Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
The Gibbs Entropy Pair of the Hubbard-Stratonovich Ising Field is Lambda-Displacement Convex, with Lambda Uniform in the Cutoff when the Shifted Symbol Stays below One
lemmalem:hubbard-stratonovich-ising-convex-torus-hilbert-2026aProbabilityPDEThe Gibbs entropy pair of the Hubbard-Stratonovich Ising field is (1/r-bar_s - K_s)-displacement convex for every admissible split; when the shifted symbol stays below q-bar < 1 it is lambda(q-bar)-displacement convex with lambda(q-bar) = min{1/(1+q-bar), (1-q-bar)/q-bar} > 0, in…Well-Posedness of the Hamilton-Jacobi Equation of the Hubbard-Stratonovich Ising Field on the Lattice Torus at Every Cutoff, Interaction and Shift: Comparison, Existence and Uniqueness
corollarycor:hubbard-stratonovich-ising-hamilton-jacobi-well-posed-torus-hilbert-2026aProbabilityPDEFor every cube cutoff, reflection-symmetric periodic pair interaction and shift, the Hamilton-Jacobi equation driven by the Gibbs score of the Hubbard-Stratonovich Ising field satisfies comparison and has exactly one bounded viscosity solution, for every bounded uniformly continu…The Hamilton-Jacobi Equation of the Hubbard-Stratonovich Ising Field on the Lattice Torus with White Noise
equationeq:hubbard-stratonovich-ising-hamilton-jacobi-torus-hilbert-2026aProbabilityPDEThe Hamilton-Jacobi equation of the Hubbard-Stratonovich Ising field is the Hamilton-Jacobi equation with Gibbs score drift whose Gibbs measure is the Hubbard-Stratonovich field law, written through any admissible split of its Gaussian. Its penalty pair, and hence its operator, i…The Galerkin Wick-Ordered Potential and Its Wick Constant on the Torus
definitiondef:free-field-phi4-galerkin-potential-torus-hilbert-2026aAnalysisProbabilityPDEDefines the Galerkin Wick-ordered phi^4 potential on the torus together with the inseparable ingredients of its construction: the head dimension of the cutoff (the Galerkin head dimension and active indices), the Galerkin field, the Wick constant and the Wick polynomial, as well…Well-Posedness of the Galerkin Stochastic-Quantisation Hamilton-Jacobi Equation on the Torus for Every Cutoff, Coupling and Mass: Comparison, Existence and Uniqueness
corollarycor:free-field-phi4-galerkin-hamilton-jacobi-well-posed-torus-hilbert-2026aAnalysisProbabilityPDEFor every cube cutoff, positive coupling, real mass and positive temperature, the Hamilton-Jacobi equation with the Gibbs score drift of the Galerkin Wick-ordered phi^4 measure on the torus satisfies comparison and has exactly one bounded viscosity solution, for every bounded uni…The Galerkin Wick-Ordered Potential on the Torus is an Admissible Cylindrical Potential, with Semiconvexity Constant the Positive Part of Three Times the Coupling Times the Wick Constant Minus the Mass
lemmalem:free-field-phi4-galerkin-potential-torus-hilbert-2026aAnalysisProbabilityPDEAt a cube cutoff, the Wick-ordered quartic plus Wick-ordered mass term of the Galerkin free field on the torus is an admissible cylindrical potential in the white-noise geometry; its Hessian is bounded below by the mass minus three times the coupling times the Wick constant, meas…Well-Posedness of the Hamilton-Jacobi Equation with Penalty Drift for a Lambda-Displacement Convex Noise Penalty Pair, Lambda Any Real Number: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution
theoremthm:noise-penalty-drift-well-posed-lambda-convex-hilbert-2026aAnalysisProbabilityPDEFor a noise-closed noise penalty pair with closed score and regular penalised maxima that is lambda-displacement convex for some real lambda, possibly negative, whose penalty domain has the noise map property and whose penalty controls the squared distance to the reference measur…The Hamilton-Jacobi Operator with Penalty Drift on the Noise Wasserstein Space Satisfies the Hypotheses of the First-Order Comparison Principle for a Lambda-Displacement Convex Noise Penalty Pair, Lambda Any Real Number
lemmalem:noise-penalty-drift-hypotheses-lambda-convex-hilbert-2026aAnalysisProbabilityPDEFor a noise penalty pair that is lambda-displacement convex for some real lambda, possibly negative, with lower semicontinuous penalty controlling the squared noise Wasserstein distance to the reference measure, control cost at most one and a bounded uniformly continuous running…Well-Posedness of the Hamilton-Jacobi Equation with Gibbs Score Drift for a Semiconvex Cylindrical Potential: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution
corollarycor:gibbs-score-drift-hamilton-jacobi-well-posed-hilbert-2026aAnalysisProbabilityPDEFor every admissible cylindrical potential, however nonconvex, every temperature, discount, control cost in (0,1] and bounded uniformly continuous running cost, the Hamilton-Jacobi equation with Gibbs score drift satisfies comparison and has exactly one bounded viscosity solution…On Integrals of Cylindrical Functions the Gibbs Score Drift is the Gibbs Ornstein-Uhlenbeck Functional, and the Gibbs Score-Drift Hamilton-Jacobi Operator Takes Its Pointwise Form
