Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
Well-Posedness of the Langevin Hamilton-Jacobi Equation on Euclidean Space under a Dissipation Inequality
theoremthm:langevin-well-posed-euclidean-2026aAnalysisPDEFor a C^2 potential V on R^n with compact sublevel sets satisfying (kappa/2) tr D^2V <= (1-eps)|DV|^2 + lambda V + C, and bounded continuous g, the Langevin Hamilton-Jacobi equation satisfies comparison in the class of V-subordinate growth and has exactly one viscosity solution t…The Hamilton-Jacobi Equation for Controlled Langevin Dynamics in a Potential on Euclidean Space
equationeq:langevin-hamilton-jacobi-euclidean-2026aAnalysisPDEThe discounted Hamilton-Jacobi equation lambda u + (theta/2)|Du|^2 + DV.Du - (kappa/2) tr D^2u = g on R^n of a controlled Langevin particle in a C^2 potential V.The Gradient of a Function on Euclidean Space is One-Sided Lipschitz on Bounded Sets
lemmalem:c2-gradient-one-sided-lipschitz-euclidean-2026aAnalysisFor a C^2 function V on R^n and a bounded set B there is c >= 0 with (DV(x)-DV(y)).(x-y) >= -c|x-y|^2 for all x, y in B.Well-Posedness of the Penalty-Drift Hamilton-Jacobi Equation under a Dissipation Inequality
theoremthm:penalty-drift-well-posed-euclidean-2026aAnalysisPDEIf P is a penalty whose gradient is one-sided Lipschitz on sublevel sets and which satisfies the dissipation inequality (kappa/2) tr D^2P <= (1-eps)|DP|^2 + lambda P + C, then for bounded continuous g the penalty-drift Hamilton-Jacobi equation satisfies comparison in the class of…The Penalty-Drift Hamilton-Jacobi Equation on an Open Subset of Euclidean Space
equationeq:penalty-drift-hamilton-jacobi-euclidean-2026bAnalysisPDEThe discounted viscous Hamilton-Jacobi equation lambda u + (theta/2)|Du|^2 + DP.Du - (kappa/2) tr D^2u = g on an open set D, whose drift is the gradient of a C^2 potential P.The Viscous Hamilton-Jacobi Operator with a Drift that is One-Sided Lipschitz on Sublevel Sets Satisfies the Operator Hypotheses of the Weighted-Penalty Comparison Principle
lemmalem:quadratic-operator-weighted-penalty-euclidean-2026aAnalysisPDEThe operator lambda r + (theta/2)|p|^2 + b(x).p - (kappa/2) tr X - G(x) on the domain of a penalty, with continuous b and G and a drift b that is one-sided Lipschitz on each sublevel set of the penalty (so it may blow up at the boundary), is continuous, strictly proper, convex an…Gauge Covariance under Subtraction of a Multiple of a Function: Viscosity and Classical Sub- and Supersolutions and the Structural Properties of the Operator
lemmalem:gauge-weighted-penalty-euclidean-2026aAnalysisPDESubtracting cP from the unknown turns sub- and supersolutions of F, viscosity or classical, into those of the operator F_c obtained by shifting the value, gradient and Hessian arguments by cP, cDP and cD^2P; continuity, degenerate ellipticity, strict properness and convexity pass…Perron's Method with a Weighted Penalty for Operators Convex in the Value, Gradient and Matrix Variables: Existence and Uniqueness of a Viscosity Solution of Given Weight
theoremthm:perron-weighted-penalty-convex-euclidean-2026aAnalysisPDEUnder the hypotheses of the weighted-penalty comparison principle (in particular F convex in (r,p,X), with no convexity in x), if a viscosity subsolution and a viscosity supersolution of weight w exist, then exactly one function of weight w is both; it is continuous and lies betw…Comparison Principle with a Weighted Penalty for Operators Convex in the Value, Gradient and Matrix Variables
theoremthm:comparison-weighted-penalty-convex-euclidean-2026aAnalysisPDELet F be continuous, strictly proper, convex in (r,p,X) at each x (no convexity in x), and satisfy the structure condition on the sublevel sets of a penalty P, and let sP-K be a classical subsolution for some s<w. Then a viscosity subsolution u with u-wP of P-subordinate growth f…A Convex Combination of a Viscosity Subsolution and a Classical Subsolution is a Viscosity Subsolution, for an Operator Convex in the Value, Gradient and Matrix Variables
