Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
Perturbed Maximisers of the Doubled Difference Penalised by the Wall-Confined Free Energy
lemmalem:nc-penalised-doubling-perturbed-maximiser-2026bAnalysisPDEFor bounded u and v the penalised doubled difference has the usual bounds and monotonicity; when u minus delta times the free energy is Wasserstein upper semicontinuous and v plus delta times the free energy lower semicontinuous on the domain, after a small series of squared-dist…The Structure Condition and Uniform Continuity on Bounded Sets Imply Their Forms at Bounded Positions
lemmalem:nc-hamiltonian-bounded-conditions-2026aAnalysisPDEA Hamiltonian satisfying the structure condition satisfies it at bounded positions, and a Hamiltonian uniformly continuous on bounded sets is uniformly continuous in the momentum at bounded positions.The Structure Condition at Bounded Positions for a Hamiltonian on Phase-Space Noncommutative Laws
definitiondef:nc-hamiltonian-structure-condition-bounded-2026aAnalysisPDEThe structure condition required only at pairs of positions whose laws have a given norm bound: exchanging the positions under the doubling momentum raises the Hamiltonian by little when the doubling cost is small.Hamiltonians on Phase-Space Noncommutative Laws that are Uniformly Continuous in the Momentum at Bounded Positions
definitiondef:nc-hamiltonian-momentum-continuous-2026aAnalysisPDEA Hamiltonian is uniformly continuous in the momentum at bounded positions if, for positions whose law has a given norm bound and momenta in a given L2 ball, small L2 changes of the momentum change its value uniformly little.Uniqueness of a Continuous Viscosity Solution of the Mean-Field Control Equation on the Torus
corollarycor:torus-control-uniqueness-2026aProbabilityPDEFor the control Hamiltonian of a control datum, a continuous viscosity subsolution lies below a continuous viscosity supersolution, so the mean-field control equation has at most one continuous viscosity solution.The Discounted Hamilton-Jacobi-Bellman Equation of Mean-Field Control with Idiosyncratic Noise on the Torus
equationeq:torus-control-hjb-2026aProbabilityPDEThe discounted Hamilton-Jacobi-Bellman equation on the torus Wasserstein space with the control Hamiltonian of a control datum.The Control Hamiltonian is Continuous Along Couplings and Satisfies the Score-Perturbation Bound and the Structure Condition
lemmalem:torus-control-hamiltonian-conditions-2026aProbabilityPDEThe control Hamiltonian of a control datum satisfies all three conditions of the torus comparison theory, with explicit constants in the score-perturbation bound.Control Data and the Control Hamiltonian on the Torus Wasserstein Space
definitiondef:torus-control-hamiltonian-2026aProbabilityPDEA control datum is a bounded drift and a bounded running cost on the torus that are Lipschitz in the position and the measure. Its control Hamiltonian integrates half the squared momentum minus the drift times the momentum minus the running cost.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.Existence and Uniqueness of a Bounded Continuous Plan-Jet Viscosity Solution of the Linear-Quadratic Hamilton-Jacobi Equation on Square-Integrable Noncommutative Laws
corollarycor:nc-lq-existence-2026aAnalysisPDEThe linear-quadratic Hamilton-Jacobi equation on square-integrable noncommutative laws has exactly one bounded continuous plan-jet viscosity solution.Perron's Method for Plan-Jet Viscosity Solutions of the Discounted Hamilton-Jacobi Equation on Square-Integrable Noncommutative Laws: Existence and Well-Posedness
theoremthm:nc-plan-perron-existence-2026aAnalysisPDEPerron's method: between a bounded usc subsolution and a bounded lsc supersolution there is a bounded continuous plan-jet viscosity solution; if H(X,0) is bounded, the equation has exactly one bounded continuous solution.