Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
Relative Entropy and Relative Score with Respect to a Diagonal Gaussian Measure: Comparison with the Entropy, the Score and the Relative Free Energy of a Quadratic Potential
lemmalem:diagonal-gaussian-relative-entropy-score-euclidean-2026aAnalysisProbabilityFor measures with finite second moment, relative entropy with respect to a diagonal Gaussian is finite exactly when the entropy is, and equals entropy plus half the weighted second moment plus the log-normalizer; the relative score is the score plus the scaling map; consequently…Finite Fisher Information Relative to a Diagonal Gaussian Measure: the Ornstein-Uhlenbeck Functional, the Relative Score and the Relative Fisher Information
definitiondef:relative-score-diagonal-gaussian-euclidean-2026aAnalysisProbabilityFor a measure with finite second moment, the Ornstein-Uhlenbeck functional integrates Laplacian minus scaling-map drift of test functions; the measure has finite Fisher information relative to the diagonal Gaussian when this functional is bounded by the L^2 norm of gradients, its…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.Lifting Shifted Galerkin Solutions of the Wick-Square Problem to Viscosity Solutions of the Cut-Off Shifted Equation on the Sobolev Triple
lemmalem:wick-square-galerkin-lift-2026aAnalysisPDEA viscosity sub- or supersolution of the shifted Galerkin equation, composed with the coordinate projection, is a viscosity sub- or supersolution of the cut-off shifted equation on the Sobolev triple; continuity is preserved.The Cut-Off Shifted Operators of the Wick-Square Problem: the Hypotheses of the Convergence Theorem with Constants Independent of the Cutoff, Convergence on Bounded Test Data, and Compactness of the Triple
lemmalem:wick-square-shifted-cutoff-data-2026aAnalysisPDEThe cut-off shifted operators satisfy the hypotheses of the Hilbert-triple convergence theorem with constants independent of the cutoff and converge to the uncut shifted operator on bounded test data; and bounded sets of the form space of the Sobolev triple are precompact in the…Convergence of Solutions of Viscous Hamilton-Jacobi Equations with Monotone Nonlinearities on a Hilbert Triple under Convergence of the Operators
theoremthm:convergence-viscous-hamilton-jacobi-hilbert-triple-2026aAnalysisPDEFor viscous Hamilton-Jacobi equations with monotone nonlinearities on a Hilbert triple whose form-space bounded sets are precompact in the ambient space, bounded continuous solutions are uniformly Lipschitz and converge, pointwise and uniformly on bounded sets of the form space,…Stability of Viscosity Sub- and Supersolutions on a Hilbert Triple: Limits of Uniformly Bounded, Equicontinuous Sequences under Convergence of the Operators on Bounded Test Data
theoremthm:stability-equicontinuous-hilbert-triple-2026aAnalysisPDEOn a Hilbert triple whose bounded sets of the form space are precompact in the ambient space, the pointwise limit superior of uniformly bounded, equicontinuous viscosity subsolutions of operators converging on bounded test data is a viscosity subsolution of the limit operator, an…Convergence of Second-Order Equation Operators on a Hilbert Triple on Bounded Test Data
definitiondef:operator-convergence-bounded-test-data-hilbert-triple-2026aAnalysisPDETest data bounded by R on a Hilbert triple, and convergence of a sequence of second-order equation operators to a limit operator uniformly on such data, for both delta-shifts.In a Diagonal Hilbert Triple whose Weights Tend to Infinity, Bounded Sequences of the Form Space Have Subsequences Converging in the Ambient Space
lemmalem:diagonal-hilbert-triple-compact-2026aAnalysisIn a diagonal Hilbert triple whose weights tend to infinity, every sequence bounded in the form space has a subsequence converging in the ambient space to a point of the form space with the same bound.Limits of a Uniformly Bounded Sequence of Real Functions with a Common Modulus of Continuity: Bounds, Extraction and Uniform Convergence on Sequentially Compact Sets
