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e4d9d2b | 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 | import OAI.Geometry.RCD.Transport.DominatedOptimalPlan
namespace OAI
/-! # Bounded optimal recovery of an arbitrary finite-cost geodesic plan
Cutting off the original plan preserves optimality for the resulting
endpoint marginals. All integrable moments converge along the recovery.
-/
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
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)
end RegularPointCoordinates
end
end OAI
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