Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
Viscosity Subsolution, Supersolution and Solution on the Lift of the Wasserstein Space
definitiondef:viscosity-lift-wasserstein-2026bAnalysisPDEA bounded-near-each-point function on the Wasserstein space is a viscosity subsolution (supersolution) on the lift if, wherever its delta-envelope composed with the law map minus a lifted test function has a local maximum (minimum) on the lift of the penalty domain, the shifted o…The Second-Order Structure Condition at Optimally Coupled Pairs on the Lift of the Wasserstein Space
definitiondef:second-order-structure-condition-lift-wasserstein-2026bAnalysisProbabilityPDEA pair of symmetric matrices is admitted at a doubling strength when it satisfies the ordering, norm and quadratic-form bounds of Lions' lemma; the operator satisfies the second-order structure condition when, at every level measured by the delta-scaled penalty and by the value s…The Shift-Semicontinuity Condition for an Equation Operator on the Lift of the Wasserstein Space
definitiondef:shift-semicontinuity-condition-lift-wasserstein-2026aAnalysisProbabilityPDESays that the lower shift of the operator passes to the limit from below, and the upper shift from above, along sequences of test data with bounded score whose lifted scores converge weakly.The Shift-Coercivity Condition for an Equation Operator on the Lift of the Wasserstein Space
definitiondef:shift-coercivity-condition-lift-wasserstein-2026aAnalysisProbabilityPDESays that on the test data admitted at a level, the shifts of the operator force the score of the penalty pair to be bounded.Locally Strictly Proper Second-Order Equation Operator on the Lift of the Wasserstein Space
definitiondef:locally-strictly-proper-lift-wasserstein-2026aAnalysisProbabilityPDESays that the operator increases in its value argument at a strictly positive rate, uniform over each bounded range of values.Test Data for a Second-Order Equation Operator on the Lift of the Wasserstein Space and the Admissible Sets
definitiondef:test-data-lift-wasserstein-2026aAnalysisProbabilityPDEFixes the quadruples of arguments that a second-order equation operator on the lift and its shifts accept, what it means for such a datum to be bounded at a level, and the two sets of data admitted at a level.Existence and Basic Estimates at a Maximiser of the Gauge-Doubled Difference on the Wasserstein Space
lemmalem:doubling-maximiser-wasserstein-2026aAnalysisProbabilityPDEFor a coercive penalty pair the difference of the delta-envelopes penalised by the squared centred heat gauge attains its supremum; at any maximiser the penalty and the second moment are bounded independently of the doubling strength, and if the two functions are Lipschitz for th…The Centred Heat Gauge: Metric Properties, Comparison with the Wasserstein Distance, Behaviour Under Translations, the Squared Gauge as a Test Function, and the Polarisation Inequality
lemmalem:centred-heat-gauge-wasserstein-2026aAnalysisProbabilityPDEThe centred heat gauge is a metric on the Wasserstein space dominated by a constant times W_2, is exactly quadratic along translations of either argument, and its square in one argument is a test function with an explicit gradient (a constant plus a bounded smooth field of size c…The Hamilton-Jacobi Equation of the Controlled Heat Equation with Square-Integrable White Noise on the Torus
equationeq:hamilton-jacobi-white-noise-heat-torus-2026bAnalysisPDEThe discounted viscous Hamilton-Jacobi equation on the Sobolev space of order -(s+1) of the torus whose second-order term is the trace along the trigonometric white noise, whose Hamiltonian is half the squared norm of the gradient, and whose drift is minus the Laplacian, written…What the Hamilton-Jacobi Equation of the Heat Equation with Square-Integrable White Noise Models, and What Is Not Proved
remarkrem:white-noise-heat-equation-torus-2026bAnalysisProbabilityPDEReads the well-posedness corollary for the white-noise heat equation as the dynamic programming equation of a controlled stochastic heat equation on the torus, explains why the noise is square-integrable white noise and why the equation lives on a negative-order Sobolev space, an…Well-Posedness of the Hamilton-Jacobi Equation of the Controlled Heat Equation with Square-Integrable White Noise on the Torus
