File size: 4,045 Bytes
110f4f3 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 | import Mathlib
namespace OAI
noncomputable section
open Set MeasureTheory Manifold Bundle
open scoped ContDiff Manifold ENNReal NNReal Topology
namespace WeakMTWTransport
abbrev Model (n : ℕ) := EuclideanSpace ℝ (Fin n)
section MetricDefinitions
variable {M : Type*} [MetricSpace M]
/-- Squared-distance cost with factor one half. -/
def cost (x y : M) : ℝ := dist x y ^ 2 / 2
/-- Calibration of Hausdorff measure to Euclidean volume. -/
def euclideanVolumeFactor (n : ℕ) : ℝ≥0∞ :=
volume (Metric.ball (0 : Model n) 1) /
(Measure.hausdorffMeasure (n : ℝ) : Measure (Model n))
(Metric.ball (0 : Model n) 1)
variable [MeasurableSpace M]
/-- Riemannian volume calibrated to Lebesgue measure in Euclidean space. -/
def metricVolume [BorelSpace M] (n : ℕ) : Measure M :=
euclideanVolumeFactor n • Measure.hausdorffMeasure (n : ℝ)
/-- Almost-everywhere measurable probability densities between fixed bounds. -/
def AdmissibleDensity (vol : Measure M) (lam cap : ℝ) (rho : M → ℝ) : Prop :=
AEMeasurable rho vol ∧ Integrable rho vol ∧
(∫ x, rho x ∂vol) = 1 ∧ (∀ᵐ x ∂vol, lam ≤ rho x ∧ rho x ≤ cap)
/-- Multiplication by a density, ignoring exceptional negative values on null sets. -/
def densityMeasure (vol : Measure M) (rho : M → ℝ) : Measure M :=
vol.withDensity (fun x => ENNReal.ofReal (rho x))
/-- Optimality for the Monge minimization problem. -/
def IsOptimalMap (vol : Measure M) (rho0 rho1 : M → ℝ) (T : M → M) : Prop :=
Measurable T ∧ Measure.map T (densityMeasure vol rho0) = densityMeasure vol rho1 ∧
∀ S : M → M, Measurable S →
Measure.map S (densityMeasure vol rho0) = densityMeasure vol rho1 →
(∫ x, cost x (T x) * rho0 x ∂vol) ≤
(∫ x, cost x (S x) * rho0 x ∂vol)
/-- The two global estimates have the very same exponent and constant. -/
def BiHolderEstimate (alpha C : ℝ) (T : M ≃ₜ M) : Prop :=
(∀ x x', dist (T x) (T x') ≤ C * (dist x x') ^ alpha) ∧
(∀ y y', dist (T.symm y) (T.symm y') ≤ C * (dist y y') ^ alpha)
end MetricDefinitions
variable {n : ℕ} {M : Type*} [MetricSpace M] [ChartedSpace (Model n) M]
[IsManifold 𝓘(ℝ, Model n) ∞ M]
[RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ, Model n) x)]
/-- Complete affinely parametrized geodesic with prescribed initial tangent.
Local metric minimization is an intrinsic definition of Riemannian geodesics. -/
def IsGeodesicWithInitialData (x : M) (v : TangentSpace 𝓘(ℝ, Model n) x)
(gamma : ℝ → M) : Prop :=
ContMDiff 𝓘(ℝ, ℝ) 𝓘(ℝ, Model n) ∞ gamma ∧
(⟨gamma 0, mfderiv 𝓘(ℝ, ℝ) 𝓘(ℝ, Model n) gamma 0 (1 : ℝ)⟩ :
TangentBundle 𝓘(ℝ, Model n) M) = ⟨x, v⟩ ∧
∀ t : ℝ, ∃ eps : ℝ, 0 < eps ∧
∀ s u : ℝ, |s - t| < eps → |u - t| < eps →
dist (gamma s) (gamma u) = |s - u| * ‖v‖
/-- Evaluation at time one of the geodesic determined by initial data. -/
def riemannianExp (x : M) (v : TangentSpace 𝓘(ℝ, Model n) x) : M := by
classical
exact if h : ∃ gamma, IsGeodesicWithInitialData x v gamma then (Classical.choose h) 1 else x
/-- The open injectivity domain, defined by extendible minimizing vectors. -/
def injectivityDomain (x : M) : Set (TangentSpace 𝓘(ℝ, Model n) x) :=
{p | ∃ a : ℝ, 1 < a ∧ dist x (riemannianExp x (a • p)) = a * ‖p‖}
/-- The Ma–Trudinger–Wang fourth derivative with factor -3/2. -/
def mtw (x : M) (p xi eta : TangentSpace 𝓘(ℝ, Model n) x) : ℝ :=
-(3 / 2 : ℝ) *
iteratedDeriv 2 (fun s : ℝ =>
iteratedDeriv 2 (fun t : ℝ =>
cost (riemannianExp x (t • xi)) (riemannianExp x (p + s • eta))) 0) 0
/-- Weak MTW only at interior velocities and only for orthogonal test vectors. -/
def WeakMTW : Prop :=
∀ (x : M) (p : TangentSpace 𝓘(ℝ, Model n) x), p ∈ injectivityDomain x →
∀ xi eta : TangentSpace 𝓘(ℝ, Model n) x,
inner ℝ xi eta = 0 → 0 ≤ mtw x p xi eta
end WeakMTWTransport
end
end OAI
|