Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
Integrals of Lipschitz Semiconvex Viscosity Subsolutions and Lipschitz Semiconcave Viscosity Supersolutions of the Ornstein-Uhlenbeck Hamilton-Jacobi Equation Are Viscosity Sub- and Supersolutions of the Lifted Equation
propositionprop:semiconvex-lift-ou-hamilton-jacobi-euclidean-2026aAnalysisProbabilityPDEIntegrating a bounded Lipschitz semiconvex viscosity subsolution of the pointwise Ornstein-Uhlenbeck Hamilton-Jacobi equation against laws of finite relative entropy gives a viscosity subsolution of the lifted equation with the integrated running cost; semiconcave supersolutions…Penalised Extrema of the Integral of a Lipschitz Semiconvex Function Relative to the Gaussian Free-Energy Pair: Finite Relative Fisher Information and the First-Order Condition
lemmalem:semiconvex-integral-penalised-extrema-gaussian-euclidean-2026aAnalysisProbabilityIf the integral of a bounded Lipschitz semiconvex function, minus a multiple of the Gaussian free energy and an intrinsic test function, has a local maximum at a law, then that law has finite Fisher information relative to the Gaussian and the gradient of the function equals the…The Almost-Everywhere Gradient of a Lipschitz Semiconvex or Semiconcave Function Is Tangent at Every Absolutely Continuous Measure
lemmalem:semiconvex-gradient-tangent-wasserstein-2026aAnalysisProbabilityThe almost-everywhere gradient of a bounded Lipschitz function is a bounded Borel field defined off a null set of every absolutely continuous measure, and when the function is semiconvex or semiconcave this gradient lies in the Wasserstein tangent space at every absolutely contin…The Wick-Square Potential Relative to a Diagonal Gaussian Measure
definitiondef:wick-square-potential-diagonal-gaussian-euclidean-2026aAnalysisProbabilityThe Wick-square potential with variances c and couplings beta is the sum over coordinates of beta_i times x_i squared minus c_i: each square with its mean under the diagonal Gaussian measure removed.Changing the Variances of a Diagonal Gaussian Reference Measure: Equal Domains, the Shift of the Relative Entropy and the Shift of the Relative Score
lemmalem:change-of-variances-diagonal-gaussian-euclidean-2026aAnalysisProbabilityTwo diagonal Gaussian reference measures give the same measures of finite relative entropy and of finite relative Fisher information; the relative entropies differ by half the integral of the difference of the weighted squares plus a constant, and the relative scores differ by th…Gaussian Integration by Parts Relative to a Diagonal Gaussian Measure for Functions with a Bounded Hessian
lemmalem:gaussian-integration-by-parts-diagonal-euclidean-2026aAnalysisProbabilityFor a measure of finite Fisher information relative to a diagonal Gaussian, the pairing of its relative score with the gradient of a function with bounded Hessian equals minus the integral of the Ornstein-Uhlenbeck generator applied to the function.Well-Posedness of the Lifted Ornstein-Uhlenbeck Hamilton-Jacobi Equation Relative to a Diagonal Gaussian Measure: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution
theoremthm:lifted-ou-hamilton-jacobi-well-posed-euclidean-2026aAnalysisProbabilityPDEFor a bounded, Wasserstein-uniformly continuous running cost and control cost theta in (0,1], the lifted Ornstein-Uhlenbeck Hamilton-Jacobi equation relative to a diagonal Gaussian satisfies comparison and has exactly one bounded viscosity solution on the finite-entropy measures,…A Convergent Series of Squared Wasserstein Distances to Measures of Bounded Second Moment is an Intrinsic Test Function
lemmalem:w2-squared-series-intrinsic-test-function-wasserstein-2026aAnalysisProbabilityA convergent positive series of squared Wasserstein distances to measures with uniformly bounded second moments defines an intrinsic test function on any set with the map property; its gradient is the corresponding series of displacement fields, bounded by twice the series of dis…The Gaussian Free-Energy Pair is Wasserstein-Closed
lemmalem:gaussian-free-energy-pair-w2-closed-euclidean-2026aAnalysisProbabilityThe Gaussian free-energy pair (relative entropy with respect to a diagonal Gaussian, with the relative score) is Wasserstein-closed, although it is not Wasserstein-coercive.Basic Properties of a Wasserstein-Closed Penalty Pair: Lower Bound, Lower Semicontinuity, Complete Sublevel Sets, Bounded Distances, and Coercive Pairs
