Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
The Bundle of Noise Fields over a Set of Measures, First-Order Equation Operators on the Noise Wasserstein Space, and Their Delta-Shifts
definitiondef:noise-first-order-operator-hilbert-2026bAnalysisProbabilityPDEA first-order equation operator over a set of noise-connected measures is a real function of a measure, a real number and a square-integrable noise field; relative to a noise penalty pair its delta-shifts add or subtract delta times the penalty in the real slot and delta times th…At Each Cutoff the Lifted Wick-Square Solution Relative to the Free-Field Gaussian Is the Integral of the Galerkin Solution, and Its Pointwise Recovery Converges to the Renormalised Solution
corollarycor:lifted-wick-square-galerkin-identification-torus-2026aAnalysisProbabilityPDEAt each cutoff, the lifted Wick-square solution relative to the free-field Gaussian is the integral of the Galerkin solution; evaluating it along shrinking Gaussians recovers the Galerkin solution, and removing the cutoff afterwards gives the renormalised viscosity solution.The Galerkin Wick-Square Problem at a Cutoff as a Gaussian-Dressed Problem Relative to the Free-Field Gaussian: the Coupling Condition, the Riccati Coefficients, the Riccati Potential, the Profile and the Running Cost
lemmalem:wick-square-galerkin-gaussian-dictionary-torus-2026aAnalysisProbabilityPDEAt each cutoff, the Galerkin Wick-square data match the Gaussian dressing relative to the free-field Gaussian with temperature nu/2: the coupling condition holds, the dressing Riccati coefficients are twice the Galerkin ones, the Galerkin Riccati potential is the dressed Gaussian…The Lifted Ornstein-Uhlenbeck Hamilton-Jacobi Equation with a Linear Running Cost: Its Viscosity Solution Is the Integral of the Pointwise Solution, Which It Determines
theoremthm:lifted-ou-linear-cost-integral-representation-euclidean-2026aAnalysisProbabilityPDEFor a bounded uniformly continuous pointwise cost, the unique bounded viscosity solution of the lifted Ornstein-Uhlenbeck Hamilton-Jacobi equation with the integrated cost is the integral of the unique bounded viscosity solution of the pointwise equation, and the pointwise soluti…Integrals of Lipschitz Semiconvex Viscosity Subsolutions and Lipschitz Semiconcave Viscosity Supersolutions of the Ornstein-Uhlenbeck Hamilton-Jacobi Equation Are Viscosity Sub- and Supersolutions of the Lifted Equation
propositionprop:semiconvex-lift-ou-hamilton-jacobi-euclidean-2026aAnalysisProbabilityPDEIntegrating a bounded Lipschitz semiconvex viscosity subsolution of the pointwise Ornstein-Uhlenbeck Hamilton-Jacobi equation against laws of finite relative entropy gives a viscosity subsolution of the lifted equation with the integrated running cost; semiconcave supersolutions…Sup- and Inf-Convolutions of Bounded Viscosity Sub- and Supersolutions of the Penalty-Drift Equation on the Whole Space with a Monotone Drift
lemmalem:sup-inf-convolution-monotone-drift-euclidean-2026aAnalysisPDEFor the penalty-drift equation on the whole space with a potential of monotone gradient, the sup-convolution of a bounded viscosity subsolution (inf-convolution of a bounded supersolution) is attained within a fixed radius, lies above (below) the function, is semiconvex (semiconc…Well-Posedness of the Lifted Hamilton-Jacobi Equation with a Wick-Square Cost Relative to a Diagonal Gaussian Measure, by Gaussian Dressing
theoremthm:lifted-wick-square-hamilton-jacobi-well-posed-euclidean-2026aAnalysisPDEFor the lifted Hamilton-Jacobi equation with a Wick-square cost relative to a diagonal Gaussian measure, viscosity solutions relative to the quadratic Riccati profile satisfy comparison, exist and are unique among functions differing from the profile by a bounded function; the di…Gaussian Dressing of the Wick-Square Cost: the Riccati Coefficients, the Dressed Variances, the Quadratic Profile, the Shifted Pair and the Shifted Operator
lemmalem:gaussian-dressing-wick-square-euclidean-2026aAnalysisPDESolving a scalar Riccati equation in each coordinate gives a quadratic profile and narrower dressed variances; the Gaussian free-energy pair of the dressed variances is the original pair shifted by the profile, and the lifted Wick-square operator shifted by the profile is the lif…The Lifted Hamilton-Jacobi Equation with a Wick-Square Cost Relative to a Diagonal Gaussian Measure
equationeq:lifted-wick-square-hamilton-jacobi-euclidean-2026aAnalysisPDEThe lifted Ornstein-Uhlenbeck Hamilton-Jacobi equation relative to a diagonal Gaussian measure whose running cost is the integral of a Wick-square potential plus a function on measures; viscosity solutions are taken relative to the Gaussian free-energy pair and a profile.Shifting the Penalised Viscosity Notion by an Intrinsic Test Function: Composition of Shifts, the Shift Identity, Invariance under a Change of Profile, and a Change of the Penalty
