Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
Cross-Level Comparison through One-Particle Marginals: N Times a Mean-Field Subsolution Lies below a Lifted N-Particle Supersolution under Cost Domination
theoremthm:n-particle-marginal-comparison-wasserstein-2026aAnalysisProbabilityPDEFor the Langevin Hamilton-Jacobi equations with common noise, if the configuration cost dominates N times the mean-field cost of the one-particle marginal, then N times a bounded-above viscosity subsolution of the mean-field equation, evaluated at the one-particle marginal, is at…Cross-Level Comparison at Tensor Powers: a Lifted N-Particle Subsolution Lies below N Times a Mean-Field Supersolution with the Tensor-Averaged Cost
theoremthm:n-particle-tensor-comparison-wasserstein-2026aAnalysisProbabilityPDEFor the Langevin Hamilton-Jacobi equations with common noise, a bounded-above viscosity subsolution of the lifted N-particle equation, evaluated at the tensor power of a one-particle law, is at most N times a bounded-below viscosity supersolution of the mean-field equation whose…Cross-Level Inequalities between the Shifted Lifted N-Particle and Mean-Field Langevin Operators at Tensor Powers and through One-Particle Marginals
lemmalem:n-particle-cross-level-operator-wasserstein-2026aAnalysisProbabilityPDEAt a tensor power, the shifted supersolution operator of the lifted N-particle equation dominates N times the mean-field one evaluated at the one-particle projection of the field, when the mean-field cost is the tensor-averaged cost. Along product fields, the lifted operator is d…The Langevin Free-Energy Pair across the Particle and Configuration Levels: Tensor Powers, One-Particle Marginals and Diagonal Translation Hessians
lemmalem:langevin-pair-tensor-marginal-wasserstein-2026aAnalysisProbabilityFor a confining potential, the configuration-level Langevin pair of the N-particle potential contains the tensor powers of the particle-level domains, with N times the penalty and the product score; it maps to the particle level under one-particle marginals, with the penalty supe…The Tensor-Averaged Cost of a Bounded Uniformly Continuous Cost is Bounded and Uniformly Continuous
lemmalem:tensor-averaged-cost-wasserstein-2026aAnalysisProbabilityIf a configuration-space cost is bounded by b and uniformly continuous, its tensor-averaged cost is bounded by b over N and uniformly continuous for the Wasserstein distance.The Tensor-Averaged Running Cost of a Configuration-Space Cost
definitiondef:tensor-averaged-cost-wasserstein-2026aAnalysisProbabilityFor a bounded Borel cost on the configuration space, the tensor-averaged cost of a one-particle law is one N-th of the integral of the cost against its N-th tensor power.Intrinsic Test Functions at a Maximiser of the Doubled Difference Linked through One-Particle Marginals
lemmalem:marginal-doubling-test-functions-wasserstein-2026aAnalysisProbabilityAt a maximiser of N times a particle-level envelope minus a configuration-level envelope minus half of N times a multiple of the squared distance to the one-particle marginal, there is a maximising pair near which both envelopes are touched by intrinsic test functions whose gradi…Intrinsic Test Functions at a Maximiser of the Doubled Difference Linked through Tensor Powers
lemmalem:tensor-doubling-test-functions-wasserstein-2026aAnalysisProbabilityAt a maximiser of a configuration-level envelope minus N times a particle-level envelope minus half a multiple of the squared distance from the configuration law to the tensor power, there is a maximising pair near which both envelopes are touched by intrinsic test functions whos…Existence, Penalty Bounds and Monotonicity of the Maximum of a Two-Space Doubled Difference with a Continuous Link
lemmalem:two-space-doubling-maximiser-wasserstein-2026aAnalysisProbabilityFor Wasserstein-coercive penalty pairs at the particle dimension or at the configuration level, a bounded-above function, a bounded-below function and a continuous nonnegative link, the weighted doubled difference of their delta-envelopes minus a multiple of the link attains its…Closure of Approximate Test Data for Viscosity Sub- and Supersolutions on the Wasserstein Space
lemmalem:viscosity-closure-wasserstein-2026aAnalysisProbabilityPDEIf a bounded viscosity subsolution (supersolution) is touched from above (below) by intrinsic test functions at points converging along couplings to a limit point, with gradients converging along those couplings to a field and translation Hessians converging to a matrix, then the…A Configuration-Level Majorant of the Distance from a One-Particle Marginal, Touching at a Given Configuration Law