lemmalem:w2-closed-penalty-pair-basic-wasserstein-2026aAnalysisProbabilityFor a Wasserstein-closed penalty pair the penalty is bounded below and lower semicontinuous, each sublevel set is a complete metric space on which squared Wasserstein distances are bounded by four times the moment bound, and every Wasserstein-coercive pair is Wasserstein-closed.The Lifted Ornstein-Uhlenbeck Hamilton-Jacobi Equation Relative to a Diagonal Gaussian Measure
equationeq:lifted-ou-hamilton-jacobi-euclidean-2026aAnalysisProbabilityPDEThe lifted Ornstein-Uhlenbeck Hamilton-Jacobi equation: the penalty-drift Hamilton-Jacobi equation of the Gaussian free-energy pair without common noise, lambda r + (theta/2)|q|^2 + a<relative score, q> = g, with viscosity solutions relative to the Gaussian free-energy pair.Wasserstein-Closed Penalty Pairs
definitiondef:w2-closed-penalty-pair-wasserstein-2026aAnalysisProbabilityA penalty pair is Wasserstein-closed if each sublevel set of its penalty is closed in the Wasserstein space (limits of sequences with penalty at most c stay in the domain with penalty at most c) and has bounded second moments. It weakens Wasserstein-coercivity, which asks for com…The Diagonal Gaussian Density on Euclidean Space and Its Notation
definitiondef:diagonal-gaussian-density-euclidean-2026aAnalysisProbabilityFor a vector of positive variances, fixes the scaling map x -> (x_i/c_i), the weighted square sum x_i^2/c_i, the log-normalizer, and the diagonal Gaussian density exp(-weighted square/2 - log-normalizer) on R^d.Completeness of the Quadratic Wasserstein Space over Euclidean Space
theoremthm:wasserstein-space-complete-euclidean-2026aAnalysisProbabilityThe space of probability measures on R^d with finite second moment, with the quadratic Wasserstein distance, is a complete metric space.The Gaussian Free-Energy Pair is Uniformly Displacement Convex, with Closed Score Along Couplings and Regular Penalised Maxima
lemmalem:gaussian-free-energy-pair-regularity-euclidean-2026aAnalysisProbabilityThe Gaussian free-energy pair is (a/c_max)-displacement convex, has closed score along couplings, and has regular penalised maxima.The Gaussian Free-Energy Pair is a Penalty Pair: Translations, the First Variation, Lower Semicontinuity, the Moment Bound and the Map Property
lemmalem:gaussian-free-energy-penalty-pair-euclidean-2026aAnalysisProbabilityThe Gaussian free-energy pair is a penalty pair: translations change the penalty by an explicit quadratic, so its translation Hessian is diagonal with entries a/c_i; its first variation along gradients of test functions is minus the temperature times the Ornstein-Uhlenbeck functi…The Gaussian Free-Energy Pair: Relative Entropy and Relative Score with Respect to a Diagonal Gaussian Measure
definitiondef:gaussian-free-energy-pair-euclidean-2026aAnalysisProbabilityThe Gaussian free-energy pair has as penalty the temperature times the relative entropy with respect to a diagonal Gaussian, on measures of finite second moment and finite relative entropy, and as score the temperature times the relative score, on those of finite relative Fisher…The Entropy Inequality for Nonnegative Functions, and Quadratic Moments Bounded by the Relative Entropy with Respect to a Diagonal Gaussian Measure
lemmalem:entropy-quadratic-moment-diagonal-gaussian-euclidean-2026aAnalysisProbabilityThe integral of a nonnegative function is at most the relative entropy plus the logarithm of its exponential moment under the reference measure; relative to a diagonal Gaussian this bounds diagonal quadratic moments, and the second moment, linearly by the relative entropy with ex…Relative Entropy and Relative Score with Respect to a Diagonal Gaussian Measure: Comparison with the Entropy, the Score and the Relative Free Energy of a Quadratic Potential
lemmalem:diagonal-gaussian-relative-entropy-score-euclidean-2026aAnalysisProbabilityFor measures with finite second moment, relative entropy with respect to a diagonal Gaussian is finite exactly when the entropy is, and equals entropy plus half the weighted second moment plus the log-normalizer; the relative score is the score plus the scaling map; consequently…Finite Fisher Information Relative to a Diagonal Gaussian Measure: the Ornstein-Uhlenbeck Functional, the Relative Score and the Relative Fisher Information
definitiondef:relative-score-diagonal-gaussian-euclidean-2026aAnalysisProbabilityFor a measure with finite second moment, the Ornstein-Uhlenbeck functional integrates Laplacian minus scaling-map drift of test functions; the measure has finite Fisher information relative to the diagonal Gaussian when this functional is bounded by the L^2 norm of gradients, its…