lemmalem:gibbs-score-drift-cylindrical-integral-hilbert-2026aAnalysisProbabilityPDEFor the Gibbs entropy pair and a bounded C^2 function f of finitely many coordinates, the integral U_phi of phi = f(p_n) is a noise intrinsic test function on the penalty domain with gradient the noise gradient of phi; the pairing of the Gibbs score drift with it equals beta time…The Hamilton-Jacobi Equation with Gibbs Score Drift on a Hilbert Space
equationeq:gibbs-score-drift-hamilton-jacobi-hilbert-2026aAnalysisProbabilityPDEThe discounted first-order Hamilton-Jacobi equation on the noise Wasserstein space whose drift is the score of the Gibbs entropy pair: the Gaussian score field times the temperature plus the noise gradient of the potential.Well-Posedness of the Hamilton-Jacobi Equation with the Free-Field Score Drift and the Wick-Square Cost for Laws of Distributions on the Torus of Dimension at Most Three, and the Forced Counterterm
corollarycor:free-field-wick-square-hamilton-jacobi-well-posed-torus-hilbert-2026aAnalysisPDEFor laws of distributions on the torus of dimension at most three relative to the free field with square-integrable white noise, the Hamilton-Jacobi equation with the Wick-square cost is well posed relative to the dressing profile, and its cutoff counterterms are forced to be the…Well-Posedness of the Hamilton-Jacobi Equation with Gaussian Score Drift and a Score-Paired Wick-Square Cost on a Hilbert Space, by Gaussian Dressing
theoremthm:wick-square-score-drift-hamilton-jacobi-well-posed-hilbert-2026aAnalysisPDEFor the Hamilton-Jacobi equation with Gaussian score drift and a score-paired Wick-square cost on laws over a Hilbert space, viscosity solutions relative to the dressing profile satisfy comparison, exist and are unique among those differing from the profile by a bounded function;…Gaussian Dressing of the Score-Paired Wick-Square Cost on a Hilbert Space: Riccati Coefficients, Dressed Variances, the Quadratic Profile, the Constant, the Dressed Pair and the Shifted Operator
lemmalem:gaussian-dressing-wick-square-hilbert-2026aAnalysisPDEPer-mode Riccati coefficients give a dressed diagonal Gaussian and a quadratic profile; shifting the unknown by the profile turns the Wick-square Hamilton-Jacobi operator into the Gaussian score-drift operator relative to the dressed Gaussian with the running cost shifted by an a…The Hamilton-Jacobi Equation with Gaussian Score Drift and a Score-Paired Wick-Square Cost on a Hilbert Space
equationeq:wick-square-score-drift-hamilton-jacobi-hilbert-2026aAnalysisPDEThe first-order Hamilton-Jacobi equation on laws relative to a diagonal Gaussian in which the Gaussian score is paired with the momentum minus the gradient of the Wick-square corrector, so that the singular Wick-square cost enters through the score.Changing the Penalty of a Noise Penalty Pair by a Noise Intrinsic Test Function and a Constant Does Not Change the Viscosity Notion
lemmalem:noise-penalty-change-viscosity-hilbert-2026aAnalysisPDEIf two noise penalty pairs share their domains and their penalties differ by a noise intrinsic test function plus a constant, with scores differing by its gradient, then they define the same viscosity subsolutions and supersolutions.Viscosity Subsolutions, Supersolutions and Solutions on the Noise Wasserstein Space Relative to a Noise Penalty Pair and a Profile
definitiondef:noise-profile-viscosity-solution-hilbert-2026aAnalysisPDEA function is a viscosity sub- or supersolution relative to a noise penalty pair and a profile, a noise intrinsic test function, when its difference with the profile is one for the operator shifted by the profile's value and gradient.Well-Posedness of the Hamilton-Jacobi Equation with the Free-Field Score Drift for Laws of Distributions on the Torus: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution
corollarycor:free-field-score-drift-hamilton-jacobi-well-posed-torus-hilbert-2026aProbabilityPDEFor laws of distributions on the torus with finite entropy relative to the free field, and a bounded running cost uniformly continuous for the Wasserstein distance with L^2 cost, the Hamilton-Jacobi equation with the free-field score drift has comparison and exactly one bounded v…On Functions of Finitely Many Fourier Modes the Free-Field Score Drift is Minus the Ornstein-Uhlenbeck Generator with Drift the Laplacian Minus One
lemmalem:free-field-score-drift-fourier-torus-hilbert-2026aProbabilityPDEFor functions of finitely many Fourier modes, the score drift of the free field equals minus the integral of the Ornstein-Uhlenbeck generator with second-order part the sum of second mode derivatives and drift coefficient minus the Fourier weight, which is the Fourier symbol of t…Well-Posedness of the Hamilton-Jacobi Equation with Gaussian Score Drift on a Hilbert Space: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution
corollarycor:gaussian-score-drift-hamilton-jacobi-well-posed-hilbert-2026aProbabilityPDEFor a bounded, uniformly continuous running cost, the Hamilton-Jacobi equation with Gaussian score drift on laws of finite relative entropy with respect to a diagonal Gaussian measure satisfies comparison and has exactly one bounded viscosity solution.