| import OAI.Geometry.RCD.Sobolev.BoundedLocalRepresentative |
| import OAI.Geometry.RCD.Metric.EuclideanPairCost |
|
|
| namespace OAI |
|
|
| /-! # Optimality from local dual potentials on bounded endpoint supports -/ |
|
|
| open Set Filter MeasureTheory Metric |
| open scoped Topology BoundedContinuousFunction NNReal |
| noncomputable section |
| namespace RegularPointCoordinates |
|
|
| theorem bounded_pair_cost_integrable {n : ℕ} {B : ℝ} (hB : 0≤B) |
| (π : Measure (Euclidean n × Euclidean n)) [IsFiniteMeasure π] |
| (hb : ∀ᵐ z ∂π, z∈closedBall (0:Euclidean n) B ×ˢ closedBall (0:Euclidean n) B) : |
| Integrable (fun z => dist z.1 z.2^2) π := by |
| apply ((boundedPairCost B).integrable π).congr |
| filter_upwards [hb] with z hz |
| exact boundedPairCost_eq hB hz.1 hz.2 |
|
|
| theorem bounded_pair_transportCost {n : ℕ} {B : ℝ} (hB : 0≤B) |
| (π : Measure (Euclidean n × Euclidean n)) [IsFiniteMeasure π] |
| (hb : ∀ᵐ z ∂π, z∈closedBall (0:Euclidean n) B ×ˢ closedBall (0:Euclidean n) B) : |
| transportCost π=ENNReal.ofReal (∫ z, dist z.1 z.2^2 ∂π) := by |
| rw [ofReal_integral_eq_lintegral_ofReal (bounded_pair_cost_integrable hB π hb) |
| (Eventually.of_forall fun _ => sq_nonneg _)] |
| simp only [transportCost,ENNReal.ofReal_pow dist_nonneg] |
|
|
| / |
| theorem bounded_coupling_optimal_of_dual {n : ℕ} {B : ℝ} (hB : 0≤B) |
| (Q : ProbabilityMeasure (Euclidean n × Euclidean n)) |
| (hQ : ∀ᵐ z ∂(Q : Measure (Euclidean n × Euclidean n)), |
| z∈closedBall (0:Euclidean n) B ×ˢ closedBall (0:Euclidean n) B) |
| {f g : Euclidean n → ℝ} {L M : ℝ≥0} |
| (hf : LipschitzWith L f) (hg : LipschitzWith M g) |
| (had : ∀ x∈closedBall 0 B, ∀ y∈closedBall 0 B, f x+g y≤dist x y^2/2) |
| (hc : (∫ z, f z.1+g z.2 ∂(Q : Measure (Euclidean n × Euclidean n)))= |
| (∫ z, dist z.1 z.2^2 ∂(Q : Measure (Euclidean n × Euclidean n)))/2) : |
| IsOptimalCoupling ((Q : Measure (Euclidean n × Euclidean n)).map Prod.fst) |
| ((Q : Measure (Euclidean n × Euclidean n)).map Prod.snd) Q := by |
| obtain ⟨F,hF⟩ := bounded_local_representative hB hf.lipschitzOnWith |
| obtain ⟨G,hG⟩ := bounded_local_representative hB hg.lipschitzOnWith |
| let H : (Euclidean n × Euclidean n) →ᵇ ℝ := |
| F.compContinuous ⟨Prod.fst,continuous_fst⟩+G.compContinuous ⟨Prod.snd,continuous_snd⟩ |
| have hsep (τ : Measure (Euclidean n × Euclidean n)) [IsFiniteMeasure τ] : |
| (∫ z, H z ∂τ)=(∫ x, F x ∂τ.map Prod.fst)+(∫ y, G y ∂τ.map Prod.snd) := by |
| have hFi : Integrable (fun z : Euclidean n × Euclidean n => F z.1) τ := |
| (F.compContinuous ⟨Prod.fst,continuous_fst⟩).integrable τ |
| have hGi : Integrable (fun z : Euclidean n × Euclidean n => G z.2) τ := |
| (G.compContinuous ⟨Prod.snd,continuous_snd⟩).integrable τ |
| calc |
| (∫ z, H z ∂τ)=(∫ z, F z.1 ∂τ)+(∫ z, G z.2 ∂τ) := integral_add hFi hGi |
| _ = _ := by |
| rw [integral_map measurable_fst.aemeasurable F.continuous.aestronglyMeasurable, |
| integral_map measurable_snd.aemeasurable G.continuous.aestronglyMeasurable] |
| have hHQ : (∫ z, H z ∂(Q : Measure (Euclidean n × Euclidean n)))= |
| (∫ z, dist z.1 z.2^2 ∂(Q : Measure (Euclidean n × Euclidean n)))/2 := by |
| rw [← hc] |
| apply integral_congr_ae |
| filter_upwards [hQ] with z hz |
| exact congrArg₂ (·+·) (hF hz.1) (hG hz.2) |
| refine ⟨rfl,rfl,?_⟩ |
| intro π hfst hsnd |
| let : IsProbabilityMeasure π := coupling_probability_of_fst hfst |
| have hfQ : ∀ᵐ x ∂(Q : Measure (Euclidean n × Euclidean n)).map Prod.fst, |
| x∈closedBall (0:Euclidean n) B := |
| (ae_map_iff measurable_fst.aemeasurable measurableSet_closedBall).mpr |
| (hQ.mono fun _ hz => hz.1) |
| have hgQ : ∀ᵐ x ∂(Q : Measure (Euclidean n × Euclidean n)).map Prod.snd, |
| x∈closedBall (0:Euclidean n) B := |
| (ae_map_iff measurable_snd.aemeasurable measurableSet_closedBall).mpr |
| (hQ.mono fun _ hz => hz.2) |
| have hπ : ∀ᵐ z ∂π, z∈closedBall (0:Euclidean n) B ×ˢ closedBall (0:Euclidean n) B := by |
| have hh0 := ae_of_ae_map measurable_fst.aemeasurable (hfst.symm ▸ hfQ) |
| have hh1 := ae_of_ae_map measurable_snd.aemeasurable (hsnd.symm ▸ hgQ) |
| exact hh0.and hh1 |
| have heq : (∫ z, H z ∂(Q : Measure (Euclidean n × Euclidean n)))=∫ z, H z ∂π := by |
| rw [hsep,hsep,hfst,hsnd] |
| have hle : (∫ z, H z ∂π)≤(∫ z, dist z.1 z.2^2 ∂π)/2 := by |
| rw [← integral_div] |
| apply integral_mono_ae (H.integrable π) ((bounded_pair_cost_integrable hB π hπ).div_const 2) |
| filter_upwards [hπ] with z hz |
| change F z.1+G z.2≤_ |
| rw [hF hz.1,hG hz.2] |
| exact had _ hz.1 _ hz.2 |
| rw [bounded_pair_transportCost hB _ hQ,bounded_pair_transportCost hB π hπ] |
| apply ENNReal.ofReal_le_ofReal |
| rw [← heq,hHQ] at hle |
| linarith |
|
|
| end RegularPointCoordinates |
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| end |
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| end OAI |
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