Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field
definitiondef:noise-gaussian-entropy-pair-hilbert-2026aAnalysisProbabilityDefines the Gaussian entropy pair on the noise Wasserstein space: the penalty is beta times relative entropy with respect to the diagonal Gaussian reference measure, and the score is beta times the noise score field on measures of finite weighted Fisher information.Noise Gradients of Bounded Cylindrical Functions Are Dense in the Noise Tangent Space
lemmalem:c2-cylindrical-noise-gradients-dense-hilbert-2026aAnalysisProbabilityEvery bounded C^2 cylindrical function is a bounded C^1 cylindrical function, and for every Borel probability measure the noise gradients of bounded C^2 cylindrical functions approximate every element of the noise tangent space.Gaussian Smoothing of a Bounded Borel Function on Euclidean Space: the Two Forms, Derivatives through the Kernel and through the Function, and Pointwise Convergence
lemmalem:gaussian-smoothing-bounded-function-euclidean-2026aAnalysisThe convolution of a bounded Borel function with a Gaussian kernel is bounded by the same constant, continuously differentiable with derivatives computed through the kernel, commutes with differentiation when the function is bounded C^1, and converges to the function at its point…Bounded Cylindrical Functions on a Hilbert Space with an Orthonormal Basis
definitiondef:bounded-c2-cylindrical-functions-hilbert-2026aAnalysisDefines the bounded C^2 cylindrical functions on a Hilbert space with a fixed orthonormal basis: functions of finitely many coordinates given by a bounded C^2 function with bounded first and second derivatives.Bounded Twice Continuously Differentiable Functions with Bounded First and Second Partial Derivatives on Euclidean Space
definitiondef:bounded-c2-functions-euclidean-2026aAnalysisDefines the set of twice continuously differentiable real functions on Euclidean space that are bounded together with all their first and second partial derivatives.The Entropy Domain of a Diagonal Gaussian Reference Measure Has the Noise Map Property
corollarycor:noise-map-property-entropy-diagonal-gaussian-hilbert-2026aAnalysisProbabilityWhen the reference measure is a diagonal Gaussian measure whose variances are dominated by a multiple of the noise weights, the measures of finite relative entropy form a set with the noise map property.Tangency of Noise-Optimal Displacements out of a Measure with a Density Relative to a Diagonal Gaussian Measure, and the Noise Map Property
theoremthm:noise-tangency-absolutely-continuous-hilbert-2026aAnalysisProbabilityOut of a measure with a density relative to a diagonal Gaussian measure, every noise-optimal displacement lies in the noise tangent space; hence the set of such measures has the noise map property.Strong Convergence of Noise Displacement Fields from Weak Convergence of Their Graph Couplings and an Upper Bound on Their Norms
lemmalem:graph-coupling-strong-convergence-hilbert-2026aAnalysisProbabilityIf the graph couplings of noise displacement fields converge weakly to the graph coupling of a limit field, and their norms are eventually at most the norm of the limit plus any margin, then the fields converge strongly among the square-integrable noise fields.Measures Whose Heads Agree with a Fixed Measure and Whose Tails Vanish Converge in the Quadratic Wasserstein Distance
lemmalem:head-matched-w2-convergence-hilbert-2026aAnalysisProbabilityIf the n-th measure of a sequence has the same law of the first n coordinates as a fixed measure and its tail second moments tend to zero, the sequence converges to the fixed measure in the quadratic Wasserstein distance.Lifting Euclidean Tangent Fields of the Rescaled Head to Noise Tangent Fields on a Hilbert Space
lemmalem:noise-head-lift-tangent-hilbert-2026aAnalysisProbabilityRescaling the first n coordinates by the noise weights lifts Euclidean vector fields on the head to noise fields on the Hilbert space. The lift is a linear isometry sending gradients of test functions to noise gradients of cylindrical functions, hence Euclidean tangent fields int…Talagrand's Inequality in the Noise Norm for a Diagonal Gaussian Reference Measure on a Hilbert Space
