Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
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 .Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation
settingset:gaussian-analysis-hilbert-2026aAnalysisProbabilityStanding notation for analysis on the Lebesgue spaces of a diagonal Gaussian reference measure on a Hilbert space with noise weights: integer and series conventions, the spaces , Cameron-Martin vectors and Paley-Wiener functionals, the rates , and…Hermite Polynomials: Low Orders, Derivatives, Recursion, Scaling, the Addition Formula, Gaussian Orthogonality and the Mehler Identity
lemmalem:hermite-polynomials-real-2026aAnalysisProbabilityBasic properties of the Hermite polynomials : explicit low-order forms, polynomial growth and regularity, the derivative rule , the three-term recursion, the eigenfunction equation, scaling, the addition formula, orthogonality in of the c…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.