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  "title": "When Gaussian kernel spaces are invariant under stochastic dynamics: transition rigidity, a Gaussian-component noise boundary and a discontinuous diffusion limit",
  "shortTitle": "The factor is 4, not 2: when Gaussian kernel spaces survive random dynamics",
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  "oneLine": "A Gaussian kernel space is invariant under an Ornstein–Uhlenbeck process exactly when AP + PAᵀ ⪯ 4BBᵀ, twice the range of the earlier sufficient condition, and among Gaussian or fixed-noise transitions only affine maps can preserve it at all.",
  "abstract": "Kernel methods such as kernel EDMD learn how a random system evolves by working inside the reproducing kernel Hilbert space (RKHS) of a Gaussian kernel, and their error analysis often assumes the dynamics keep that space invariant. Philipp et al. (2025, PAMM) proved invariance for linear stochastic differential equations (Ornstein–Uhlenbeck processes) when AP + PAᵀ ⪯ 2BBᵀ and left open whether this is necessary. This candidate shows that the exact condition is AP + PAᵀ ⪯ 4BBᵀ; a stable controllable shear separates the two. More generally, a Markov transition with Gaussian conditional laws N(F(x), Q(x)) that maps one global Gaussian RKHS boundedly into another must have constant Q, affine F and MRMᵀ ⪯ P + 4Q; a transition of the form map plus fixed noise, for noise whose characteristic function has a Gaussian lower bound (every infinitely divisible law, pure jumps included), must have an affine map, so Euler steps of nonlinear SDEs never preserve the space. For affine maps with infinitely divisible noise only the Gaussian covariance component counts, and a compound-Poisson approximation of a stable linear diffusion converges at sharp order ε² in prediction while every approximant sends one fixed Gaussian observable out of the space. Consequences include an exact bandwidth rule (twice the earlier range) and the invariance hypothesis of an existing kEDMD error bound for multivariate OU processes. The fixed affine criterion specialises classical Gaussian-Fock theory, and the separating witness with a proposed derivation was posted first, explicitly unproven, in a GitHub issue. AI-assisted, unrefereed candidate; no external peer review, independent reproduction or formal verification is claimed.",
  "datePublished": "2026-11-04",
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    "detail": "Unrefereed AI-assisted analysis candidate. All theorems have written proofs. The fixed affine criterion and the Ornstein–Uhlenbeck answer are a specialisation of classical Gaussian-Fock operator theory; the separating witness and a proposed derivation appeared first, explicitly unproven, in GitHub issue the-omega-institute/trureturing #23376 (28 October 2026). Internal AI-assisted review in five roles, a separate-model adversarial review and audit, a clean replay, one revision round and a re-review. External specialist review, independent reproduction and formal verification have not been established."
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  "keywords": [
    "Koopman operator",
    "reproducing kernel Hilbert space",
    "Gaussian kernel",
    "Ornstein–Uhlenbeck process",
    "Lévy noise",
    "kernel EDMD",
    "bandwidth selection",
    "Gaussian Fock space"
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    "Gaussian transition rigidity (Theorem 5, proof): a transition with Gaussian conditional laws N(F(x), Q(x)), arbitrary Borel fields, maps one global Gaussian RKHS boundedly into another only if Q is constant, F is affine with full-column-rank matrix M, and MRMᵀ ⪯ P + 4Q.",
    "Additive-noise rigidity (Theorem 9, proof): with fixed noise whose characteristic function has a Gaussian lower bound (every infinitely divisible law, pure jumps included) the drift must be affine; for affine maps only the Gaussian covariance component enters the boundary (Theorems 8, 11).",
    "Ornstein–Uhlenbeck answer (Theorem 6, proof): all-time invariance holds exactly when AP + PAᵀ ⪯ 4BBᵀ; the shear A = [[−1, 5], [0, −1]], B = P = I satisfies it but not the factor-2 condition of Philipp et al. (2025). This specialises classical Gaussian-Fock theory; the witness was proposed first in a GitHub issue.",
    "Discontinuous diffusion limit (Theorem 12, Proposition 13, proof and quadrature): compound-Poisson approximants converge at sharp order ε² from the RKHS to bounded continuous functions, yet at t = 1/10 each sends exp(−8|x|²/5) outside the space.",
    "Bandwidth rule (Proposition 17, Corollary 18): isotropic widths σ² ≤ 4/λmax are admissible, twice the earlier range (shear: 4/3, not 2/3), so a published kEDMD error bound applies to multivariate OU processes."
