Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
The Discounted Stationary Hopf-Lax Equation on an Open Subset of a Metric Space
equationeq:discounted-hopf-lax-metric-2026aAnalysisPDENames the discounted stationary equation rho u + (1/2)|grad u|^2 = f on an open subset of a metric space, as the slope-based equation of the Hamiltonian rho r + p^2/2 - f(x).- Names the metric eikonal equation |grad u| = f on an open subset of a metric space, as the slope-based equation of the Hamiltonian p - f(x).
Slope-Based Viscosity Solutions on a Metric Space: Standing Notation
settingset:metric-slope-viscosity-2026aAnalysisPDEStanding notation for slope-based viscosity solutions on a metric space: balls, distance to a set, semicontinuity, local slopes, test classes, Hamiltonians and s-solutions, with the basic slope lemma and Ekeland's principle in force.Slope-Based Viscosity Subsolutions, Supersolutions and Solutions on a Metric Space
definitiondef:slope-viscosity-solution-metric-2026aAnalysisPDEDefines Hamiltonians nondecreasing in the slope variable and the slope-based (Gangbo-Swiech) viscosity sub- and supersolutions of H = 0 on an open subset of a metric space, tested by a function of a test class plus a locally Lipschitz perturbation.Mean-Field Limit of the N-Particle Dyson Game: the Value per Particle Converges to the Solution of the Dyson Hamilton-Jacobi Equation on the Wasserstein Space
theoremthm:dyson-n-particle-mean-field-limit-2026aAnalysisProbabilityPDEFor N particles on the line with logarithmic repulsion beta/(2(N-1)), a confining potential, idiosyncratic noise sigma^2/(N-1) below the collision threshold sigma^2 < beta and mean-field running cost, the solution of the N-particle Hamilton-Jacobi equation divided by N converges,…The Half-Relaxed Limits of the Regularised N-Particle Dyson Solutions are Viscosity Sub- and Supersolutions of the Limit Dyson Equation up to a Cost Defect
propositionprop:dyson-half-relaxed-limits-viscosity-2026aAnalysisProbabilityPDEThe upper half-relaxed limit of the sup-convolved N-particle Dyson solutions divided by N is a viscosity subsolution, and the lower half-relaxed limit of the inf-convolved ones a viscosity supersolution, of the Dyson Hamilton-Jacobi equation on the Wasserstein space without commo…Gibbs Maximisers for the Regularised N-Particle Dyson Solutions Tested at a Local Extremum: Energy Bound, Concentration and the Score Identity
lemmalem:dyson-localised-gibbs-concentration-2026aAnalysisProbabilityPDEAt a point where the half-relaxed limit minus the penalty is locally dominated by a linear function plus a small multiple of the distance, the Gibbs maximisers of the regularised N-particle solution minus N times the quadratic block test function minus the penalty have bounded en…Half-Relaxed Limits of the Sup- and Inf-Convolved N-Particle Dyson Solutions: Bounds, Continuity and Limits along Configurations of Bounded Energy
lemmalem:dyson-regularised-half-relaxed-limits-2026aAnalysisProbabilityPDERegularise the N-particle Dyson solution v_N by a sup-convolution and an inf-convolution with wall weight 1/N and fixed parameter tau. The upper and lower half-relaxed limits of the regularised solutions divided by N are bounded, satisfy an explicit W2 modulus on the finite-energ…The N-Particle Dyson Game and its Mean-Field Limit: Standing Data and Equations
settingset:dyson-n-particle-2026aAnalysisProbabilityPDEStanding data for the mean-field limit of the N-particle Dyson game: interaction beta, noise sigma with sigma^2 < beta, a confining potential V, a discount lambda and a bounded uniformly continuous cost g; the N-particle Hamilton-Jacobi equation on the Weyl chamber with its solut…The N-Particle Dyson Hamilton-Jacobi Equation with a Mean-Field Running Cost
equationeq:dyson-n-particle-hamilton-jacobi-2026aAnalysisProbabilityPDEThe Hamilton-Jacobi equation of N controlled particles on the Weyl chamber with logarithmic repulsion of strength beta/(2(N-1)), confinement V, idiosyncratic noise of intensity sigma^2/(N-1), quadratic control cost and running cost N times a function of the empirical measure.The N-Particle Dyson Potential on the Weyl Chamber: Penalty, Monotone Gradient, Dissipation with Constants Uniform in N, and Gibbs Integrability
