Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
The Bump Construction on the Noise Wasserstein Space: the Local Maximum of a Viscosity Subsolution and a Penalised Noise Intrinsic Test Function
lemmalem:noise-perron-bump-hilbert-2026bAnalysisProbabilityPDEInside a noise Wasserstein ball, replace a viscosity subsolution v by the maximum of v and a test function minus a multiple of the penalty, which stays below v near the edge of the ball. If the test function satisfies the shifted subsolution inequality wherever it wins, the resul…Exact Inequalities at a Touching Point of a Penalised Viscosity Subsolution or Supersolution on the Noise Wasserstein Space
lemmalem:noise-penalised-exact-jets-hilbert-2026bAnalysisProbabilityPDEFor a noise-closed penalty pair with closed score and a first-order operator satisfying shift-coercivity and shift-semicontinuity, a point where a noise intrinsic test function touches the delta-envelope of a penalised viscosity subsolution (or supersolution) lies in the score do…The Pointwise Supremum of a Uniformly Subordinate Family of Viscosity Subsolutions on the Noise Wasserstein Space is a Viscosity Subsolution
lemmalem:noise-sup-of-subsolutions-hilbert-2026bAnalysisProbabilityPDEFor a noise-closed noise penalty pair whose penalty domain has the noise map property, the pointwise supremum of a nonempty family of viscosity subsolutions of a first-order equation that is uniformly subordinate from above is again a viscosity subsolution.Bounds, Monotonicity and Borwein-Preiss Perturbed Maximisers of the Doubled Difference in the Noise Wasserstein Distance for a Noise-Closed Penalty Pair
lemmalem:noise-doubling-perturbed-maximiser-hilbert-2026bAnalysisProbabilityPDEFor a noise-closed penalty pair and bounded u, v, the difference of their delta-envelopes doubled with the squared noise Wasserstein distance has a finite supremum, monotone in the weight and in the doubling strength with slack, and a Borwein-Preiss perturbed maximiser: a near-ma…The First-Order Condition at a Penalised Extremum of a Noise Intrinsic Test Function on the Noise Wasserstein Space
lemmalem:noise-penalised-extremum-hilbert-2026bAnalysisProbabilityAt a point of the score domain at which a noise intrinsic test function minus a positive multiple of the penalty has a local maximum on the penalty domain, the gradient along noise couplings equals that multiple of the score; at a local minimum of the test function plus the penal…A Toolkit for Penalised Comparison on the Noise Wasserstein Space: Constant Test Functions and Linear Combinations of Test Functions, the Identity as a Unique Noise-Optimal Map, and Discrepancies Along the Push-Forward Under (id, id) and Along Glued Couplings
lemmalem:noise-penalty-toolkit-hilbert-2026bAnalysisProbabilityConstant functions are noise intrinsic test functions with zero gradient, and real linear combinations of noise intrinsic test functions are again such functions, with the corresponding linear combination of gradients; the identity is a noise-optimal map from a measure to itself…Basic Properties of the Delta-Envelopes on the Noise Wasserstein Space, and the Envelopes of Bounded Functions for a Noise-Closed Penalty Pair
lemmalem:noise-delta-envelopes-hilbert-2026bAnalysisProbabilityThe delta-envelopes relative to a noise penalty pair are semicontinuous and placed on the correct side of the penalised function, are exchanged by negation, are exact for continuous functions and a lower semicontinuous penalty, and inherit bounds by a smaller multiple of the pena…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.