Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
Hermite Polynomials with a Given Variance
definitiondef:hermite-polynomials-real-2026aAnalysisProbabilityDefines the Hermite polynomial of degree with variance as the explicit finite sum over with of .The Gibbs Ornstein-Uhlenbeck Functional of a Probability Measure at a Bounded Function of Finitely Many Coordinates
definitiondef:gibbs-ou-functional-hilbert-2026aAnalysisProbabilityFor an admissible cylindrical potential V, a temperature beta, a measure mu of finite second moment integrating V, and a bounded C^2 function g of n coordinates, defines the Gibbs Ornstein-Uhlenbeck functional: the sum over k up to n of a_k times the mu-integral of (x_k/c_k + d_k…The Galerkin Wick-Ordered Potential and Its Wick Constant on the Torus
definitiondef:free-field-phi4-galerkin-potential-torus-hilbert-2026aAnalysisProbabilityPDEDefines the Galerkin Wick-ordered phi^4 potential on the torus together with the inseparable ingredients of its construction: the head dimension of the cutoff (the Galerkin head dimension and active indices), the Galerkin field, the Wick constant and the Wick polynomial, as well…The Weighted Fisher Information Relative to the Gibbs Measure of an Admissible Cylindrical Potential
definitiondef:weighted-fisher-information-gibbs-hilbert-2026aAnalysisProbabilityDefines, for a measure with a relative score with respect to the Gibbs measure of an admissible cylindrical potential, finite Fisher information relative to that Gibbs measure with the noise weights and its value as the weighted series of squared score norms.Well-Posedness of the Galerkin Stochastic-Quantisation Hamilton-Jacobi Equation on the Torus for Every Cutoff, Coupling and Mass: Comparison, Existence and Uniqueness
corollarycor:free-field-phi4-galerkin-hamilton-jacobi-well-posed-torus-hilbert-2026aAnalysisProbabilityPDEFor every cube cutoff, positive coupling, real mass and positive temperature, the Hamilton-Jacobi equation with the Gibbs score drift of the Galerkin Wick-ordered phi^4 measure on the torus satisfies comparison and has exactly one bounded viscosity solution, for every bounded uni…The Galerkin Wick-Ordered Potential on the Torus is an Admissible Cylindrical Potential, with Semiconvexity Constant the Positive Part of Three Times the Coupling Times the Wick Constant Minus the Mass
lemmalem:free-field-phi4-galerkin-potential-torus-hilbert-2026aAnalysisProbabilityPDEAt a cube cutoff, the Wick-ordered quartic plus Wick-ordered mass term of the Galerkin free field on the torus is an admissible cylindrical potential in the white-noise geometry; its Hessian is bounded below by the mass minus three times the coupling times the Wick constant, meas…Well-Posedness of the Hamilton-Jacobi Equation with Penalty Drift for a Lambda-Displacement Convex Noise Penalty Pair, Lambda Any Real Number: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution
theoremthm:noise-penalty-drift-well-posed-lambda-convex-hilbert-2026aAnalysisProbabilityPDEFor a noise-closed noise penalty pair with closed score and regular penalised maxima that is lambda-displacement convex for some real lambda, possibly negative, whose penalty domain has the noise map property and whose penalty controls the squared distance to the reference measur…The Hamilton-Jacobi Operator with Penalty Drift on the Noise Wasserstein Space Satisfies the Hypotheses of the First-Order Comparison Principle for a Lambda-Displacement Convex Noise Penalty Pair, Lambda Any Real Number
lemmalem:noise-penalty-drift-hypotheses-lambda-convex-hilbert-2026aAnalysisProbabilityPDEFor a noise penalty pair that is lambda-displacement convex for some real lambda, possibly negative, with lower semicontinuous penalty controlling the squared noise Wasserstein distance to the reference measure, control cost at most one and a bounded uniformly continuous running…Well-Posedness of the Hamilton-Jacobi Equation with Gibbs Score Drift for a Semiconvex Cylindrical Potential: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution
corollarycor:gibbs-score-drift-hamilton-jacobi-well-posed-hilbert-2026aAnalysisProbabilityPDEFor every admissible cylindrical potential, however nonconvex, every temperature, discount, control cost in (0,1] and bounded uniformly continuous running cost, the Hamilton-Jacobi equation with Gibbs score drift satisfies comparison and has exactly one bounded viscosity solution…On Integrals of Cylindrical Functions the Gibbs Score Drift is the Gibbs Ornstein-Uhlenbeck Functional, and the Gibbs Score-Drift Hamilton-Jacobi Operator Takes Its Pointwise Form
