Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
Noise Penalty Pairs with Closed Score Along Noise Couplings
definitiondef:noise-closed-score-hilbert-2026bAnalysisProbabilityA noise penalty pair has closed score along noise couplings if, whenever measures in the score domain with uniformly bounded scores are joined to a measure of the penalty domain by couplings of vanishing noise cost, the limit lies in the score domain and the scores converge weakl…Noise-Closed Noise Penalty Pairs
definitiondef:noise-closed-penalty-pair-hilbert-2026bAnalysisProbabilityA noise penalty pair is noise-closed if each sublevel set of the penalty is closed under limits in the noise Wasserstein metric and has bounded noise Wasserstein distance to the reference measure.First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation
settingset:noise-penalty-hilbert-2026bAnalysisProbabilityPDEStanding notation for first-order equations and their penalised viscosity solutions on the noise Wasserstein space: couplings of finite noise cost and their pairings, noise-optimal maps, gradients along noise couplings and test functions, noise penalty pairs, operators, shifts an…The Bundle of Noise Fields over a Set of Measures, First-Order Equation Operators on the Noise Wasserstein Space, and Their Delta-Shifts
definitiondef:noise-first-order-operator-hilbert-2026bAnalysisProbabilityPDEA first-order equation operator over a set of noise-connected measures is a real function of a measure, a real number and a square-integrable noise field; relative to a noise penalty pair its delta-shifts add or subtract delta times the penalty in the real slot and delta times th…The Delta-Envelopes of a Function on the Domain of a Noise Penalty Pair
definitiondef:noise-delta-envelopes-hilbert-2026bAnalysisProbabilityFor a function on the penalty domain of a noise penalty pair with penalty-subordinate growth, the delta-envelopes are the upper semicontinuous envelope of the function minus delta times the penalty and the lower semicontinuous envelope of the function plus delta times the penalty…Penalty-Subordinate Growth of a Function on the Domain of a Noise Penalty Pair
definitiondef:noise-penalty-subordinate-growth-hilbert-2026bAnalysisProbabilityA function on the penalty domain of a noise penalty pair has penalty-subordinate growth from above (below) if, for every positive delta, it is bounded above (below) by a constant plus (minus) delta times the penalty.Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains
definitiondef:noise-penalty-pair-hilbert-2026bAnalysisProbabilityA noise penalty pair on the measures noise-connected to the reference measure consists of a penalty domain, a nonempty score domain, a penalty bounded below by minus a multiple of one plus the squared noise distance to the reference, and a score in the noise tangent space that is…Linearity of the Noise Gradient, and the Noise Tangent Space is a Closed Linear Subspace
lemmalem:noise-tangent-space-hilbert-2026aAnalysisProbabilityThe noise gradient is linear on bounded C^1 cylindrical functions, so the noise gradients form a linear subspace of the square-integrable noise fields and the noise tangent space, their closure, is a closed linear subspace.The Noise Map Property of a Set of Probability Measures
definitiondef:noise-map-property-hilbert-2026aAnalysisProbabilityA set of measures noise-connected to the reference measure has the noise map property if every noise-optimal coupling between its members is induced by a map whose displacement lies in the noise tangent space.The Measures Noise-Connected to the Reference Measure
definitiondef:noise-wasserstein-space-hilbert-2026aAnalysisProbabilityThe set of Borel probability measures on a Hilbert space that are noise-connected to the fixed reference measure.- For a noise-connected ordered pair of Borel probability measures on a Hilbert space, a coupling of finite noise cost is noise-optimal if its noise cost equals the square of the noise Wasserstein distance.
Squared Noise Wasserstein Distances, Their Convergent Series and Linear Combinations are Noise Intrinsic Test Functions
lemmalem:noise-w2-squared-test-function-hilbert-2026aAnalysisProbabilityOn a set with the noise map property, the squared noise Wasserstein distance to a fixed measure and convergent weighted series of squared distances to measures at bounded distance from the reference measure are noise intrinsic test functions, with gradients given by optimal displ…Noise Intrinsic Test Functions on the Noise Wasserstein Space
definitiondef:noise-intrinsic-test-function-hilbert-2026aAnalysisProbabilityA real function on the noise Wasserstein space is a noise intrinsic test function on a set of measures if it is continuous, differentiable along noise couplings at each point of the set with gradient in the noise tangent space, and its gradient is continuous along couplings of va…The Noise-Optimal Map: Transport Cost, Stability Along Nearly Optimal Couplings and Stability Under Perturbation of the Source
theoremthm:noise-optimal-map-stability-hilbert-2026aAnalysisProbabilityThe displacement of a noise-optimal map has squared norm equal to the squared noise Wasserstein distance; for a uniquely noise-mapped pair, couplings of nearly optimal noise cost concentrate on the graph of the map in mean square; and the optimal displacements to a fixed target a…Noise-Optimal Maps and Uniquely Noise-Mapped Pairs
definitiondef:noise-optimal-map-hilbert-2026aAnalysisProbabilityDefines a noise-optimal transport map between two measures (a Borel map pushing one to the other whose displacement lies in the noise space with finite noise cost and whose induced coupling is noise-optimal), uniquely noise-mapped pairs, and the noise map property of a set of mea…Strong and Weak Convergence of Noise Fields Along Couplings of Vanishing Noise Cost
definitiondef:noise-field-convergence-hilbert-2026aAnalysisProbabilityDefines sequences of couplings of vanishing noise cost, and strong and weak convergence of noise fields along them, via the discrepancy and the cross pairing.Completeness of the Quadratic Wasserstein Space and of the Noise Wasserstein Space over a Hilbert Space
theoremthm:noise-wasserstein-complete-hilbert-2026aAnalysisProbabilityThe quadratic Wasserstein space over the Hilbert space is complete, and so is the noise Wasserstein space based at the reference measure.Differentiability of a Function on the Noise-Connected Measures Along Noise Couplings, and Its Gradient
definitiondef:noise-differentiable-along-couplings-hilbert-2026aAnalysisProbabilityA real function on the measures noise-connected to the reference measure is differentiable along noise couplings at a measure if its increments are approximated, to first order in the square root of the noise cost, by the noise displacement pairing with a noise field; that field…Cauchy Sequences in the Quadratic Wasserstein Space and Weakly Convergent Sequences on a Hilbert Space are Tight
lemmalem:w2-cauchy-tight-hilbert-2026aAnalysisProbabilityEvery Cauchy sequence for the quadratic Wasserstein distance of probability measures with finite second moment on a separable Hilbert space is a tight sequence of measures.Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion
lemmalem:noise-displacement-pairing-hilbert-2026aAnalysisProbabilityAlong a coupling of finite noise cost the displacement is a square-integrable noise field whose squared norm is the noise cost; pairing it with a noise field gives a bounded linear functional, computed explicitly along displacement couplings. Also: the cross pairing of two noise…