Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
Diagonal Gaussian Measures on Euclidean Space
definitiondef:diagonal-gaussian-measure-euclidean-2026aAnalysisProbabilityThe diagonal Gaussian measure with given variances is the probability measure on R^d whose density with respect to Lebesgue measure is the diagonal Gaussian density.The Diagonal Gaussian Density on Euclidean Space: Regularity, Gradient, Normalization, Second Moments and Exponential Moments of Diagonal Quadratic Forms
lemmalem:diagonal-gaussian-density-euclidean-2026aAnalysisProbabilityThe diagonal Gaussian density is smooth, positive and Borel, its gradient is minus the density times the scaling map, it integrates to one, its i-th coordinate has second moment c_i, and exponential moments of diagonal quadratic forms are explicit products.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.The Counterterm Is Forced at the Level of Galerkin Solutions: Other Counterterms Shift the Solutions by a Constant, and the Bare Solutions Diverge
corollarycor:wick-square-galerkin-counterterm-2026aProbabilityPDEChanging the counterterm shifts each Galerkin solution by a constant, so the Galerkin solutions converge exactly when the counterterm differs from the free-field variances by a cube-summable family; in dimension at least two the bare Galerkin solutions are unbounded and converge…The Galerkin Solutions of the Wick-Square Problem Converge to the Renormalised Viscosity Solution for Every Bounded Lipschitz Running Cost
theoremthm:wick-square-galerkin-convergence-2026aProbabilityPDEFor a bounded running cost that is Lipschitz in the distance of H^-3, each Wick-ordered Galerkin equation has exactly one solution differing from the free Galerkin quadratic by a bounded function; these differences are bounded and Lipschitz uniformly in the cutoff, and the Galerk…The Riccati Shift of the Galerkin Equations of the Wick-Square Problem: a Penalty-Drift Equation with Bounded Cost, and Well-Posedness at Each Cutoff
lemmalem:wick-square-galerkin-shift-2026aProbabilityPDESubtracting the free Galerkin quadratic turns the Wick-ordered Galerkin equation into a penalty-drift equation with bounded running cost whose quadratic potential meets the hypotheses of the Euclidean well-posedness theorem; so each Galerkin equation has exactly one solution diff…The Shifted Hamilton-Jacobi-Bellman Equation of the Wick-Square Problem with Noise, Control and Cost Cut Off to a Cube of Modes
equationeq:wick-square-hjb-shifted-cutoff-2026aProbabilityPDEThe shifted Wick-square operator on the Sobolev triple with the noise, the gradient form, the Riccati drift and the running cost cut off to a cube of modes, the full linear drift being kept, and its equation.The Galerkin Hamilton-Jacobi-Bellman Equations of the Wick-Square Problem in the Coordinates of a Cube of Modes
equationeq:wick-square-galerkin-2026aProbabilityPDEThe Galerkin Hamilton-Jacobi-Bellman operators of the Wick-square problem in the coordinates of a cube of modes, with a general counterterm (Wick-ordered and bare as special cases), the shifted Galerkin operator of penalty-drift form, and their equations.The Free-Field Variances of the Fourier Modes
definitiondef:free-field-variances-torus-2026aAnalysisProbabilityThe free-field variance of a mode is the noise intensity divided by twice its Fourier weight.How the Wick-Square Problem Is Renormalised: Equipartition, the Corrector Identity and the Gaussian Penalty
remarkrem:wick-square-renormalisation-2026bAnalysisProbabilityPDEExplains the Wick-square results: the counterterm is the equipartition energy of the free field, the singular cost is the dissipation of the Gaussian penalty under the free dynamics, and this is why the renormalised equation is written with the generator acting on the unknown min…The Wick Square with a Mode Cutoff and the Wick Domain
definitiondef:wick-square-torus-2026aAnalysisProbabilityThe Wick square at cutoff N subtracts the free-field variance from the square of each kept coordinate and sums; the Wick domain is the set of states at which these cutoff Wick squares converge, and the Wick square is their limit there.Polynomial Controls on Unitary Laws and Their Energy
definitiondef:nc-unitary-polynomial-controls-2026aProbabilityPDEA polynomial control is a d-tuple of self-adjoint word polynomials; its energy under a unitary law is the sum of the law's values on the squares of its entries.Rotations and Cyclic Derivatives of Words in Unitary Letters
definitiondef:nc-unitary-cyclic-derivative-2026aAlgebraProbabilityThe i-th cyclic derivative of a word is the word polynomial summing, over the occurrences of the i-th unitary or its inverse, the rotation of the word starting at the insertion point, with coefficient i times the sign of the letter.Evaluation of Word Polynomials by a Unitary Law
definitiondef:nc-unitary-law-evaluation-2026aAlgebraProbabilityA unitary law is extended linearly to word polynomials: lambda(X) is the sum of X(w) lambda(w) over a support set of X.Well-Posedness for Polynomial Sources in the Quadratic Control Problem on Unitary Laws
corollarycor:nc-unitary-quadratic-polynomial-source-2026aProbabilityPDEA source given by the real part of a finite combination of word moments is Gamma-Lipschitz in the gauge distance with an explicit Gamma, so the quadratic control equation is well posed with a Gamma/rho-Lipschitz solution when Gamma is at most rho beta^2/(4K).Well-Posedness and Lipschitz Regularity for the Quadratic Control Problem on Unitary Laws under Strong Free Noise
theoremthm:nc-unitary-quadratic-well-posedness-2026aProbabilityPDEUnder strong free noise, L_f at most rho beta^2/(4K), the quadratic control equation on unitary laws has exactly one viscosity solution; it is L_f/rho-Lipschitz in the gauge distance and bounded by sup|f|/rho.Lipschitz-Cone Comparison for the Quadratic Control Problem on Unitary Laws under Strong Free Noise
theoremthm:nc-unitary-quadratic-cone-comparison-2026aProbabilityPDEIf the source is L_f-Lipschitz in the gauge distance with L_f at most rho beta^2/(4K), every subsolution minus every supersolution of the quadratic control equation is bounded by L times the gauge distance for L between L_f/rho and beta^2/(4K); hence comparison, uniqueness, and t…The Trilinear Estimate for Cyclic Gradients of Differences of Unitary Laws
lemmalem:nc-unitary-cyclic-gradient-trilinear-2026aAnalysisProbabilityFor two unitary laws with gauge difference x, the difference of the laws evaluated on the squared truncated cyclic gradients of x is bounded by a constant times the gauge norm of x times the square of its length-weighted gauge, uniformly in the truncation.The Quadratic Control Hamiltonian: Legendre Formula, Bounds, Differences in the Law and Admissibility
lemmalem:nc-unitary-quadratic-hamiltonian-2026aProbabilityPDEThe quadratic control Hamiltonian equals the limit of half the sum of the law's values on the squares of the truncated cyclic gradients, lies between 0 and kappa_d^2 |p|^2 / 2, has the corresponding limit formula for differences in the law, and is an admissible Hamiltonian.The Quadratic Control Hamiltonian on Unitary Laws
definitiondef:nc-unitary-quadratic-hamiltonian-2026aProbabilityPDEThe quadratic control Hamiltonian at a unitary law and a momentum is the supremum, over polynomial feedback controls, of minus the pairing of the momentum with the feedback drift minus half the control energy.