Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
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.The Gibbs Measure of an Admissible Cylindrical Potential Relative to a Diagonal Gaussian Measure
definitiondef:gibbs-measure-cylindrical-potential-hilbert-2026aAnalysisProbabilityThe Gibbs measure of an admissible cylindrical potential V at temperature beta is the probability measure with density proportional to exp(-V/beta) with respect to the diagonal Gaussian reference measure.Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient
lemmalem:admissible-cylindrical-potential-hilbert-2026aAnalysisProbabilityAn admissible cylindrical potential is continuous with a continuous noise gradient, grows at most exponentially under noise translations, has integrable slope wherever it is integrable, is integrable with its squared slope against the Gaussian reference measure, satisfies the sem…Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry
definitiondef:admissible-cylindrical-potential-hilbert-2026aAnalysisProbabilityAn admissible cylindrical potential is a twice continuously differentiable function of finitely many coordinates that is bounded below, semiconvex in the noise norm, has slope controlled by its value and weighted Laplacian controlled by its squared slope; its noise gradient is fi…Well-Posedness of the Hamilton-Jacobi Equation with the Free-Field Score Drift and the Wick-Square Cost for Laws of Distributions on the Torus of Dimension at Most Three, and the Forced Counterterm
corollarycor:free-field-wick-square-hamilton-jacobi-well-posed-torus-hilbert-2026aAnalysisPDEFor laws of distributions on the torus of dimension at most three relative to the free field with square-integrable white noise, the Hamilton-Jacobi equation with the Wick-square cost is well posed relative to the dressing profile, and its cutoff counterterms are forced to be the…