Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
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 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…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…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…Well-Posedness of the Langevin Hamilton-Jacobi Equation with Common Noise and a Density Cost: Existence and Uniqueness of a Bounded Viscosity Solution
theoremthm:langevin-density-cost-well-posed-wasserstein-2026bAnalysisProbabilityPDEFor a confining potential, positive noise, a common-noise matrix, control cost at most one, a bounded uniformly continuous running cost and the density cost of a convex Lipschitz integrand, in any dimension, the Langevin Hamilton-Jacobi equation with common noise and density cost…The Langevin Hamilton-Jacobi Operator with a Density Cost Satisfies the Hypotheses of the Comparison Principle and of Perron's Method
lemmalem:langevin-density-cost-comparison-hypotheses-wasserstein-2026cAnalysisProbabilityPDEFor a confining potential, positive noise, a common-noise matrix, control cost at most one, a bounded uniformly continuous running cost and the density cost of a convex Lipschitz integrand, the Langevin Hamilton-Jacobi operator with common noise and density cost is degenerate ell…The Langevin Hamilton-Jacobi Equation with Common Noise and a Density Cost on the Wasserstein Space
equationeq:langevin-density-cost-hamilton-jacobi-wasserstein-2026bAnalysisProbabilityPDEThe Langevin Hamilton-Jacobi equation in a confining potential with common noise along a common-noise matrix, with the density cost of a convex Lipschitz integrand added to the running cost.Well-Posedness of the Singular-Cost Dyson Hamilton-Jacobi Equation above the Hardy Threshold
theoremthm:dyson-singular-cost-well-posed-weyl-chamber-2026aAnalysisPDEFor a pair cost c times the sum of inverse squared gaps with c > -kappa^2/(8 theta), a harmonic confinement cost omega|x|^2 and bounded continuous g, the singular-cost Dyson equation on the Weyl chamber satisfies comparison in the class of u with u - P of P-subordinate growth, wh…The Dyson Hamilton-Jacobi Equation in Singular-Cost Form on the Weyl Chamber
equationeq:dyson-singular-cost-hamilton-jacobi-weyl-chamber-2026aAnalysisPDEThe Hamilton-Jacobi equation lambda u + (theta/2)|Du|^2 - (kappa/2) tr D^2u = c S + omega |x|^2 + g on the Weyl chamber, with S the sum of the inverse squared gaps: a pair cost with reciprocal squares, a harmonic confinement cost and a running cost g, with no drift.Well-Posedness of the Dyson Hamilton-Jacobi Equation in the Weyl Chamber below the Collision Threshold
theoremthm:dyson-well-posed-weyl-chamber-2026aAnalysisPDEFor N particles with logarithmic repulsion of strength beta > kappa/2 and a convex confinement of at least linear and regular growth, and bounded continuous running cost g, the Dyson Hamilton-Jacobi equation on the Weyl chamber satisfies comparison in the class of P-subordinate g…The Dyson Hamilton-Jacobi Equation for N Controlled Particles in the Weyl Chamber
equationeq:dyson-hamilton-jacobi-weyl-chamber-2026aAnalysisPDEThe drift-form Hamilton-Jacobi equation on the Weyl chamber for N controlled particles with logarithmic repulsion of strength beta and confinement V_1: the penalty-drift equation with potential P = H_beta + sum V_1(x_k).Well-Posedness of the Langevin Hamilton-Jacobi Equation on Euclidean Space under a Dissipation Inequality
theoremthm:langevin-well-posed-euclidean-2026aAnalysisPDEFor a C^2 potential V on R^n with compact sublevel sets satisfying (kappa/2) tr D^2V <= (1-eps)|DV|^2 + lambda V + C, and bounded continuous g, the Langevin Hamilton-Jacobi equation satisfies comparison in the class of V-subordinate growth and has exactly one viscosity solution t…The Hamilton-Jacobi Equation for Controlled Langevin Dynamics in a Potential on Euclidean Space
equationeq:langevin-hamilton-jacobi-euclidean-2026aAnalysisPDEThe discounted Hamilton-Jacobi equation lambda u + (theta/2)|Du|^2 + DV.Du - (kappa/2) tr D^2u = g on R^n of a controlled Langevin particle in a C^2 potential V.