Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
The Bump Construction on the Wasserstein Space: the Local Maximum of a Viscosity Subsolution and a Penalised Intrinsic Test Function
lemmalem:perron-bump-wasserstein-2026cAnalysisProbabilityPDEInside a Wasserstein ball, replace a viscosity subsolution v on the penalty domain by the maximum of v and an intrinsic test function, bounded above on the ball, minus a multiple of the penalty, which stays below v near the edge of the ball. If the test function satisfies the shi…The Pointwise Supremum of a Uniformly Subordinate Family of Viscosity Subsolutions on the Wasserstein Space is a Viscosity Subsolution
lemmalem:sup-of-subsolutions-wasserstein-2026cAnalysisProbabilityPDEFor a Wasserstein-closed penalty pair whose penalty domain has the map property, the pointwise supremum of a nonempty family of viscosity subsolutions on the penalty domain that is uniformly subordinate from above (for each positive delta, one constant plus delta times the penalt…Penalty Pairs with Regular Penalised Maxima
definitiondef:regular-penalised-maxima-wasserstein-2026bAnalysisProbabilityPDEA penalty pair has regular penalised maxima when every local maximiser, on the penalty domain, of an intrinsic test function minus a positive multiple of the penalty lies in the score domain.The Structure Estimate at a Maximiser of the Wasserstein-Doubled Difference
lemmalem:comparison-estimate-wasserstein-2026dAnalysisPDEFor a bounded-above viscosity subsolution and a bounded-below viscosity supersolution on the penalty domain, a nonnegative supremum of the Wasserstein-doubled difference of their delta-envelopes is attained at a pair in the score domain at which the properness constant times the…Intrinsic Test Functions at a Maximiser of the Wasserstein-Doubled Difference
lemmalem:doubling-test-functions-wasserstein-2026cAnalysisPDEAt a maximiser of the Wasserstein-doubled difference of the delta-envelopes of two bounded functions over a penalty domain with the map property, there is a maximising pair and a pair of matrices admitted by Ishii's lemma such that each side is touched, arbitrarily close by, by i…A Comparison Principle for Viscosity Solutions on the Wasserstein Space
theoremthm:comparison-wasserstein-2026dAnalysisProbabilityPDEFor a Wasserstein-coercive penalty pair with closed score along couplings whose penalty domain has the map property, and an operator that is locally strictly proper and satisfies the shift-coercivity, shift-semicontinuity and second-order structure conditions: if u is a viscosity…Existence, Penalty Bounds and Monotonicity of the Maximum of the Wasserstein-Doubled Difference of Bounded Functions
lemmalem:w2-doubling-maximiser-wasserstein-2026bAnalysisPDEFor a Wasserstein-coercive penalty pair and functions bounded above and below on the penalty domain, the Wasserstein-doubled difference of their delta-envelopes attains its supremum; the penalty is bounded by a constant over delta wherever that difference is nonnegative; and the…The Second-Order Structure Condition at Uniquely Mapped Pairs on the Wasserstein Space
definitiondef:second-order-structure-condition-wasserstein-2026bAnalysisPDEThe intrinsic twin of the structure condition on the lift: at pairs of measures uniquely mapped in both directions, with the doubling momenta given by the optimal displacements and a pair of matrices admitted by Ishii's lemma, the difference of the shifted operators is bounded be…The Shift-Semicontinuity Condition for an Equation Operator on the Wasserstein Space
definitiondef:shift-semicontinuity-condition-wasserstein-2026cAnalysisPDEAn intrinsic equation operator is shift-semicontinuous if its lower shift is lower semicontinuous and its upper shift upper semicontinuous along bounded test data converging along couplings of vanishing cost, the fields strongly and the scores weakly.The Shift-Coercivity Condition for an Equation Operator on the Wasserstein Space
definitiondef:shift-coercivity-condition-wasserstein-2026bAnalysisPDEAn intrinsic equation operator satisfies the shift-coercivity condition if, for every shift size and every level, the scores of the measures carrying admissible test data are uniformly bounded.Locally Strictly Proper Second-Order Equation Operator on the Wasserstein Space
