| import OAI.Geometry.RCD.Transport.DominatedOptimalPlan |
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| namespace OAI |
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| /-! # Bounded optimal recovery of an arbitrary finite-cost geodesic plan |
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| Cutting off the original plan preserves optimality for the resulting |
| endpoint marginals. All integrable moments converge along the recovery. |
| -/ |
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|
| open Set Filter MeasureTheory Metric TopologicalSpace |
| open scoped ENNReal Topology |
| noncomputable section |
| namespace RegularPointCoordinates |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [SeparableSpace X] |
| attribute [local instance] pathBorel pathBorelSpace |
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| omit [SeparableSpace X] in |
| theorem bounded_optimal_recovery |
| (ρ : ProbabilityMeasure C(Time,X)) (p : X) |
| (hg : ∀ᵐ γ ∂(ρ : Measure C(Time,X)), IsGeodesic γ) |
| (hp : IsOptimalCoupling ((ρ : Measure C(Time,X)).map (fun γ => γ 0)) |
| ((ρ : Measure C(Time,X)).map (fun γ => γ 1)) |
| ((ρ : Measure C(Time,X)).map (fun γ => (γ 0,γ 1)))) |
| (hc : transportCost ((ρ : Measure C(Time,X)).map (fun γ => (γ 0,γ 1)))≠∞) : |
| ∃ (ρs : ℕ → ProbabilityMeasure C(Time,X)) (R : ℕ → ℝ), |
| (∀ j, 0≤R j) ∧ |
| (∀ j, (ρs j : Measure C(Time,X))≤(2 : ℝ≥0∞) • (ρ : Measure C(Time,X))) ∧ |
| (∀ j, ∀ᵐ γ ∂(ρs j : Measure C(Time,X)), IsGeodesic γ) ∧ |
| (∀ j, ∀ᵐ γ ∂(ρs j : Measure C(Time,X)), ∀ t : Time, dist (γ t) p≤R j) ∧ |
| (∀ j, IsOptimalCoupling ((ρs j : Measure C(Time,X)).map (fun γ => γ 0)) |
| ((ρs j : Measure C(Time,X)).map (fun γ => γ 1)) |
| ((ρs j : Measure C(Time,X)).map (fun γ => (γ 0,γ 1)))) ∧ |
| Tendsto ρs atTop (𝓝 ρ) ∧ |
| ∀ f : C(Time,X) → ℝ, Integrable f (ρ : Measure C(Time,X)) → |
| Tendsto (fun j => ∫γ,f γ ∂(ρs j : Measure C(Time,X))) |
| atTop (𝓝 (∫γ,f γ ∂(ρ : Measure C(Time,X)))) := by |
| obtain ⟨ρs,R,w,he,hw,hb,hl,hle,hR,hlim,hint⟩ := |
| weighted_probability_recovery ρ (ContinuousMap.const Time p) |
| refine ⟨ρs,fun j => max (R j) 0,fun _ => le_max_right _ _,hle,?_,?_,?_,hlim,hint⟩ |
| · intro j |
| exact (hg.filter_mono Measure.smul_absolutelyContinuous.ae_le).filter_mono |
| (hle j).absolutelyContinuous.ae_le |
| · intro j |
| filter_upwards [hR j] with γ hγ |
| intro t |
| have hd := ContinuousMap.dist_apply_le_dist (f:=ContinuousMap.const Time p) (g:=γ) t |
| have ht : dist (γ t) p≤R j := by |
| simpa only [ContinuousMap.const_apply,dist_comm p (γ t)] using hd.trans hγ |
| exact ht.trans (le_max_left _ _) |
| · intro j |
| exact dominated_optimal_plan (ρ : Measure C(Time,X)) (ρs j : Measure C(Time,X)) hp hc (hle j) |
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| end RegularPointCoordinates |
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| end |
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| end OAI |
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