, it is non-constant in space on a set of positive probability, and it moves non-trivially in time on a set of positive probability, as stated in Theorem 1.2.","A concrete choice $h(x)=0.1\\cos(x)$ on $D=[0,2\\pi]$ yields the quantitative statement that more than $22.98\\%$ of the trajectories exhibit non-trivial space–time dynamics, from the lower bound in Section 6.","The stochastic SME admits stationary martingale solutions, and besides the space-independent stationary spherical Brownian motion, there are stationary solutions with non-zero gradient, as stated in Theorem 1.5.","Integration in space maps each trajectory to a solution of the binormal curvature flow, so the existence transfers: the vortex filament equation also has non-trivial statistically stationary solutions, as stated in Theorem 1.6."],"supporting_citations":[{"why":"Supplies the invariant measures for the stochastic LLG and the identity $\\mathbb{E}\\|\\mathrm{z}^{\\nu}\\times\\partial_x^2\\mathrm{z}^{\\nu}\\|_{L^2}^2=\\|\\partial_x h\\|_{L^2}^2$ (Proposition 3.3) that drives every uniform bound and the non-triviality criterion.","marker":"[21]"},{"why":"Supplies the limiting procedure—stationary solutions of a stochastic approximation converging to stationary solutions of the deterministic equation—that the paper adapts to the geometric setting.","marker":"[28]"},{"why":"Establishes existence and uniqueness of strong pathwise solutions to the stochastic LLG in one dimension, the solution framework on which the whole construction rests.","marker":"[13]"},{"why":"Provides the geometric correspondence between the Schrödinger map equation and the binormal curvature flow, used to transfer stationarity to the vortex filament equation.","marker":"[27]"},{"why":"Supplies the compactness and stationarity tools (Skorokhod–Jakubowski theorem and a criterion for stationarity of limits) used in the limiting procedure.","marker":"[12]"}],"fun_headline_variants":["Stationary random solutions for 1D Schrödinger map","Random initial data yields steady Schrödinger-map flows","Non-trivial statistical steady states for Schrödinger maps","Existence of random stationary states for 1D SME","Schrödinger maps admit random steady states"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction depends on an imported balance law for the stochastic Landau–Lifschitz–Gilbert equation—every stationary solution satisfies $\\mathbb{E}\\|\\mathrm{z}^{\\nu}\\times\\partial_x^2\\mathrm{z}^{\\nu}\\|_{L^2}^2=\\|\\partial_x h\\|_{L^2}^2$—and on being allowed to read stationary solutions on the whole time line and to condition on the event that they stay flat in space; if either step requires smoother functions than $h\\in W^{1,\\infty}$, the non-triviality theorems weaken, although a stationary limit may still exist.","fun_headline_variants_meta":{"raw":{"variants":["Stationary random solutions for 1D Schrödinger map","Random initial data yields steady Schrödinger-map flows","Non-trivial statistical steady states for Schrödinger maps","Existence of random stationary states for 1D SME","Schrödinger maps admit random steady states"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000708,"raw_usage":{"total_tokens":3295,"prompt_tokens":1156,"completion_tokens":2139,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":772,"completion_tokens_details":{"reasoning_tokens":2074}},"tokens_in":772,"tokens_out":2139,"duration_ms":15318,"temperature":1.0,"reasoning_tokens":2074,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T12:50:40.218544+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run a structure-preserving discretization of the stochastic LLG with $h(x)=0.1\\cos(x)$ on $D=[0,2\\pi]$, wait until the law is stationary, and estimate both $\\mathbb{E}\\|\\mathrm{z}^{\\nu}\\times\\partial_x^2\\mathrm{z}^{\\nu}\\|_{L^2}^2$ and $\\mathbb{P}(\\|\\partial_x \\mathrm{z}^{\\nu}_t\\|_{L^2}>0\\ \\forall t\\ge 0)$; Corollary 1.4 and Proposition 3.3 predict the probability is $1$ and the identity equals $0.01\\pi\\approx0.0314$, so a reproducible violation of either prediction would settle the central claim.","supporting_citations":[{"cited_title":"Gussetti","cited_arxiv_id":null,"evidence_quote":"Supplies the invariant measures for the stochastic LLG and the identity $\\mathbb{E}\\|\\mathrm{z}^{\\nu}\\times\\partial_x^2\\mathrm{z}^{\\nu}\\|_{L^2}^2=\\|\\partial_x h\\|_{L^2}^2$ (Proposition 3.3) that drives every uniform bound and the non-triviality criterion."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the limiting procedure—stationary solutions of a stochastic approximation converging to stationary solutions of the deterministic equation—that the paper adapts to the geometric setting."},{"cited_title":"Brze´ zniak, B","cited_arxiv_id":null,"evidence_quote":"Establishes existence and uniqueness of strong pathwise solutions to the stochastic LLG in one dimension, the solution framework on which the whole construction rests."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the geometric correspondence between the Schrödinger map equation and the binormal curvature flow, used to transfer stationarity to the vortex filament equation."},{"cited_title":"Breit, E","cited_arxiv_id":null,"evidence_quote":"Supplies the compactness and stationarity tools (Skorokhod–Jakubowski theorem and a criterion for stationarity of limits) used in the limiting procedure."}],"review_version":1}