| import OAI.Geometry.RCD.Transport.CompressedInitialInverse |
| import OAI.Geometry.RCD.Metric.InverseMeasureUnique |
|
|
| namespace OAI |
|
|
| /-! # Uniqueness among bounded compressed optimal dynamical plans -/ |
|
|
| open Set Filter MeasureTheory Metric TopologicalSpace |
| open scoped ENNReal NNReal Topology |
| noncomputable section |
| namespace RegularPointCoordinates.NoncollapsedRCD |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [ProperSpace X] [SeparableSpace X] {K : ℝ} {n : ℕ} |
| [IsLocallyFiniteMeasure (hausdorff n X)] (hX : NoncollapsedRCD K n X) |
| include hX |
| attribute [local instance] pathBorel pathBorelSpace |
|
|
| theorem compressed_plan_unique (hn : 2 ≤ n) (p : X) {R : ℝ} (hR : 0 ≤ R) |
| {ν₀ ν₁ : Measure X} |
| (π ρ : Measure C(Time,X)) [IsProbabilityMeasure π] [IsProbabilityMeasure ρ] |
| (hgπ : ∀ᵐ γ ∂π, IsGeodesic γ) (hgρ : ∀ᵐ γ ∂ρ, IsGeodesic γ) |
| (hpπ : IsOptimalCoupling ν₀ ν₁ (π.map (fun γ => (γ 0,γ 1)))) |
| (hpρ : IsOptimalCoupling ν₀ ν₁ (ρ.map (fun γ => (γ 0,γ 1)))) |
| {C : ℝ≥0} (hC : 0 < C) |
| (hcπ : ∀ t : Time, π.map (fun γ => γ t) ≤ ENNReal.ofReal C • hausdorff n X) |
| (hcρ : ∀ t : Time, ρ.map (fun γ => γ t) ≤ ENNReal.ofReal C • hausdorff n X) |
| (hs₀ : ∀ᵐ x ∂ν₀, dist x p ≤ R) (hs₁ : ∀ᵐ x ∂ν₁, dist x p ≤ R) : π=ρ := by |
| let ξ := (2:ℝ≥0∞)⁻¹ • (π+ρ) |
| obtain ⟨hprob,hgξ,hpξ,hπξ,hρξ⟩ := half_mixture_plan_data π ρ hgπ hgρ hpπ hpρ |
| let : IsProbabilityMeasure ξ := hprob |
| have heπ := optimalPlan_endpoints ν₀ ν₁ π hpπ |
| have heρ := optimalPlan_endpoints ν₀ ν₁ ρ hpρ |
| have heξ := optimalPlan_endpoints ν₀ ν₁ ξ hpξ |
| have hcξ := half_mixture_compression (hausdorff n X) π ρ hcπ hcρ |
| have hpξ' : IsOptimalCoupling (ξ.map (fun γ => γ 0)) (ξ.map (fun γ => γ 1)) |
| (ξ.map (fun γ => (γ 0,γ 1))) := by rwa [heξ.1,heξ.2] |
| obtain ⟨g,hg,hgi⟩ := hX.compressed_initial_inverse hn p hR ξ hgξ hpξ' hC hcξ |
| (by rwa [heξ.1]) (by rwa [heξ.2]) |
| have hhalf : (2:ℝ≥0∞)⁻¹ ≠ 0 := by norm_num |
| exact measure_eq_of_common_ae_inverse ξ π ρ (measurable_path_eval 0) hg hgi |
| ((Measure.absolutelyContinuous_smul hhalf).trans hπξ.absolutelyContinuous) |
| ((Measure.absolutelyContinuous_smul hhalf).trans hρξ.absolutelyContinuous) |
| (heπ.1.trans heρ.1.symm) |
|
|
| end RegularPointCoordinates.NoncollapsedRCD |
|
|
| end |
|
|
| end OAI |
|
|
