Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
The Mollified Mean-Field Cost and the Mollified N-Particle Cost of a Local Coupling
definitiondef:n-particle-mollified-cost-wasserstein-2026aAnalysisProbabilityPDEDefines the mollified mean-field cost g_eps(nu) = int f dnu + G_{Phi,eps}(nu) on P_2(R^d) and the mollified N-particle cost c_{N,eps}(x) = N g_eps(mu^N_x) on R^{dN}, the empirical-measure form of a mollified local coupling.The Mean Distance Between the Mollified Empirical Measure and the Mollified Measure
lemmalem:mollified-empirical-fluctuation-euclidean-2026aAnalysisProbabilityFor N particles drawn from the tensor power of a measure mu with finite second moment, the expected Lebesgue L^1 distance between the mollified empirical measure and the mollified measure at scale eps <= 1 is at most the square root of kappa_d R^d S eps^{-d}/N plus 2 M_2(mu)/(R-1…Mean-Square Deviation of the Empirical Average of a Bounded Borel Function under a Tensor Power
lemmalem:empirical-average-variance-euclidean-2026aAnalysisProbabilityUnder the N-fold tensor power of a probability measure rho, the integral of a bounded Borel function phi against the empirical measure has mean-square deviation (int phi^2 d rho - (int phi d rho)^2)/N from its mean, which is at most (int phi^2 d rho)/N.The Mollified Density Cost is Bounded, Lipschitz for the Wasserstein Distance, and Below the Density Cost
lemmalem:mollified-density-cost-properties-wasserstein-2026aAnalysisProbabilityThe mollified density cost with a convex Lipschitz integrand of constant L takes values in [0,L], is Lipschitz on the Wasserstein space with an explicit constant depending on L, the kernel and eps, and never exceeds the density cost of an absolutely continuous measure.The Mollified Density Cost of a Probability Measure with Finite Second Moment
definitiondef:mollified-density-cost-euclidean-2026aAnalysisProbabilityFor a convex Lipschitz integrand, a mollifier kernel and a scale eps, the mollified density cost of a probability measure with finite second moment is the Lebesgue integral of the integrand applied to its mollified density.Mollifying a Probability Measure on Euclidean Space: Kernel Bounds, the Translation Estimate, the Mollified Density and Its Cost
lemmalem:mollified-measure-density-euclidean-2026aAnalysisProbabilityIntegrating a rescaled mollifier kernel of radius eps against a probability measure on R^d gives a continuous probability density bounded by eps^{-d} times a bound of the kernel; translating the rescaled kernel by h changes it in L^1 by at most an explicit constant over eps times…Supporting Lines, Composition and Jensen's Inequality for a Convex Lipschitz Integrand
lemmalem:jensen-integral-convex-integrand-2026aAnalysisProbabilityA convex Lipschitz integrand has a supporting line at every point of the half-line; composed with a nonnegative measurable function it stays measurable and lies between 0 and L times the function; and on a probability space it satisfies Jensen's inequality.Stability of Comparison on the Wasserstein Space under Locally Vanishing Cost Defects
theoremthm:stability-cost-defect-wasserstein-2026bAnalysisProbabilityPDELet F satisfy the hypotheses of the comparison principle, and let F_n, F'_n be operators with F - h_n <= F_n and F'_n <= F + h'_n, where the defects h_n, h'_n take values in [0,H] and tend to zero uniformly on every set of measures with bounded penalty and bounded score. If u_n i…Lipschitz Dependence of the Bounded Viscosity Solution of the Langevin Hamilton-Jacobi Equation with a Density Cost on the Running Cost
corollarycor:langevin-density-cost-cost-lipschitz-wasserstein-2026aAnalysisProbabilityPDEFor two bounded uniformly continuous running costs g, g' with sup|g-g'| <= k, the bounded viscosity solutions u, u' of the corresponding Langevin Hamilton-Jacobi equations with common noise and density cost satisfy |u-u'| <= k/lambda_0 on the penalty domain. Proof: u'+k/lambda_0…Adding a Constant to a Viscosity Subsolution or Supersolution on the Wasserstein Space
lemmalem:constant-shift-viscosity-wasserstein-2026aAnalysisProbabilityPDEIf u is a viscosity subsolution of F relative to a penalty pair and F'(nu,r+s,q,Y) <= F(nu,r,q,Y) everywhere, then u+s is a viscosity subsolution of F'; symmetrically for supersolutions with the reverse inequality. The delta-envelopes of u+s are those of u shifted by s.The Potential Gradient Paired with the Score, and a Fisher Information Bound on the Domain of the Langevin Free-Energy Pair
lemmalem:langevin-fisher-bound-euclidean-2026aAnalysisProbabilityOn the domain of the Langevin free-energy pair, the pairing of the potential gradient with the score equals minus the mean Laplacian of the potential, that is minus the trace of the translation Hessian; consequently sigma^4/4 times the Fisher information is at most the squared no…Translation and Mollification Estimates in the Integrable Norm for a Density of Finite Fisher Information, and the Mollified Density Cost
lemmalem:score-translation-l1-euclidean-2026aAnalysisProbabilityFor an absolutely continuous measure with finite second moment and finite Fisher information, translating its density by z moves it by at most |z| times the square root of the Fisher information in the integrable norm; mollifying at scale eps moves it by at most eps times that ro…The Structure Estimate at a Maximiser for Operators within a Cost Defect of a Reference Operator
lemmalem:comparison-estimate-cost-defect-wasserstein-2026bAnalysisProbabilityPDEFor a subsolution of one operator and a supersolution of another, each within a bounded zeroth-order defect of a reference operator satisfying the comparison hypotheses, a nonnegative supremum of the Wasserstein-doubled difference of their delta-envelopes is attained at a pair in…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…