Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
Comparison and Uniqueness for Continuous Viscosity Solutions of the Discounted Hamilton-Jacobi-Bellman Equation on the Torus Wasserstein Space
corollarycor:torus-comparison-uniqueness-2026aProbabilityPDEUnder continuity along couplings, the score-perturbation bound and the structure condition, a continuous viscosity subsolution lies below a continuous viscosity supersolution, so the equation has at most one continuous viscosity solution.Viscosity Sub- and Supersolutions on the Torus Wasserstein Space are Entropy-Penalised Viscosity Sub- and Supersolutions
theoremthm:torus-viscosity-penalised-bridge-2026aProbabilityPDEIf the Hamiltonian is continuous along couplings and satisfies the score-perturbation bound, every upper semicontinuous viscosity subsolution is an entropy-penalised viscosity subsolution, and likewise for lower semicontinuous supersolutions. The proof regularises the test functi…Comparison Principle for Entropy-Penalised Viscosity Solutions of the Discounted Hamilton-Jacobi-Bellman Equation on the Torus Wasserstein Space
theoremthm:torus-penalised-comparison-2026aProbabilityPDEIf the Hamiltonian satisfies the structure condition, an upper semicontinuous entropy-penalised subsolution lies below a lower semicontinuous entropy-penalised supersolution at every measure of finite entropy.Entropy-Penalised Viscosity Subsolutions, Supersolutions and Solutions of the Discounted Hamilton-Jacobi-Bellman Equation on the Torus Wasserstein Space
definitiondef:torus-penalised-viscosity-solution-2026aProbabilityPDESub- and supersolutions tested at strict extrema of the function minus an intrinsic test function and a small multiple of the entropy. The extremal measure must have finite Fisher information, and the score form of the equation must hold there up to a tolerance proportional to th…Viscosity Subsolutions, Supersolutions and Solutions of the Discounted Hamilton-Jacobi-Bellman Equation on the Torus Wasserstein Space
definitiondef:torus-viscosity-solution-2026aProbabilityPDEViscosity sub- and supersolutions tested by Laplacian test functions at global maxima and minima, with the Laplacian form of the equation holding as an inequality at the touching point.The Discounted Hamilton-Jacobi-Bellman Equation with Idiosyncratic Noise on the Torus Wasserstein Space
equationeq:torus-hjb-2026aProbabilityPDEThe discounted equation: discount rate times U, plus the Hamiltonian at the gradient, minus half the squared noise intensity times the Laplacian, equals zero. It is also recorded in score form, where the Laplacian is replaced by minus the pairing of the gradient with the score.The Structure Condition Along Optimal Maps for a Hamiltonian on the Torus Wasserstein Space
definitiondef:torus-hamiltonian-structure-2026aProbabilityPDEA Hamiltonian satisfies the structure condition if, at two absolutely continuous measures joined by optimal maps, its values at the scaled displacement momenta differ by an amount that is small when the scaled squared distance plus the distance is small.The Score-Perturbation Bound for a Hamiltonian on the Torus Wasserstein Space
definitiondef:torus-hamiltonian-score-perturbation-2026aProbabilityPDEA Hamiltonian satisfies the score-perturbation bound if a small multiple of a tangent field changes it by at most a quarter of the noise intensity squared times that multiple times the squared norm of the field, plus a term linear in the multiple that is uniform over bounded mome…Continuity Along Couplings of a Hamiltonian on the Torus Wasserstein Space
definitiondef:torus-hamiltonian-continuous-2026aProbabilityPDEA Hamiltonian is continuous along couplings if its values converge whenever measures with tangent fields converge along couplings.The Discounted Hamilton-Jacobi-Bellman Equation with Idiosyncratic Noise on the Torus Wasserstein Space: Standing Notation
settingset:torus-hjb-2026aProbabilityPDEStanding notation for the discounted Hamilton-Jacobi-Bellman equation on the torus Wasserstein space: a discount rate, a noise intensity, and a Hamiltonian defined on measures paired with tangent fields.Heat Regularisation of the Entropy: a Laplacian Test Function whose Laplacian is Minus the Fisher Information