definitiondef:locally-strictly-proper-wasserstein-2026aAnalysisPDEAn intrinsic equation operator on the Wasserstein space is locally strictly proper if on every bounded range of the value argument it increases in that argument at least linearly with a positive rate, uniformly in the measure, the field and the matrix.Test Data for an Intrinsic Second-Order Equation Operator on the Wasserstein Space and the Admissible Sets
definitiondef:test-data-wasserstein-2026bAnalysisProbabilityPDEThe test data of an intrinsic second-order equation operator are a measure in the score domain, a vector field against it, a real number and a symmetric matrix; bounded test data and the admissible sets on which the structure conditions are imposed are defined from them.Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians
definitiondef:intrinsic-test-function-wasserstein-2026cAnalysisProbabilityPDEA function on the Wasserstein space is an intrinsic test function on a set of measures when it is continuous, differentiable along couplings at each point of the set with a tangent gradient, its gradient is continuous along couplings of vanishing cost within the set, and it is tw…The Intrinsic Calculus on the Wasserstein Space: Standing Notation
settingset:wasserstein-intrinsic-2026bAnalysisProbabilityPDEStanding notation for the intrinsic theory of second-order equations on the Wasserstein space: couplings and the displacement pairing, optimal maps and the map property, differentiability along couplings, and the operator and penalty vocabulary. No probability space is assumed to…A Comparison Principle for Viscosity Solutions on the Lift of the Wasserstein Space
theoremthm:comparison-lift-wasserstein-2026aAnalysisProbabilityPDEFor a Wasserstein-coercive penalty pair with closed score and an operator that is locally strictly proper and satisfies the shift-coercivity, shift-semicontinuity and second-order structure conditions, a bounded viscosity subsolution on the lift lies below a bounded viscosity sup…The Structure Estimate at a Maximiser of the Wasserstein-Doubled Difference on the Lift
lemmalem:comparison-estimate-lift-wasserstein-2026aAnalysisProbabilityPDEAt a maximiser of the Wasserstein-doubled difference of the delta-envelopes, the optimally coupled lift lies in the score domain, and the maximal value is bounded by the two moduli of a second-order structure pair evaluated at the penalised distance and at the scaled penalty.The Doubled Difference on the Lift of a Wasserstein-Coercive Penalty Pair: Bounds, Closed Superlevel Sets and the Least Penalty
lemmalem:doubled-difference-lift-wasserstein-2026aAnalysisProbabilityPDEFor a Wasserstein-coercive penalty pair, the penalty attains a least value, and the lifts of the delta-envelopes of a bounded upper semicontinuous function and a bounded lower semicontinuous function are bounded and have closed superlevel sets, as does the quadratically doubled d…Lifted Test Data from Second-Order Data on the Space of Square-Integrable Random Vectors: Compression along the Constant Tuple
lemmalem:compression-lifted-test-data-wasserstein-2026aAnalysisProbabilityPDEEvery function of class on the lift is a lifted test function whose translation Hessian is the compression of its Hilbert Hessian along the constant tuple; quadruples approximable by test data therefore produce lifted test functions with compressed matrices, and the second-…Lions' Lemma on a Hilbert Space over a Translation-Closed Set: Test Data and Form Bounds at a Sequentially Strict Maximum of a Quadratically Penalised Difference
lemmalem:lions-doubling-translation-closed-hilbert-2026aAnalysisPDEThe infinite-dimensional counterpart of Ishii's lemma, carried from a linear subspace to any set closed under the translations of a finite orthonormal tuple: at a sequentially strict maximum of a quadratically penalised difference, both functions admit test data whose second-orde…Existence, Penalty Bounds and Optimal Realisation at a Maximiser of the Wasserstein-Doubled Difference on the Lift
lemmalem:doubling-maximiser-lift-wasserstein-2026bAnalysisPDEFor a Wasserstein-coercive pair the Wasserstein-doubled difference of the delta-envelopes attains its supremum, the penalty and second moment are bounded at a near-maximiser independently of the doubling strength, the penalised distance vanishes along a doubling sequence, and the…