Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
The Gaussian Sobolev Space of the Noise Gradient
definitiondef:gaussian-sobolev-space-hilbert-2026bAnalysisProbabilityDefines the Gaussian Sobolev space as the set of in that are -limits of bounded cylindrical functions whose noise gradients form a Cauchy sequence, and defines the noise gradient of such as the limit of those gradients, whic…Gaussian Integration by Parts for Products of Cylindrical Functions, and Closability of the Noise Gradient in the Gaussian Lebesgue Space
lemmalem:noise-gradient-closable-hilbert-2026aAnalysisProbabilityGaussian integration by parts for bounded cylindrical functions: ; and as a consequence the noise gradient is closable, i.e. if in and…Moment and Tail Bounds on the Wiener Chaos up to a Given Order
corollarycor:wiener-chaos-moment-bounds-hilbert-2026aAnalysisProbabilityEvery in the Wiener chaos up to order has finite moments of all orders, with for ; and, for and , it has the tail bound fo…Hypercontractivity of the Mehler Semigroup with Noise Weights (Nelson's Theorem)
theoremthm:gaussian-hypercontractivity-hilbert-2026aAnalysisProbabilityNelson's hypercontractivity for the Mehler semigroup with noise weights: if for all and with , then for every continuous cylindrical function of polynomial growth.A Log-Sobolev Inequality for the Galerkin Wick-Ordered Gibbs Measure on the Torus at Temperatures above the Semiconvexity Constant
corollarycor:free-field-phi4-galerkin-log-sobolev-torus-hilbert-2026aAnalysisProbabilityOn the torus, the Gibbs measure of the Galerkin Wick-ordered potential with cutoff at a temperature above the semiconvexity constant satisfies a log-Sobolev inequality in entropy form with constant and in Gross's form with constan…Log-Sobolev and Talagrand Inequalities for the Gibbs Measure of a Semiconvex Cylindrical Potential below the Critical Curvature (Bakry-Emery)
corollarycor:gibbs-log-sobolev-hilbert-2026aAnalysisProbabilityIf for all and is an admissible cylindrical potential with semiconvexity constant where , then the Gibbs measure of at temperature satisfies log-Sobolev inequalities in entropy form and in Gross's form and a Talagrand ine…A Log-Sobolev Inequality Survives a Bounded Perturbation of the Measure (Holley-Stroock)
lemmalem:holley-stroock-log-sobolev-hilbert-2026aAnalysisProbabilityHolley-Stroock perturbation: if a probability measure satisfies a log-Sobolev inequality with constant for bounded cylindrical functions and noise gradients, then the normalized measure with density proportional to , for a bounded Borel with values in…The Gaussian Log-Sobolev, HWI and Poincare Inequalities in the Noise Geometry, in Entropy Form and in Gross's Functional Form
corollarycor:gaussian-log-sobolev-poincare-hilbert-2026aAnalysisProbabilityIf for all , the diagonal Gaussian measure satisfies, in the noise geometry, the log-Sobolev inequality , an HWI inequality, Gross's functional log-Sobolev inequality with constant…Talagrand, HWI and Log-Sobolev Inequalities for a Uniformly Displacement Convex Noise Penalty Pair with a Point of Zero Score
theoremthm:noise-hwi-log-sobolev-talagrand-hilbert-2026aAnalysisProbabilityFor a -displacement convex noise penalty pair with and a measure at which the score vanishes, the Talagrand inequality , the HWI inequality, and the log-Sobolev inequality…The Mehler Law Flow: Duality, Finite Second Moments, Invariance of the Gaussian Measure, Generation by the Noise Ornstein-Uhlenbeck Functional, and Decrease of Relative Entropy
theoremthm:mehler-law-flow-hilbert-2026aAnalysisProbabilityFor a probability measure with finite second moment, the Mehler law flow stays in , is a flow leaving invariant, is dual to the Mehler semigroup on cylindrical functions of at most quadratic growth, has time derivative given by minus t…The Mehler Semigroup: Cylindrical Form, the Semigroup Law, Invariance, Positivity, Contraction, Symmetry, Hermite Eigenfunctions, Commutation with Partial Derivatives, and the Generator
