Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
Basic Properties of a Noise-Closed Noise Penalty Pair: Lower Bound, Lower Semicontinuity, Complete Sublevel Sets and Bounded Distances
lemmalem:noise-closed-penalty-pair-basic-hilbert-2026bAnalysisProbabilityFor a noise-closed noise penalty pair the penalty is bounded below and lower semicontinuous in the noise Wasserstein distance, each sublevel set is a complete metric space, and noise Wasserstein distances on a sublevel set are at most twice its bound on the distance to the refere…The Discounted Hamilton-Jacobi Equation with a Penalty Drift on the Noise Wasserstein Space
equationeq:noise-penalty-drift-hamilton-jacobi-hilbert-2026bAnalysisProbabilityPDEThe first-order discounted Hamilton-Jacobi equation lambda_0 r + (theta/2)|q|^2 + <Sigma(nu), q> = g(nu) on the score domain of a noise penalty pair, with the score of the pair acting as a drift; its viscosity solutions are taken relative to the pair.Lambda-Displacement Convexity of a Noise Penalty Pair
definitiondef:noise-displacement-convex-pair-hilbert-2026bAnalysisProbabilityA noise penalty pair is lambda-displacement convex if the penalty lies above its first-order expansion along every noise-optimal coupling out of a score-domain measure, with a lambda/2 times noise-cost correction.Viscosity Subsolution, Supersolution and Solution of a First-Order Equation on the Noise Wasserstein Space Relative to a Noise Penalty Pair
definitiondef:noise-viscosity-solution-hilbert-2026bAnalysisProbabilityPDEA function on the penalty domain is a viscosity subsolution of a first-order operator relative to a noise penalty pair if, wherever its lower delta-envelope minus a noise intrinsic test function has a local maximum, the lower delta-shift of the operator is almost nonpositive at a…First-Order Equation Operators on the Noise Wasserstein Space with Momentum-Continuous Shifts
definitiondef:noise-momentum-continuous-shifts-hilbert-2026bAnalysisProbabilityPDEA first-order operator has momentum-continuous shifts relative to a noise penalty pair if, on data with bounded score, real argument and field norm, its delta-shifts are uniformly continuous in the noise field.The First-Order Structure Condition at Uniquely Noise-Mapped Pairs
definitiondef:noise-structure-condition-hilbert-2026bAnalysisProbabilityPDEA first-order operator satisfies the structure condition at uniquely noise-mapped pairs if the difference of its shifts, evaluated at two score-domain measures with momenta given by alpha times their optimal displacements towards each other, is bounded below by moduli of alpha ti…The Shift-Semicontinuity Condition for a First-Order Equation Operator on the Noise Wasserstein Space
definitiondef:noise-shift-semicontinuity-hilbert-2026bAnalysisProbabilityPDEA first-order operator satisfies the shift-semicontinuity condition if, along test data converging along couplings of vanishing noise cost with fields converging strongly and bounded scores converging weakly, the lower shift is upper semicontinuous and the upper shift lower semic…The Shift-Coercivity Condition for a First-Order Equation Operator on the Noise Wasserstein Space
definitiondef:noise-shift-coercivity-hilbert-2026bAnalysisProbabilityPDEA first-order operator satisfies the shift-coercivity condition relative to a noise penalty pair if, for every delta in (0,1) and every R, the scores of the measures in admissible test data are bounded.Locally Strictly Proper First-Order Equation Operators on the Noise Wasserstein Space
definitiondef:noise-locally-strictly-proper-hilbert-2026bAnalysisProbabilityPDEA first-order operator on the noise Wasserstein space is locally strictly proper if on every bounded range of its real argument it increases in that argument at least at a positive linear rate.Test Data for a First-Order Equation Operator on the Noise Wasserstein Space and the Admissible Sets
definitiondef:noise-test-data-hilbert-2026bAnalysisProbabilityPDETest data for a first-order operator over the score domain are triples of a measure, a real number and a noise field; a datum is R-bounded when the distance to the reference, the penalty, the real number and the field norm are all below R, and the admissible sets collect R-bounde…Noise Penalty Pairs with Regular Penalised Maxima
definitiondef:noise-regular-penalised-maxima-hilbert-2026bAnalysisProbabilityA noise penalty pair has regular penalised maxima if every local maximum on the penalty domain of a noise intrinsic test function minus a positive multiple of the penalty lies in the score domain.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.