Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
Comparison, Uniqueness and Convergence of Coarse Solutions Inherited by a Fine First-Order Equation on the Noise Wasserstein Space from a Consistent Family of Coarse Levels
corollarycor:inherited-comparison-first-order-hilbert-2026aAnalysisProbabilityUnder the hypotheses of the cross-level comparison principle, if every coarse equation has a viscosity solution and these are bounded uniformly in the level, then the fine equation satisfies comparison and uniqueness for bounded solutions, and a bounded fine solution is the limit…A Cross-Level Comparison Principle for First-Order Equations on a Fine Noise Wasserstein Space and a Family of Coarse Levels Linked by Mode Restrictions
theoremthm:cross-level-comparison-first-order-hilbert-2026aAnalysisProbabilityIf a family of coarse noise levels, linked to a fine level by mode restrictions, satisfies the hypotheses of the single-level comparison principle with constants uniform in the level, and the consistency defect between the fine and the coarse operators vanishes as the level grows…Bounds, Monotonicity and Borwein-Preiss Perturbed Maximisers of the Cross-Level Doubled Function in the Coarse Noise Wasserstein Distance for Compatible Noise-Closed Penalty Pairs
lemmalem:cross-level-doubling-perturbed-maximiser-hilbert-2026aAnalysisProbabilityFor compatible noise-closed penalty pairs at a fine and a coarse level and upper semicontinuous functions dominated by minus delta times the penalties, the doubled function U(mu)+V(nu)-(alpha/2)W(p_#mu,nu)^2 has a finite supremum, controlled by its diagonal, monotone in the doubl…Cross-Level Data and Eta-Consistency of a Fine and a Coarse Level at (delta, R)
definitiondef:cross-level-consistency-hilbert-2026aAnalysisProbabilityFor compatible noise penalty pairs at a fine and a coarse level linked by a mode restriction, defines cross-level data (a fine measure, a real number and a coarse tangent field) and their R-boundedness, and calls the two levels eta-sub-, eta-super- or eta-consistent at (delta, R)…Noise Intrinsic Test Functions Pulled Back along a Mode Restriction, and the Squared Coarse Noise Wasserstein Distance as a Fine Test Function
lemmalem:restricted-noise-test-function-hilbert-2026aAnalysisProbabilityComposing a coarse-level noise intrinsic test function with push-forward by the mode restriction gives a fine-level test function whose gradient is the pulled-back coarse gradient; in particular the squared coarse distance from the restricted measure to a fixed measure is a fine…Properties of a Mode-Restriction Link: Isometric Embedding, Contraction of Noise Norms and Noise Wasserstein Distances, and Pull-Back of Cylindrical Functions and Tangent Fields
lemmalem:mode-restriction-link-hilbert-2026aAnalysisProbabilityThe mode restriction p of a mode-restriction link is a left inverse of the isometric mode embedding j; both respect the noise norms and are adjoint for them, push-forward by p contracts noise costs and noise Wasserstein distances, and composing with p and j carries cylindrical fu…Mode-Restriction Links between a Fine and a Coarse Noise Level
definitiondef:mode-restriction-link-hilbert-2026aAnalysisProbabilityDefines a mode-restriction link from a fine to a coarse noise level: an injective relabelling of basis modes that preserves the noise weights, together with its mode restriction p and its mode embedding j.Finite Linear Combinations of Continuous Real-Valued Maps on a Metric Space are Continuous
lemmalem:finite-sum-continuous-metric-2026aAnalysisTopologyA finite linear combination, over a nonempty finite index set, of continuous real-valued maps on a metric space is continuous.The White-Noise Embedding of Lattice Fields: Linearity, Inversion by the Site Field, the Site Sum as the White-Noise Norm, and Coordinates
lemmalem:lattice-field-embedding-torus-hilbert-2026aAnalysisThe white-noise embedding of lattice fields into the negative Sobolev space is linear and injective with the site field as left inverse; site evaluations are continuous linear functionals, the white-noise norm of an embedded field is its site sum of squares, the site sum of a noi…The White-Noise Embedding of Lattice Fields into the Negative Sobolev Space of the Torus, and the Site Field of a Distribution
definitiondef:lattice-field-embedding-torus-hilbert-2026aAnalysisThe white-noise embedding sends a lattice field to the distribution on the torus whose Fourier coefficients on the cube of modes are its discrete Fourier coefficients times L^{n/2}, and zero elsewhere. The site field of a distribution is the lattice field obtained by evaluating i…Discrete Fourier Analysis on the Lattice Torus: Counting, Orthogonality, Inversion, Parseval, and the Spin Sum
lemmalem:lattice-torus-discrete-fourier-torus-2026aAnalysisOn the lattice torus of odd side L = 2M+1 the sites and the modes both number L^n, the sampled trigonometric system is orthogonal, the discrete Fourier transform is a linear bijection with an explicit inversion formula and satisfies Parseval's identity, and the sum over all spin…The Lattice Torus of Odd Side, Lattice Fields, and the Discrete Fourier Coefficients of a Lattice Field
definitiondef:lattice-torus-discrete-fourier-torus-2026aAnalysisThe lattice torus of odd side L = 2M+1 is the set of points w/L of the unit cell with integer coordinates w between 0 and L-1. A lattice field is a real function on it, and its discrete Fourier coefficients are its averages against the real trigonometric maps indexed by the cube…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…