lemmalem:profile-shift-viscosity-wasserstein-2026aAnalysisPDESubtracting an intrinsic test function from a function and shifting the operator accordingly preserves being a penalised viscosity subsolution or supersolution; consequently the notion relative to a profile does not depend on the profile within a growth class, and a penalty chang…Viscosity Subsolutions, Supersolutions and Solutions Relative to a Penalty Pair and a Profile
definitiondef:profile-viscosity-solution-wasserstein-2026aAnalysisPDEGiven a penalty pair, an equation operator and an intrinsic test function called the profile, a function is a viscosity subsolution, supersolution or solution relative to the profile when its difference with the profile is one, in the penalised sense, of the operator shifted by t…Well-Posedness of the Lifted Ornstein-Uhlenbeck Hamilton-Jacobi Equation Relative to a Diagonal Gaussian Measure: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution
theoremthm:lifted-ou-hamilton-jacobi-well-posed-euclidean-2026aAnalysisProbabilityPDEFor a bounded, Wasserstein-uniformly continuous running cost and control cost theta in (0,1], the lifted Ornstein-Uhlenbeck Hamilton-Jacobi equation relative to a diagonal Gaussian satisfies comparison and has exactly one bounded viscosity solution on the finite-entropy measures,…A Comparison Principle for First-Order Equations on the Wasserstein Space Relative to a Wasserstein-Closed Penalty Pair
theoremthm:comparison-first-order-w2-closed-wasserstein-2026aAnalysisPDEComparison without compactness: for a Wasserstein-closed penalty pair with closed score whose domain has the map property, and a first-order operator (independent of the matrix argument) that is locally strictly proper, shift-coercive, shift-semicontinuous, satisfies the structur…Constant Viscosity Subsolutions and Supersolutions of the Hamilton-Jacobi Equation with Common Noise and Penalty Drift
lemmalem:constant-sub-supersolutions-penalty-drift-wasserstein-2026aAnalysisPDEFor a Wasserstein-closed pair with regular penalised maxima, a constant function has penalty-subordinate growth from both sides; it is a viscosity subsolution of the penalty-drift Hamilton-Jacobi equation when discount times the constant is at most the running cost on the score d…The Hamilton-Jacobi Operator with Common Noise and Penalty Drift Has Momentum-Continuous Shifts
lemmalem:penalty-drift-momentum-continuous-wasserstein-2026aAnalysisPDEThe delta-shifts of the Hamilton-Jacobi operator with common noise and penalty drift are Lipschitz in the vector field, with constant (2 theta + 1) R on data of score and field norms at most R; hence the operator has momentum-continuous shifts.Bounds, Monotonicity and Borwein-Preiss Perturbed Maximisers of the Wasserstein-Doubled Difference for a Wasserstein-Closed Penalty Pair
lemmalem:w2-doubling-perturbed-maximiser-wasserstein-2026aAnalysisPDEFor a Wasserstein-closed pair and bounded u, v, the Wasserstein-doubled difference of their delta-envelopes has a finite supremum, monotone in the weight and in the doubling strength with slack, and a Borwein-Preiss perturbed maximiser: a near-maximising pair, localised in a subl…Exact Inequalities at a Touching Point of a Penalised Viscosity Subsolution or Supersolution on the Wasserstein Space
lemmalem:penalised-exact-jets-wasserstein-2026aAnalysisPDEFor a Wasserstein-closed pair with closed score and an operator satisfying shift-coercivity and shift-semicontinuity, a point where an intrinsic test function touches the delta-envelope of a penalised viscosity subsolution (or supersolution) lies in the score domain, and the shif…The Lifted Ornstein-Uhlenbeck Hamilton-Jacobi Equation Relative to a Diagonal Gaussian Measure
equationeq:lifted-ou-hamilton-jacobi-euclidean-2026aAnalysisProbabilityPDEThe lifted Ornstein-Uhlenbeck Hamilton-Jacobi equation: the penalty-drift Hamilton-Jacobi equation of the Gaussian free-energy pair without common noise, lambda r + (theta/2)|q|^2 + a<relative score, q> = g, with viscosity solutions relative to the Gaussian free-energy pair.Equation Operators on the Wasserstein Space with Momentum-Continuous Shifts
definitiondef:momentum-continuous-shifts-wasserstein-2026aAnalysisPDEAn equation operator on the Wasserstein space has momentum-continuous shifts if, at each weight delta in (0,1) and each level R, its two delta-shifts are uniformly continuous in the vector-field argument over data bounded by R with score bounded by R.The Galerkin Problems of the Wick-Square Problem: Standing Notation for Cubes of Modes, Coordinates, the Free Galerkin Quadratic and the Riccati Potential
settingset:wick-square-galerkin-2026aProbabilityPDEStanding notation for the Galerkin problems of the Wick-square problem: for each cutoff an enumeration of the cube of modes, the coordinate map and projection, the free Galerkin quadratic and the Riccati potential.