lemmalem:marginal-distance-majorant-wasserstein-2026aAnalysisProbabilityPushing a configuration law forward by the product of an optimal map from its one-particle marginal to a target gives a configuration law whose squared distance majorises N times the squared distance of one-particle marginals to the target, with equality at the given law; the opt…Pulling Back Intrinsic Test Functions along Tensor Powers
lemmalem:tensor-pullback-test-function-wasserstein-2026aAnalysisProbabilityIf a function on the configuration-level Wasserstein space is an intrinsic test function on a set containing the tensor powers of a set of one-particle laws, then its composition with the tensor power is an intrinsic test function on that set; its gradient is N times the one-part…Classical Solutions of the N-Particle Hamilton-Jacobi Equation Lift to Classical Solutions of the Lifted Equation, and a Bounded One is the Lifted Viscosity Solution
theoremthm:n-particle-lift-classical-wasserstein-2026aAnalysisProbabilityPDEIf a function on the configuration space with bounded first and second derivatives is a classical sub- or supersolution of the N-particle Hamilton-Jacobi equation, then its integral against the measure is a classical sub- or supersolution of the lifted equation, the Laplacian of…Well-Posedness of the Lifted N-Particle Hamilton-Jacobi Equation: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution
corollarycor:n-particle-lifted-well-posed-wasserstein-2026aAnalysisProbabilityPDEFor a confining potential, positive noise, control cost at most one, any common-noise matrix and a bounded uniformly continuous running cost on the configuration space, the lifted N-particle Hamilton-Jacobi equation satisfies comparison and has exactly one bounded viscosity solut…The Lifted N-Particle Hamilton-Jacobi Equation on the Wasserstein Space of the Configuration Space
equationeq:n-particle-lifted-hamilton-jacobi-wasserstein-2026aAnalysisProbabilityPDEThe lift of the N-particle Hamilton-Jacobi equation to probability measures on the configuration space: the Langevin Hamilton-Jacobi equation with common noise at the configuration level, with the N-particle potential, the N-particle common-noise matrix and the integral of the ru…The N-Particle Hamilton-Jacobi Equation with Individual and Common Noise on the Configuration Space
equationeq:n-particle-hamilton-jacobi-euclidean-2026aAnalysisProbabilityPDEThe discounted Hamilton-Jacobi equation on the configuration space of N particles in a confining potential: individual noise enters as the full Laplacian, common noise as the trace of the Hessian against the common-noise matrix whose rows move every particle by the same vector, w…Running Costs for N-Particle Systems: Integrals of Bounded Uniformly Continuous Functions and Costs of the Empirical Measure
lemmalem:n-particle-running-cost-wasserstein-2026aAnalysisProbabilityIntegrating a bounded uniformly continuous function against a measure gives a bounded uniformly continuous function on the Wasserstein space, with the same bound; and a bounded uniformly continuous function of the empirical measure of a configuration is a bounded uniformly contin…The N-Particle Potential of a Confining Potential is a Confining Potential on the Configuration Space
lemmalem:n-particle-potential-euclidean-2026aAnalysisProbabilitySumming a confining potential over the N particles of a configuration gives a twice continuously differentiable function on the configuration space whose gradient is the configuration of the particle gradients and whose Laplacian is the sum of the particle Laplacians, and this N-…Integrals of Functions with a Bounded Hessian are Intrinsic Test Functions on the Wasserstein Space
lemmalem:linear-functional-intrinsic-wasserstein-2026bAnalysisProbabilityFor a twice continuously differentiable function with bounded second derivatives, hence of at most quadratic growth, the integral of the function against a measure is an intrinsic test function on the whole Wasserstein space; its gradient along couplings is the gradient of the fu…N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level
settingset:n-particle-wasserstein-2026bAnalysisProbabilityStanding notation for systems of N particles in d-dimensional space: the configuration space of dimension dN with its block maps, tensor powers, one-particle marginals and empirical measures, and the convention that the Wasserstein results and notions, stated for an arbitrary dim…