lemmalem:dyson-n-particle-data-weyl-chamber-2026aAnalysisPDEFor a confining potential V on the line and sigma^2 < beta, the N-particle Dyson potential P_N = H_{b_N} + sum V(x_k) with b_N = beta/(2(N-1)) is a penalty on the Weyl chamber with monotone gradient, satisfies the dissipation inequality with noise kappa_N = sigma^2/(N-1), epsilon…Integrating a Semiconvex Viscosity Subsolution of the Penalty-Drift Equation against a Measure of Finite Relative Fisher Information
lemmalem:semiconvex-subsolution-weak-form-euclidean-2026aAnalysisProbabilityPDEFor bounded w on an open convex D with |Dw|^2 <= A + B(U - p0), the a.e. gradient of a continuous w is Borel and square-integrable against measures of finite relative free energy. A semiconvex viscosity subsolution of the penalty-drift operator satisfies, for every mu of finite r…Sup- and Inf-Convolutions of Bounded Viscosity Sub- and Supersolutions of the Penalty-Drift Equation with a Convex Penalty
lemmalem:penalty-drift-sup-inf-convolution-euclidean-2026aAnalysisPDEFor the penalty-drift equation with a penalty whose gradient is monotone and which satisfies the dissipation inequality, the penalised sup-convolution of a bounded viscosity subsolution (and the penalised inf-convolution of a bounded viscosity supersolution) is attained at points…The Mean-Field Limit of the Lifted N-Particle Hamilton-Jacobi Equation with a Mollified Local Coupling
theoremthm:n-particle-local-coupling-limit-wasserstein-2026aAnalysisProbabilityPDEWith eps_N -> 0 and N eps_N^d -> infinity, the bounded solutions U_N of the lifted N-particle equations with the mollified N-particle costs satisfy U_N(mu^{tensor N})/N -> u(mu) uniformly on energy sublevel sets, and are asymptotically bounded below by u at one-particle marginals…Lower Convergence: The Solution with the Local Cost Is Asymptotically below the Mean-Field Solutions with the Mollified Costs
corollarycor:n-particle-mollified-lower-convergence-wasserstein-2026aAnalysisProbabilityPDEIf eps_N tends to 0, the bounded solution of the equation with the local density cost is, uniformly on energy sublevel sets, asymptotically below the bounded solutions of the mean-field equations with the mollified mean-field costs.Upper Convergence: Mean-Field Solutions with the Tensor-Averaged Mollified Costs Are Asymptotically below the Solution with the Local Cost
corollarycor:n-particle-mollified-upper-convergence-wasserstein-2026aAnalysisProbabilityPDEIf N eps_N^d tends to infinity, the bounded solution of the mean-field equation with the tensor-averaged cost of the mollified N-particle cost is, locally uniformly on energy sublevel sets, asymptotically below the bounded solution of the equation with the local density cost.The Mollified N-Particle Cost Dominates N Times the Mollified Mean-Field Cost of the One-Particle Marginal
lemmalem:mollified-cost-marginal-domination-wasserstein-2026aAnalysisProbabilityPDEFor every P in P_2(R^{dN}), N g_eps(P^{[1]}) <= int c_{N,eps} dP: the linear part is exact through the one-particle marginal and the density part follows from Jensen's inequality pointwise and Tonelli's theorem.The Mollified N-Particle Cost: Regularity, Its Tensor-Averaged Cost, and the Defect against the Local Cost
lemmalem:mollified-cost-tensor-average-wasserstein-2026aAnalysisProbabilityPDEThe mollified mean-field and N-particle costs are bounded and uniformly continuous; the tensor-averaged cost of the N-particle cost is the linear cost plus the tensor average of the mollified density cost at the empirical measure; and its excess over the local cost is bounded by…The Mollified Mean-Field Cost and the Mollified N-Particle Cost of a Local Coupling
definitiondef:n-particle-mollified-cost-wasserstein-2026aAnalysisProbabilityPDEDefines the mollified mean-field cost g_eps(nu) = int f dnu + G_{Phi,eps}(nu) on P_2(R^d) and the mollified N-particle cost c_{N,eps}(x) = N g_eps(mu^N_x) on R^{dN}, the empirical-measure form of a mollified local coupling.Stability of Comparison on the Wasserstein Space under Locally Vanishing Cost Defects
theoremthm:stability-cost-defect-wasserstein-2026bAnalysisProbabilityPDELet F satisfy the hypotheses of the comparison principle, and let F_n, F'_n be operators with F - h_n <= F_n and F'_n <= F + h'_n, where the defects h_n, h'_n take values in [0,H] and tend to zero uniformly on every set of measures with bounded penalty and bounded score. If u_n i…