lemmalem:mehler-semigroup-hilbert-2026aAnalysisProbabilityThe Mehler semigroup preserves continuous cylindrical functions of polynomial growth and satisfies the semigroup law, invariance of , positivity, contraction, symmetry, and continuity in time; cylindrical Hermite polynomials are eigenfunctions with eigenvalue…The Ornstein-Uhlenbeck Operator with Noise Weights on Bounded Cylindrical Functions
definitiondef:ornstein-uhlenbeck-operator-hilbert-2026aAnalysisProbabilityDefines the Ornstein-Uhlenbeck operator with noise weights on bounded cylindrical functions, , the sum running over the finitely many coordinates on which depends; the value does not depend on the c…The Mehler Maps and the Mehler Semigroup with Noise Weights on Continuous Cylindrical Functions of Polynomial Growth
definitiondef:mehler-semigroup-hilbert-2026bAnalysisProbabilityDefines the Mehler maps , which in coordinate combine and with weights and , and the Mehler semigroup with noise weights acting on continuous cylindrical functions of polynomial growth by…Wick Powers of Paley-Wiener Functionals: Expansion in Cylindrical Hermite Polynomials, the Covariance Formula, Continuity in the Vector, and Membership in the Chaos
lemmalem:wick-powers-paley-wiener-hilbert-2026aAnalysisProbabilityWick powers of Paley-Wiener functionals are square-integrable; for finitely supported they expand explicitly in cylindrical Hermite polynomials; their inner products are for equal orders and otherwise; they depend continuously in…The Wick Powers of a Paley-Wiener Functional
definitiondef:wick-powers-paley-wiener-hilbert-2026aAnalysisProbabilityDefines the -th Wick power of the Paley-Wiener functional of a Cameron-Martin vector as the Hermite polynomial of degree with variance evaluated at .The Wiener Chaos Decomposition: Cylindrical Hermite Polynomials Are Dense and the Gaussian Lebesgue Space Is the Orthogonal Sum of the Chaoses
theoremthm:wiener-chaos-decomposition-hilbert-2026aAnalysisProbabilityThe Wiener chaoses of a diagonal Gaussian measure are mutually orthogonal closed subspaces of , with the zeroth chaos the constants and the projection onto the chaos up to order the sum of the chaos projections; the cylindrical Hermite polynomials span a dense…The Wiener Chaoses of a Diagonal Gaussian Measure on a Hilbert Space
definitiondef:wiener-chaos-hilbert-2026aAnalysisProbabilityDefines the span of the cylindrical Hermite polynomials, the Wiener chaos of order as the -closure of the span of the with , and the chaos up to order as the closure of the span of those with .Cylindrical Hermite Polynomials: Growth and Integrability, Partial Derivatives, Orthogonality, and Monomials as Hermite Combinations
lemmalem:hermite-cylindrical-polynomials-hilbert-2026aAnalysisProbabilityThe cylindrical Hermite polynomials are cylindrical functions of polynomial growth lying in every ; their partial derivatives are ; they are orthogonal in with squared norm…The Cylindrical Hermite Polynomials of a Diagonal Gaussian Measure on a Hilbert Space
definitiondef:hermite-cylindrical-polynomials-hilbert-2026aAnalysisProbabilityDefines the cylindrical Hermite polynomial on the Hilbert space, for a finitely supported multi-index , as the product over coordinates of the one-dimensional Hermite polynomials of degree and variance evaluated at the coordinate .The Entropy of a Nonnegative Function with Respect to a Probability Measure
definitiondef:entropy-functional-probability-2026aAnalysisProbabilityDefines the entropy of a nonnegative measurable function with respect to a probability measure , when and are integrable, as the integral of minus of the integral of , where and .