Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
The Gibbs Dirichlet Form of the Noise Gradient
definitiondef:gibbs-dirichlet-form-hilbert-2026aAnalysisProbabilityDefines the Gibbs Dirichlet form as the inner product, in the square-integrable fields of the Gibbs measure, of Gibbs noise gradients.Cameron-Martin Partial Derivatives on the Gaussian Sobolev Space with the Variances as Weights
definitiondef:cameron-martin-partial-derivatives-hilbert-2026aAnalysisProbabilityReads the Gaussian Sobolev space with the variances as weights, the Cameron-Martin (Malliavin) Sobolev space, and defines the partial derivatives of its elements along the basis vectors.The Potential in Dimension One: Wick Ordering Is a Finite Mass Shift and Nelson's Estimate Is Not Needed
remarkrem:free-field-phi4-one-dimension-torus-hilbert-2026aAnalysisProbabilityExplains that in dimension one Wick ordering is only a finite mass shift and Nelson's estimate is not needed, which techniques could replace the chaos route there, and why the score relative to the Gibbs measure must still be defined intrinsically.The Measure on the Torus of Dimension at Most Two as the Gibbs Measure of a Gibbs-Sobolev Potential, and Its Galerkin Approximation in Total Variation
corollarycor:free-field-phi4-gibbs-measure-torus-hilbert-2026aAnalysisProbabilityIn dimension at most two the phi^4 potential is a Gibbs-Sobolev potential for every exponent, so the phi^4 measure is its Gibbs measure, and the Galerkin Gibbs measures and their normalisers converge to it uniformly over Borel sets.Nelson's Exponential Integrability for the Wick-Ordered Potential on the Torus of Dimension at Most Two
theoremthm:nelson-exponential-integrability-torus-hilbert-2026aAnalysisProbabilityNelson's estimate on the torus of dimension at most two: for every t>0, exp(-tV_N) is bounded in mean under the free field uniformly in the cutoff, exp(-tV) is integrable, and exp(-tV_N) converges to exp(-tV) in every L^p.The Wick-Ordered Potential on the Torus of Dimension at Most Two: Limit of the Galerkin Potentials Almost Everywhere and in Every , Conditional Expectations, Cameron-Martin Derivatives, and the Lower Bound at Each Cutoff
lemmalem:free-field-phi4-potential-torus-hilbert-2026aAnalysisProbabilityThe phi^4 potential is Borel, is the almost-everywhere and L^p limit of the Galerkin potentials with rate N^(-1/8), has them as conditional expectations, and lies in the Cameron-Martin Sobolev space with derivatives in every L^p; each Galerkin potential is bounded below by a cons…The Wick-Ordered Potential of the Free Field on the Torus of Dimension at Most Two
definitiondef:free-field-phi4-potential-torus-hilbert-2026aAnalysisProbabilityDefines the Wick-ordered phi^4 potential of the free field on the torus of dimension at most two as the pointwise limit of the Galerkin potentials along dyadic cutoffs, and zero where that limit does not exist.The Galerkin Wick-Ordered Potentials on the Torus of Dimension at Most Two under the Free Field: Chaos Order, Conditional Expectations, and Increments with Their Cameron-Martin Derivatives
lemmalem:free-field-galerkin-phi4-chaos-torus-hilbert-2026aAnalysisProbabilityIn dimension at most two the Galerkin Wick-ordered phi^4 potentials lie in the chaos up to order four, each is the conditional expectation of a later one given the cube modes, and their increments and Cameron-Martin derivative increments are bounded in square mean by a constant t…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…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…