lemmalem:gibbs-score-drift-cylindrical-integral-hilbert-2026aAnalysisProbabilityPDEFor the Gibbs entropy pair and a bounded C^2 function f of finitely many coordinates, the integral U_phi of phi = f(p_n) is a noise intrinsic test function on the penalty domain with gradient the noise gradient of phi; the pairing of the Gibbs score drift with it equals beta time…The Hamilton-Jacobi Equation with Gibbs Score Drift on a Hilbert Space
equationeq:gibbs-score-drift-hamilton-jacobi-hilbert-2026aAnalysisProbabilityPDEThe discounted first-order Hamilton-Jacobi equation on the noise Wasserstein space whose drift is the score of the Gibbs entropy pair: the Gaussian score field times the temperature plus the noise gradient of the potential.The Gibbs Entropy Pair of a K-Semiconvex Potential is Lambda-Displacement Convex with Lambda the Temperature over the Variance-to-Noise Bound Minus K
lemmalem:noise-gibbs-entropy-pair-convex-hilbert-2026aAnalysisProbabilityThe Gibbs entropy pair of an admissible potential with semiconvexity constant K is (beta/kappa - K)-displacement convex, and displacement convex when K is at most beta/kappa.The Gibbs Entropy Pair Has Regular Penalised Maxima
lemmalem:noise-gibbs-entropy-pair-regular-maxima-hilbert-2026aAnalysisProbabilityEvery local maximum of a noise intrinsic test function minus a positive multiple of the Gibbs entropy penalty has finite Fisher information relative to the Gibbs measure.The Gibbs Entropy Pair Has Closed Score Along Noise Couplings
lemmalem:noise-gibbs-entropy-pair-closed-score-hilbert-2026aAnalysisProbabilityAlong couplings of vanishing noise cost, measures with uniformly bounded Gibbs scores converge to a measure that again has finite Gibbs Fisher information, and the Gibbs scores converge weakly along the couplings.The Gibbs Entropy Pair is Noise-Closed, and Its Penalty Bounds the Squared Noise Wasserstein Distance to the Reference Measure
lemmalem:noise-gibbs-entropy-pair-closed-hilbert-2026aAnalysisProbabilityPenalty sublevel sets of the Gibbs entropy pair are closed and bounded in the noise Wasserstein space, and the squared distance to the Gaussian reference measure is bounded by an affine function of the penalty.The Gibbs Entropy Pair is a Noise Penalty Pair, with Nonnegative Penalty and Dense Score Domain
lemmalem:noise-gibbs-entropy-penalty-pair-hilbert-2026aAnalysisProbabilityThe Gibbs entropy pair is a noise penalty pair: its penalty is nonnegative, its score is the first variation of the penalty along noise gradients, and its score domain is dense in its penalty domain for the noise Wasserstein distance.A Bound on the Gibbs Ornstein-Uhlenbeck Functional by Noise Gradients Gives Finite Fisher Information Relative to the Gibbs Measure
lemmalem:gibbs-ou-functional-fisher-bound-hilbert-2026aAnalysisProbabilityIf the noise Ornstein-Uhlenbeck functional corrected by the pairing with the potential gradient is bounded by R times the noise-gradient norm on bounded C^2 cylindrical functions, the measure has a relative score with respect to the Gibbs measure with weighted Fisher information…The Gibbs Entropy Pair on the Noise Wasserstein Space: Relative Entropy to the Gibbs Measure and the Gibbs Score Field
definitiondef:noise-gibbs-entropy-pair-hilbert-2026aAnalysisProbabilityThe Gibbs entropy pair has penalty beta times the relative entropy to the Gibbs measure, defined on the measures where it is finite, and score beta times the Gaussian noise score field plus the noise gradient of the potential, defined where the Gibbs Fisher information is finite.Entropy and Score Relative to the Gibbs Measure Split into Their Gaussian Parts and the Potential, with a Fisher Information Bound
lemmalem:gibbs-entropy-score-splitting-hilbert-2026aAnalysisProbabilityThe Gibbs measure has density proportional to the weight with respect to the Gaussian reference measure, with the same null sets; relative entropy with respect to it is the Gaussian relative entropy plus the averaged potential over the temperature plus the log-normaliser; a measu…The Relative Score with Respect to the Gibbs Measure of an Admissible Cylindrical Potential
definitiondef:relative-score-gibbs-hilbert-2026aAnalysisProbabilityA measure has a relative score with respect to the Gibbs measure of an admissible cylindrical potential if, coordinate by coordinate, integration by parts against bounded cylindrical functions holds with the Gibbs drift; the components are unique.