lemmalem:convex-combination-subsolution-euclidean-2026aAnalysisPDEFor an operator convex in (r,p,X), the combination (1-t)u + t phi of a viscosity subsolution u and a classical subsolution phi, with 0 <= t < 1, is again a viscosity subsolution.Restriction of Viscosity Sub- and Supersolutions to an Open Subset
lemmalem:viscosity-locality-euclidean-2026aAnalysisPDEThe restriction of a viscosity subsolution (supersolution) of an operator on an open set to an open subset is a viscosity subsolution (supersolution) of the restricted operator.Second-Order Equation Operator Convex in the Value, Gradient and Matrix Variables
definitiondef:operator-convex-value-gradient-hessian-2026aAnalysisPDEAn operator F(x,r,p,X) is convex in (r,p,X) if, at each fixed point x, it is jointly convex in the value r, the gradient p and the matrix X; nothing is assumed about its dependence on x.Penalty-Subordinate Growth of a Function on an Open Subset of Euclidean Space
definitiondef:penalty-subordinate-growth-euclidean-2026aAnalysisPDEA function on the domain of a penalty P has P-subordinate growth from above (below) if for every positive delta it is bounded above (below) by a constant plus (minus) delta times P.Basic Properties of the Sublevel Sets of a Penalty
lemmalem:penalty-open-set-basic-euclidean-2026aAnalysisPDEA penalty is bounded below; each strict sublevel set is a bounded open set whose closure is compact and lies in the closed sublevel set, and the penalty equals the level on its boundary.Penalty on an Open Subset of Euclidean Space
definitiondef:penalty-open-set-euclidean-2026aAnalysisPDEA penalty on an open subset D of Euclidean space is a real function of class C^2 on D all of whose sublevel sets are compact.Vector Fields with Test-Function Components are Dense in the Square-Integrable Vector Fields Against a Measure of Finite Second Moment
lemmalem:test-fields-dense-l2-euclidean-2026aAnalysisProbabilityVector fields whose components are smooth and compactly supported are dense in the square-integrable vector fields against any probability measure of finite second moment.Well-Posedness of the Hamilton-Jacobi Equation with Common Noise for Controlled Langevin Dynamics: Existence and Uniqueness of a Bounded Viscosity Solution
theoremthm:langevin-common-noise-well-posed-wasserstein-2026bAnalysisProbabilityPDEFor a confining potential, positive noise, control cost at most one, any common-noise matrix and a bounded uniformly continuous running cost, the Hamilton-Jacobi equation with common noise for controlled Langevin dynamics satisfies comparison for bounded sub- and supersolutions a…The Hamilton-Jacobi Equation with Common Noise for Controlled Langevin Dynamics in a Confining Potential on the Wasserstein Space
equationeq:langevin-common-noise-hamilton-jacobi-wasserstein-2026bAnalysisProbabilityPDEThe discounted Hamilton-Jacobi equation with common noise for controlled Langevin dynamics in a confining potential: the penalty-drift equation of the Langevin free-energy pair, whose drift is grad V plus sigma^2/2 times the score of the measure.The Langevin Free-Energy Pair is Displacement Convex, with Closed Score Along Couplings and Regular Penalised Maxima
lemmalem:langevin-free-energy-pair-regularity-euclidean-2026aProbabilityThe Langevin free-energy pair has closed score along couplings and regular penalised maxima, and it is displacement convex.The Langevin Free-Energy Pair is a Wasserstein-Coercive Penalty Pair: Growth Bounds, Continuity of the Translation Hessian, and the First Variation of the Penalty
lemmalem:langevin-free-energy-penalty-pair-euclidean-2026bProbabilityThe Langevin free-energy pair is a Wasserstein-coercive penalty pair whose domain has the map property; its translation Hessian is the integral of the Hessian of V, the penalty is lower semicontinuous and controls the second moment and the trace of the translation Hessian, each e…