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  "relatedWorks": [
    {
      "citation": "Alekseev, Afonin and Bogachev (2021). On Gaussian Transition Operators. Doklady Mathematics. Gaussian transition operators.",
      "url": "https://doi.org/10.1134/s1064562421050021",
      "doi": "10.1134/s1064562421050021"
    },
    {
      "citation": "Aronszajn (1950). Theory of reproducing kernels. Transactions of the AMS. Reproducing kernel theory.",
      "url": "https://doi.org/10.1090/s0002-9947-1950-0051437-7",
      "doi": "10.1090/s0002-9947-1950-0051437-7"
    },
    {
      "citation": "De Palma, Mari, Giovannetti and Holevo (2015). Normal form decomposition for Gaussian-to-Gaussian superoperators. Journal of Mathematical Physics. Gaussian-to-Gaussian superoperators.",
      "url": "https://doi.org/10.1063/1.4921265",
      "doi": "10.1063/1.4921265"
    },
    {
      "citation": "Fornaro, Metafune, Pallara and Schnaubelt (2021). $L^p$-spectrum of degenerate hypoelliptic Ornstein--Uhlenbeck operators. Journal of Functional Analysis. Spectrum of hypoelliptic OU operators, used in Section 9.",
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    },
    {
      "citation": "Holevo (2017). On a Characterization of Gaussian Transition Operators. Theory of Probability and Its Applications. Characterisation of Gaussian transition operators.",
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      "doi": "10.1137/s0040585x97t988435"
    },
    {
      "citation": "Ikeda, Ishikawa and Sawano (2022). Boundedness of composition operators on reproducing kernel Hilbert spaces with analytic positive definite functions. Journal of Mathematical Analysis and Applications. Fourier description and deterministic composition rigidity for Gaussian RKHSs.",
      "url": "https://doi.org/10.1016/j.jmaa.2022.126048",
      "doi": "10.1016/j.jmaa.2022.126048"
    },
    {
      "citation": "John and Parthasarathy (2021). A common parametrization for finite mode Gaussian states, their symmetries, and associated contractions with some applications. Journal of Mathematical Physics. Gaussian states and associated contractions.",
      "url": "https://doi.org/10.1063/5.0019413",
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    },
    {
      "citation": "Klus, Schuster and Muandet (2020). Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces. Journal of Nonlinear Science. Transfer operators in RKHSs without invariance.",
      "url": "https://doi.org/10.1007/s00332-019-09574-z",
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    },
    {
      "citation": "Neretin (1990). A semigroup of operators in the boson fock space. Functional Analysis and Its Applications. Gaussian operators in boson Fock space; contains the fixed affine criterion.",
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    },
    {
      "citation": "Paulsen and Raghupathi (2016). An Introduction to the Theory of Reproducing Kernel Hilbert Spaces. Cambridge University Press. RKHS textbook.",
      "url": "https://doi.org/10.1017/cbo9781316219232",
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    },
    {
      "citation": "Philipp, Schaller, Worthmann, Peitz and Nüske (2025a). Invariance of Gaussian RKHSs Under Koopman Operators of Stochastic Differential Equations With Constant Matrix Coefficients. PAMM. The factor-2 sufficient condition and the open necessity question answered here.",
      "url": "https://doi.org/10.1002/pamm.202400127",
      "doi": "10.1002/pamm.202400127"
    },
    {
      "citation": "Philipp, Schaller, Worthmann, Peitz and Nüske (2025b). Error Analysis of Kernel EDMD for Prediction and Control in the Koopman Framework. Journal of Nonlinear Science. kEDMD error bounds whose invariance hypothesis is verified here for multivariate OU.",
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      "doi": "10.1007/s00332-025-10182-3"
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