theoremthm:noise-talagrand-diagonal-gaussian-hilbert-2026aAnalysisProbabilityWhen the reference measure is a diagonal Gaussian whose variances are dominated by a constant multiple of the noise weights, every probability measure of finite relative entropy is noise-connected to it, and its squared noise Wasserstein distance to it is at most twice that const…Talagrand's Inequality for a Diagonal Gaussian Measure on Euclidean Space in the Cameron-Martin Cost
lemmalem:talagrand-diagonal-gaussian-euclidean-2026aAnalysisProbabilityA probability measure with finite relative entropy with respect to a diagonal Gaussian measure on Euclidean space is the image of the Gaussian under a Borel map whose transport cost, with each coordinate weighted by the inverse of its variance, is at most twice the relative entro…Conditional Kernels of a Borel Probability Measure on a Polish Space Given a Borel Map: Existence, Uniqueness and Concentration on the Fibres
corollarycor:conditional-kernel-given-map-polish-2026aAnalysisProbabilityGiven a Borel map from a complete separable metric space into a separable metric space, every Borel probability measure has a conditional kernel given the map, unique up to a null set of the image measure; almost every conditional measure is carried by its fibre, and integrals ag…Disintegration of a Probability Measure on a Product with a Polish Factor: Existence and Almost-Everywhere Uniqueness of the Conditional Kernel
theoremthm:disintegration-polish-2026aAnalysisProbabilityA probability measure on the product of a measurable space and a complete separable metric space is the composite of its first marginal with a probability kernel, and this kernel is unique up to a null set of the marginal.Integration Against a Probability Kernel: Measurable Sections, the Composite Measure on the Product and the Iterated Integral
lemmalem:probability-kernel-integration-2026aAnalysisProbabilityA probability measure and a probability kernel define a unique probability measure on the product sigma-algebra, against which nonnegative functions are integrated by first integrating against the kernel and then against the measure.Extension of a Finite Premeasure from an Algebra of Sets to the Generated Sigma-Algebra
theoremthm:premeasure-extension-2026aAnalysisProbabilityA finite, countably additive set function on an algebra of subsets extends to a measure on the generated sigma-algebra, and the extension is unique.The Borel Sigma-Algebra of a Product of Two Separable Metric Spaces is the Product Sigma-Algebra
lemmalem:borel-product-separable-metric-2026aAnalysisProbabilityFor two separable metric spaces, the Borel sets of the product metric space are exactly the sets of the product sigma-algebra of the two Borel sigma-algebras.Perron's Method on the Noise Wasserstein Space: Existence of a Viscosity Solution of a First-Order Equation Between a Subsolution and a Supersolution
theoremthm:noise-perron-existence-hilbert-2026bAnalysisProbabilityPDEFor a noise-closed noise penalty pair with regular penalised maxima whose penalty domain has the noise map property, and any first-order equation operator, the pointwise supremum of all viscosity subsolutions lying between a given subsolution and a given supersolution above it, b…Well-Posedness of the Hamilton-Jacobi Equation with Penalty Drift on the Noise Wasserstein Space: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution
theoremthm:noise-penalty-drift-well-posed-hilbert-2026bAnalysisProbabilityPDEFor a displacement convex, noise-closed noise penalty pair with closed score and regular penalised maxima whose penalty domain has the noise map property and whose penalty controls the squared distance to the reference measure, and for a bounded uniformly continuous running cost…Constant Viscosity Subsolutions and Supersolutions of the Hamilton-Jacobi Equation with Penalty Drift on the Noise Wasserstein Space
lemmalem:noise-constant-sub-supersolutions-hilbert-2026bAnalysisProbabilityPDEFor a noise-closed noise penalty pair with regular penalised maxima, a constant function has penalty-subordinate growth from both sides; it is a viscosity subsolution of the penalty-drift Hamilton-Jacobi equation when discount times the constant is at most the running cost on the…