Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
Classical Sub- and Supersolutions of a Second-Order Equation on the Wasserstein Space
definitiondef:classical-sub-supersolution-wasserstein-2026cAnalysisProbabilityAn intrinsic test function on a set of measures is a classical subsolution, supersolution or solution of a second-order equation operator there when the operator, evaluated at the measure together with the value, the gradient along couplings and the translation Hessian of the fun…Degenerate Elliptic Second-Order Equation Operators on the Wasserstein Space
definitiondef:degenerate-elliptic-wasserstein-2026aAnalysisProbabilityA second-order equation operator on the Wasserstein space is degenerate elliptic when it is nonincreasing in its matrix argument for the positive semidefinite order.Basic Properties of the Delta-Envelopes on the Wasserstein Space
lemmalem:delta-envelopes-basic-wasserstein-2026bAnalysisProbabilityThe minus delta-envelope is upper semicontinuous and dominates the penalised function, and the plus envelope is dually placed; subordinate growth and the envelopes pass to the negative of a function with the two sides exchanged; for a continuous function and a lower semicontinuou…First- and Second-Order Conditions at a Penalised Extremum of a Test Function on the Wasserstein Space
lemmalem:penalised-maximiser-wasserstein-2026aAnalysisProbabilityAt a point of the score domain at which a test function minus a positive multiple of the penalty has a local maximum on the penalty domain, the intrinsic gradient equals that multiple of the score and the translation Hessian is negative semidefinite; the mirror statement holds at…First Variations of a Test Function on the Wasserstein Space Along Gradient Displacements and Along Translations
lemmalem:test-function-first-variation-wasserstein-2026aAnalysisProbabilityA translation moves a measure a Wasserstein distance at most the length of the shift. Along a displacement by the gradient of a compactly supported smooth function a test function is differentiable at time zero, with derivative the pairing of its intrinsic gradient with that grad…Sums, Real Multiples and Differences of Test Functions on the Wasserstein Space
lemmalem:test-function-wasserstein-linear-2026aAnalysisProbabilitySums, real multiples and differences of test functions on the Wasserstein space are again test functions, and the intrinsic gradient and the translation Hessian depend linearly on the test function.The Discrepancy of Two Square-Integrable Vector Fields Along a Coupling of Their Base Measures
lemmalem:coupling-field-distance-wasserstein-2026aAnalysisProbabilityFor a coupling of two measures with finite second moment and square-integrable vector fields against each of them, the integral of the squared distance between the two fields evaluated at the two coordinates is a finite nonnegative number independent of the representatives; along…The Score of a Penalty Pair is Determined by the Penalty
lemmalem:penalty-pair-score-unique-2026cAnalysisProbabilityAt each point of the score domain, the score of a penalty pair is the only tangent vector whose inner products with gradients of test functions are the first variations of the penalty; hence two penalty pairs with the same domains and the same penalty have the same score.The Push-Forward of a Probability Measure with Finite Second Moment by the Identity Perturbed along the Gradient of a Test Function: Borel, Finite Second Moment, the Diagonal Coupling and the Wasserstein Bound
lemmalem:gradient-push-forward-wasserstein-2026aAnalysisProbabilityFor a probability measure with finite second moment and a test function, the push-forward by the identity plus a multiple of the gradient is again a measure with finite second moment, at Wasserstein distance at most the multiple times the L^2-norm of the gradient; this is the def…The Dyson Hamilton-Jacobi Equation with Common Noise on the Wasserstein Space of the Real Line
equationeq:dyson-hamilton-jacobi-wasserstein-2026cAnalysisProbabilityThe Dyson Hamilton-Jacobi equation with common noise on the Wasserstein space of the real line, posed over the measures of finite free Fisher information: discount times the value, minus half the common-noise intensity times the trace of the matrix, plus half the squared norm of…The Discounted Hamilton-Jacobi Equation with Common Noise and Score Drift on the Wasserstein Space
equationeq:hamilton-jacobi-score-drift-wasserstein-2026cAnalysisProbabilityThe discounted Hamilton-Jacobi equation with common noise and score drift on the Wasserstein space, posed over the measures of finite Fisher information: discount times the value, minus half the common-noise intensity times the trace of the matrix, plus half the control cost time…The Discounted Hamilton-Jacobi Equation with Common Noise and a Penalty Drift on the Wasserstein Space
equationeq:hamilton-jacobi-penalty-drift-wasserstein-2026dAnalysisProbabilityThe discounted Hamilton-Jacobi equation with common noise and penalty drift on the Wasserstein space: discount times the value, minus half the trace of the common-noise matrix Gram product against the matrix variable, plus half the control cost times the squared norm of the vecto…Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space
definitiondef:viscosity-sub-supersolution-wasserstein-2026gAnalysisProbabilityA function on the penalty domain with penalty-subordinate growth from above is a viscosity subsolution relative to a penalty pair if, for every penalty weight delta between 0 and 1 and at every local maximum on the penalty domain of its penalised envelope minus an intrinsic test…Test Functions on the Wasserstein Space: the Intrinsic Gradient and the Translation Hessian
definitiondef:test-function-wasserstein-2026bAnalysisProbabilityA test function on the Wasserstein space is a continuous real function whose lift to the square-integrable random vectors is continuously Frechet differentiable, has a gradient of the form tangent vector field composed with the random vector, and is twice continuously differentia…The Bundle of Vector Fields over a Set of Measures, Second-Order Equation Operators on the Wasserstein Space, and Their Delta-Shifts
definitiondef:second-order-operator-wasserstein-2026bAnalysisProbabilityThe vector fields over a set of measures are the pairs of a measure in the set and a square-integrable vector field against it. A second-order equation operator over the set is a real function of such a pair, a real number and a symmetric d by d matrix. Relative to a penalty pair…The Delta-Envelopes of a Function on the Penalty Domain Relative to a Penalty Pair
definitiondef:delta-envelopes-wasserstein-2026bAnalysisProbabilityRelative to a penalty pair and a positive parameter, the delta-envelopes of a function on the penalty domain are the upper semicontinuous envelope of the function minus the penalty and the lower semicontinuous envelope of the function plus the penalty, for the Wasserstein metric;…A -Displacement Convex Penalty Pair Has a -Monotone Score Along Optimal Couplings
lemmalem:displacement-convex-pair-monotone-2026eAnalysisProbabilityFor a lambda-displacement convex penalty pair and two measures of the score domain, the displacement pairings of the two scores along an optimal coupling and its swap sum to at most minus lambda times the squared Wasserstein distance; at pairs uniquely mapped in both directions t…-Displacement Convexity of a Penalty Pair on the Wasserstein Space
definitiondef:displacement-convex-penalty-pair-2026eAnalysisProbabilityA penalty pair is lambda-displacement convex if, along every optimal coupling from a measure of the score domain to a measure of the penalty domain, the penalty lies above its first-order expansion given by the displacement pairing of the score, plus lambda/2 times the transport…Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains
definitiondef:penalty-pair-wasserstein-2026cAnalysisProbabilityA penalty pair on the Wasserstein space is a penalty domain, a score domain inside it, a real-valued penalty on the former and a score in the tangent space at each point of the latter, such that the score domain is nonempty and dense, the penalty is bounded below by a multiple of…Translations on a Space of Square-Integrable Random Vectors: Constant Classes, Law Invariance, the Translation Derivative and the Translation Hessian
lemmalem:translation-lift-2026aAnalysisProbabilityThe constant classes form a linear isometry of Euclidean space into the space of square-integrable random vectors, so translation by a constant is an isometry of the parameter into that space. For a law-invariant function the translated function depends only on the law; if the fu…