corollarycor:white-noise-heat-hamilton-jacobi-well-posed-torus-2026bAnalysisPDEOn the Sobolev space of order -(s+1) of the torus, with s at least the dimension, the discounted viscous Hamilton-Jacobi equation driven by the trigonometric white noise, with drift minus the Laplacian and a bounded uniformly continuous running cost, has a bounded uniformly conti…The Data of the Hamilton-Jacobi Equation of the Heat Equation with Square-Integrable White Noise on the Torus: the Noise, the Nonlinearities and the Drift
lemmalem:white-noise-heat-data-torus-2026aAnalysisPDEOn the Sobolev triple of order s of the torus, the Fourier coefficient families of the trigonometric classes, enumerated along the lattice, form a sequence that is square-summable in the order -s space with an explicit bound on the sum; the ambient space is infinite-dimensional;…Well-Posedness of the Viscous Allen-Cahn Hamilton-Jacobi Equation on the Torus
corollarycor:viscous-allen-cahn-hamilton-jacobi-well-posed-torus-2026aAnalysisPDEWith a finite noise built from the trigonometric system, the Allen-Cahn Hamilton-Jacobi equation on the torus carries a second-order trace term and a Hamiltonian coefficient between zero and one, and still has a unique bounded continuous viscosity solution on the square-integrabl…The Trigonometric System is Continuously Differentiable and Orthogonal in the Sobolev Space of the Torus
lemmalem:trigonometric-system-sobolev-torus-2026aAnalysisPDEEach member of the trigonometric system is continuously differentiable and periodic, with partial derivatives again members of the system up to a factor; its classes therefore lie in the Sobolev space of the torus and are orthogonal there, with squared norm one plus four pi squar…Well-Posedness for a Viscous Hamilton-Jacobi Equation with a Monotone Nonlinearity on a Hilbert Triple
theoremthm:viscous-monotone-hamilton-jacobi-well-posed-hilbert-triple-2026aAnalysisPDEOn a Hilbert triple whose ambient space is not finite-dimensional, the viscous Hamilton-Jacobi equation with a monotone nonlinearity, a Lipschitz drift, a trace term along a square-summable sequence and a bounded uniformly continuous cost has exactly one bounded continuous viscos…A Viscous Hamilton-Jacobi Operator with a Monotone Nonlinearity Satisfies the Second-Order Comparison Hypotheses
propositionprop:viscous-monotone-hamilton-jacobi-hilbert-triple-2026bAnalysisPDEA viscous Hamilton-Jacobi operator on a Hilbert triple whose Hamiltonian is half a nonnegative form dominated by the identity, with a monotone nonlinearity, a Lipschitz drift and a trace term along a square-summable sequence in the form space, is degenerate elliptic and strictly…A Comparison Principle on a Hilbert Triple under the Second-Order Structure Condition
theoremthm:comparison-second-order-hilbert-triple-2026aAnalysisPDEFor a locally strictly proper operator satisfying the second-order structure, shift-continuity and tail-insensitivity conditions, a bounded viscosity subsolution and a bounded viscosity supersolution satisfy a uniform comparison estimate on nearby points, and the subsolution does…The Second-Order Test-Datum Estimate at a Sequentially Strict Maximum of the Doubled Function
lemmalem:second-order-doubled-test-estimate-hilbert-triple-2026bAnalysisPDEAt a sequentially strict maximum of the doubled function, both functions admit test data at nearby points of the domain of the form operator with a common gradient alpha(x1 - y1) and an admitted pair of forms, satisfying the sub- and supersolution inequalities up to the shift mod…First-Order Operators Satisfy the Second-Order Structure and Tail-Insensitivity Conditions
lemmalem:first-order-implies-second-order-hilbert-triple-2026aAnalysisPDEA structure pair for the first-order condition is a structure pair for the second-order condition, and an operator that ignores its form argument is tail-insensitive along every admissible basis.The Tail-Insensitivity Condition for an Equation Operator on a Hilbert Triple
definitiondef:tail-insensitivity-condition-hilbert-triple-2026aAnalysisPDEAn operator is tail-insensitive if, along an orthonormal basis of the ambient space lying in the form space, adding or subtracting a multiple of the tail form eventually changes the shifted operator by at most any prescribed amount on bounded convergent test data.