Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
On Integrals of Cylindrical Functions the Gaussian Score Drift is the Noise Ornstein-Uhlenbeck Functional, and the Score-Drift Hamilton-Jacobi Operator Takes Its Pointwise Form
lemmalem:gaussian-score-drift-cylindrical-integral-hilbert-2026aProbabilityPDEOn integrals of bounded C^2 cylindrical functions, the score drift of the Gaussian entropy pair equals beta times the noise Ornstein-Uhlenbeck functional, so the Hamilton-Jacobi operator with Gaussian score drift takes an explicit pointwise form.The Hamilton-Jacobi Equation with Gaussian Score Drift on a Hilbert Space
equationeq:gaussian-score-drift-hamilton-jacobi-hilbert-2026aProbabilityPDEThe discounted Hamilton-Jacobi equation on laws of finite relative entropy with respect to a diagonal Gaussian measure on a Hilbert space, whose drift term pairs the momentum with beta times the noise score field.Perron's Method on the Noise Wasserstein Space: Existence of a Viscosity Solution of a First-Order Equation Between a Subsolution and a Supersolution
theoremthm:noise-perron-existence-hilbert-2026bAnalysisProbabilityPDEFor a noise-closed noise penalty pair with regular penalised maxima whose penalty domain has the noise map property, and any first-order equation operator, the pointwise supremum of all viscosity subsolutions lying between a given subsolution and a given supersolution above it, b…Well-Posedness of the Hamilton-Jacobi Equation with Penalty Drift on the Noise Wasserstein Space: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution
theoremthm:noise-penalty-drift-well-posed-hilbert-2026bAnalysisProbabilityPDEFor a displacement convex, noise-closed noise penalty pair with closed score and regular penalised maxima whose penalty domain has the noise map property and whose penalty controls the squared distance to the reference measure, and for a bounded uniformly continuous running cost…Constant Viscosity Subsolutions and Supersolutions of the Hamilton-Jacobi Equation with Penalty Drift on the Noise Wasserstein Space
lemmalem:noise-constant-sub-supersolutions-hilbert-2026bAnalysisProbabilityPDEFor a noise-closed noise penalty pair with regular penalised maxima, a constant function has penalty-subordinate growth from both sides; it is a viscosity subsolution of the penalty-drift Hamilton-Jacobi equation when discount times the constant is at most the running cost on the…The Hamilton-Jacobi Operator with Penalty Drift on the Noise Wasserstein Space Satisfies the Hypotheses of the First-Order Comparison Principle for a Displacement Convex Noise Penalty Pair
lemmalem:noise-penalty-drift-hypotheses-hilbert-2026bAnalysisProbabilityPDEFor a displacement convex noise penalty pair with lower semicontinuous penalty whose squared noise Wasserstein distance to the reference measure is controlled by the penalty, control cost at most one and a bounded uniformly continuous running cost, the penalty-drift Hamilton-Jaco…A Comparison Principle for First-Order Equations on the Noise Wasserstein Space Relative to a Noise-Closed Penalty Pair
theoremthm:noise-comparison-first-order-hilbert-2026bAnalysisProbabilityPDEComparison without compactness on the noise Wasserstein space: for a noise-closed penalty pair with closed score whose domain has the noise map property, and a first-order operator that is locally strictly proper, shift-coercive, shift-semicontinuous, satisfies the structure cond…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 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.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…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…