lemmalem:equicontinuous-sequence-limits-metric-2026aAnalysisTopologyFor a uniformly bounded sequence of real functions on a metric space sharing a modulus of continuity, the pointwise limit superior and inferior keep the bound and the modulus and are attained along subsequences at each point; where they agree on a sequentially compact set, the co…Well-Posedness of the Renormalised Wick-Square Problem: Existence, Comparison and Uniqueness of Renormalised Viscosity Solutions Differing from the Free Solution by a Bounded Lipschitz Function
theoremthm:wick-square-viscosity-well-posed-2026aAnalysisPDEFor a bounded running cost that is Lipschitz in the norm of order -3, the renormalised Wick-square equation has exactly one renormalised viscosity solution differing from the free solution by a bounded Lipschitz function, comparison holds in that class, and for zero cost the solu…Lipschitz Regularity of Bounded Continuous Viscosity Solutions of a Viscous Hamilton-Jacobi Equation with a Monotone Nonlinearity on a Hilbert Triple
theoremthm:lipschitz-viscous-hamilton-jacobi-hilbert-triple-2026aAnalysisPDEA bounded continuous viscosity solution of a discounted viscous Hamilton-Jacobi equation on a Hilbert triple, with a Hamiltonian given by a form between zero and the identity, a monotone nonlinearity and a running cost that is Lipschitz for the norm of the large space, satisfies…Renormalised Viscosity Solutions versus Viscosity Solutions of the Shifted Equation on the Sobolev Triple of Order Two
lemmalem:wick-square-viscosity-bridge-2026aAnalysisPDEIf w is a bounded continuous viscosity sub- or supersolution of the shifted equation, then the free solution plus w is a renormalised viscosity sub- or supersolution; conversely, when w is moreover Lipschitz for the norm of order -3, the renormalised property of the free solution…Renormalised Viscosity Subsolutions, Supersolutions and Solutions of the Wick-Square Problem, Tested by the Gaussian Penalty plus a Regular Function
definitiondef:wick-square-viscosity-solution-2026aAnalysisPDEA function on the Sobolev space of order -1 is a renormalised viscosity subsolution (supersolution) if the renormalised operator is nonpositive (nonnegative) at the Gaussian penalty plus any regular function touching it from above (below), touching being local in the norm of orde…The Riccati Shift: the Renormalised Operator at the Free Solution plus a Regular Function is the Shifted Operator, Which Satisfies the Hilbert-Triple Comparison Hypotheses, and Its Touching Points Lie in the State Space
lemmalem:wick-square-riccati-shift-2026aAnalysisPDESubtracting the free solution turns the renormalised Wick-square operator into the shifted Hilbert-triple operator, which satisfies the hypotheses of the second-order comparison theory; and at every touching point of a continuous viscosity sub- or supersolution of the shifted equ…The Shifted Hamilton-Jacobi-Bellman Equation of the Wick-Square Problem on the Sobolev Triple of Order Two
equationeq:wick-square-hjb-shifted-2026aAnalysisPDENames the Hamilton-Jacobi-Bellman operator obtained from the renormalised Wick-square operator by subtracting the free solution: a discounted viscous equation on the Sobolev triple of order two with white noise, the gradient form as Hamiltonian, the drift one minus the Laplacian…The Wick-Square Problem on the Sobolev Triple of Order Two: Standing Notation
settingset:wick-square-triple-2026aAnalysisPDEStanding notation for the Wick-square problem on the Sobolev triple of orders -3 and -2: the triple and its Hilbert-triple notation, the white-noise sequence and gradient form, the Riccati data and drift, and the free solution of the renormalised equation.The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift
lemmalem:wick-square-sobolev-triple-2026aAnalysisPDEFor the Sobolev triple of orders -3 and -2 on the torus in dimension at most three: the domain of the form operator is the Sobolev space of order -1, pairings with unit families are explicit, the space is infinite-dimensional, the unit families are square-summable in the form spa…Mode Derivatives of Linear Combinations and of Restrictions of Twice Differentiable Functions on a Negative Sobolev Space
lemmalem:mode-derivative-calculus-torus-2026aAnalysisMode derivatives are linear and vanish on constants, and the restriction to the Sobolev space of order -1 of a twice continuously differentiable function on a negative Sobolev space has mode derivatives given by its gradient and Hessian evaluated on the unit families.