Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
Lattice Sums of Powers of the Fourier Weights in Dimension at Most Two: Power Sums, the Growth of the Wick Constants, and the Quartic Convolution Sums
lemmalem:lattice-quartic-sums-torus-2026aAnalysisIn dimension at most two, powers of the Fourier weights are summable above half the dimension, the Wick constants grow slower than any power of the cutoff, and the quartic convolution sums with one weight relaxed by a small power are finite.The Gibbs Sobolev Space: Linearity, Closedness, the Chain Rule, and the Score of a Measure of Finite Intrinsic Fisher Information
lemmalem:gibbs-sobolev-space-hilbert-2026aAnalysisProbabilityThe Gibbs Sobolev space is closed and linear with a chain rule; the Gibbs measure has zero intrinsic Fisher information; and for a measure of finite intrinsic Fisher information the field 2 grad(sqrt f)/sqrt f is square-integrable, has squared norm equal to the Fisher information…The Intrinsic Fisher Information Relative to the Gibbs Measure of a Gibbs-Sobolev Potential
definitiondef:intrinsic-gibbs-fisher-information-hilbert-2026aAnalysisProbabilityA measure has finite intrinsic Fisher information relative to the Gibbs measure if it has a density whose square root lies in the Gibbs Sobolev space; the information is four times the Dirichlet energy of that square root.The Gibbs Sobolev Space of the Noise Gradient
definitiondef:gibbs-sobolev-space-hilbert-2026aAnalysisProbabilityDefines the Gibbs Sobolev space as the closure of the bounded C^1 cylindrical functions under the noise-gradient graph norm in the square-integrable space of the Gibbs measure, together with its Gibbs noise gradient.Integration by Parts for the Gibbs Measure of a Gibbs-Sobolev Potential, and Closability of the Noise Gradient in Its Square-Integrable Space
lemmalem:gibbs-integration-by-parts-hilbert-2026aAnalysisProbabilityFor a Gibbs-Sobolev potential the Gibbs measure has finite second moments, satisfies the Gaussian integration by parts formula with the extra drift given by the partial derivatives of the potential over the temperature, and the noise gradient on cylindrical functions is closable…Gibbs-Sobolev Potentials: Exponentially Integrable Potentials with Cameron-Martin Derivatives of Finite Moments
definitiondef:gibbs-sobolev-potential-hilbert-2026aAnalysisProbabilityA Gibbs-Sobolev potential is a Borel potential whose Boltzmann weight is integrable to a power q>1, which lies in the Cameron-Martin Sobolev space, and whose partial derivatives have moments of order 2q/(q-1).The Gibbs Measure of a Borel Potential Relative to a Diagonal Gaussian Measure
definitiondef:gibbs-measure-potential-hilbert-2026aAnalysisProbabilityDefines the Gibbs measure of a Borel potential at a positive temperature relative to a diagonal Gaussian measure: the normalised measure with density exp(-V/beta), whenever that weight is integrable.The Dirichlet Form of the Noise Gradient on the Gaussian Sobolev Space
definitiondef:gaussian-dirichlet-form-hilbert-2026aAnalysisProbabilityDefines the Dirichlet form of the noise gradient on the Gaussian Sobolev space as , the inner product in of the noise gradients.The Cameron-Martin Inner Product of Two Cameron-Martin Vectors
definitiondef:cameron-martin-inner-product-hilbert-2026aAnalysisProbabilityDefines the Cameron-Martin inner product of two vectors in the Cameron-Martin space of a variance sequence as the convergent series .The Mehler Law Flow of a Probability Measure on a Hilbert Space
definitiondef:mehler-law-flow-hilbert-2026aAnalysisProbabilityDefines the Mehler law flow from a Borel probability measure on the Hilbert space: for each , is the push-forward of the product measure under the Mehler map .Continuous Cylindrical Functions of Polynomial Growth on a Hilbert Space
definitiondef:cylindrical-polynomial-growth-hilbert-2026aAnalysisProbabilityDefines continuous cylindrical functions of polynomial growth on a Hilbert space: functions of finitely many coordinates, with continuous on and bounded by a polynomial in the Euclidean norm, together with the notion of a representation of…Polynomial Functions Are Dense in the Square-Integrable Functions of a Diagonal Gaussian Measure on Euclidean Space
lemmalem:polynomials-dense-diagonal-gaussian-euclidean-2026aAnalysisProbabilityFor a diagonal Gaussian measure on , every polynomial function has finite moments of all orders, and polynomial functions are dense in the square-integrable functions: every square-integrable Borel function can be approximated arbitrarily well in by a polynomi…Finitely Supported Multi-Indices: Order, Factorial, Powers and Weighted Order
definitiondef:finitely-supported-multi-indices-2026aAnalysisCombinatoricsDefines multi-indices as finitely supported sequences of nonnegative integers, together with their order, factorial, powers of a real sequence, weighted order , the sets of multi-indices of a given order, and the unit multi-indices with th…The Gaussian Sobolev Space: Linear Structure, Closedness, Hermite Polynomials and the Number-Operator Form of the Dirichlet Form, and the Log-Sobolev and Poincare Inequalities
lemmalem:gaussian-sobolev-space-hilbert-2026aAnalysisProbabilityThe Gaussian Sobolev space is a linear subspace of on which the noise gradient is linear and closed; it contains the cylindrical Hermite polynomials, with explicit gradient coordinates, and on them the Dirichlet form is diagonal with…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…