Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
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…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…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.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.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…- At cutoff N the gradient energy of a function at a state is the sum over the kept modes of its squared first mode derivatives; the uncut gradient energy is the limit in N where it exists.
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…