lemmalem:torus-heat-regularised-entropy-2026aAnalysisProbabilityThe entropy of the heat-smoothed measure is a Laplacian test function of the measure. Its gradient is the heat average of the score of the smoothed measure, and its Laplacian is minus the Fisher information of the smoothed measure.Heat Regularisation of an Intrinsic Test Function: a Laplacian Test Function whose Laplacian is Minus the Pairing of the Gradient with the Score
lemmalem:torus-heat-regularised-test-function-2026aAnalysisProbabilityComposing an intrinsic test function on the absolutely continuous measures with the heat semigroup gives a Laplacian test function. Its gradient is the heat average of the gradient at the smoothed measure, and its Laplacian is minus the pairing of that gradient with the score. As…Heat Smoothing, Entropy, Fisher Information and Laplacian Test Functions on the Torus Wasserstein Space: Standing Notation
settingset:torus-heat-calculus-2026aAnalysisProbabilityStanding notation for the heat semigroup and heat kernel of the torus, heat averages of vector fields, the entropy and Fisher information, and intrinsic and Laplacian test functions with their gradients, gradient fields and Laplacians.The Laplacian of a Laplacian Test Function on the Torus Wasserstein Space
definitiondef:torus-laplacian-2026aAnalysisProbabilityThe Laplacian of a Laplacian test function at a measure is the integral against that measure of the divergence of its gradient field.Laplacian Test Functions on the Torus: Uniqueness of the Gradient Field, a Criterion, and Linearity of the Test Classes
lemmalem:torus-laplacian-test-function-basic-2026aAnalysisProbabilityA function has at most one gradient field. A continuous function with a tangent-valued gradient field is a Laplacian test function. Intrinsic and Laplacian test functions are closed under linear combinations, with linear gradients and gradient fields.Gradient Fields and Laplacian Test Functions on the Torus Wasserstein Space
definitiondef:torus-laplacian-test-function-2026aAnalysisProbabilityA gradient field of a function on the torus Wasserstein space is a periodic, continuously differentiable representative of its gradient along couplings that depends continuously on the measure together with its derivatives. A Laplacian test function is an intrinsic test function…Lower Semicontinuity of the Torus Fisher Information and Closedness of the Score Along Couplings
lemmalem:torus-fisher-closed-2026aAnalysisProbabilityIf measures of bounded torus Fisher information converge, the limit has Fisher information at most the bound, and pairings of fields converging along couplings with the scores converge to the pairing with the score of the limit.Convergence Along Couplings of Measures Carrying Square-Integrable Vector Fields on the Torus
definitiondef:torus-convergence-along-couplings-2026aAnalysisProbabilityA sequence of measures carrying vector fields converges along couplings to a limit measure and field if some couplings have vanishing torus cost and vanishing mean squared field mismatch.Heat Averages of Vector Fields on the Torus: Regularity, Contraction, Tangency, the Divergence Identity and Synchronous Couplings
lemmalem:torus-heat-average-2026aAnalysisProbabilityAveraging a square-integrable field of a heat-smoothed measure against the torus heat kernel gives a continuously differentiable periodic field. The average does not increase the norm and maps tangent fields to tangent fields; its divergence integrates to minus the pairing with t…The Torus Heat Kernel: Regularity and Bounds, the Density of a Heat-Smoothed Measure, and Its Lipschitz Dependence on the Measure
lemmalem:torus-heat-kernel-2026aAnalysisProbabilityThe torus heat kernel is twice continuously differentiable, even, positive and bounded with bounded derivatives. Convolving it with a measure gives the smooth positive density of the heat-smoothed measure, whose score is the logarithmic gradient of that density, and this density…