| import OAI.Geometry.RCD.Transport.StrongEntropyMeasure |
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|
| namespace OAI |
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| /-! # Strong entropy convexity on every bounded compressed optimal plan -/ |
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|
| 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 |
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| theorem compressed_plan_entropy_convexity (hn : 2 ≤ n) (p : X) {R : ℝ} (hR : 0 ≤ R) |
| (π : Measure C(Time,X)) [IsProbabilityMeasure π] (hg : ∀ᵐ γ ∂π, IsGeodesic γ) |
| (hp : IsOptimalCoupling (π.map (fun γ => γ 0)) (π.map (fun γ => γ 1)) |
| (π.map (fun γ => (γ 0,γ 1)))) |
| {C : ℝ≥0} (hcomp : ∀ t : Time, π.map (fun γ => γ t) ≤ ENNReal.ofReal C • hausdorff n X) |
| (hs₀ : ∀ᵐ x ∂π.map (fun γ => γ 0), dist x p ≤ R) |
| (hs₁ : ∀ᵐ x ∂π.map (fun γ => γ 1), dist x p ≤ R) (t : Time) : |
| boltzmannEntropy (hausdorff n X) (π.map (fun γ => γ t)) ≤ |
| (1-(t:ℝ))*boltzmannEntropy (hausdorff n X) (π.map (fun γ => γ 0))+ |
| (t:ℝ)*boltzmannEntropy (hausdorff n X) (π.map (fun γ => γ 1))- |
| (K/2)*(t:ℝ)*(1-(t:ℝ))* |
| (wassersteinSq (π.map (fun γ => γ 0)) (π.map (fun γ => γ 1))).toReal := by |
| let : CompleteSpace X := hX.complete |
| let : Nonempty X := nonempty_of_isProbabilityMeasure (π.map (fun γ => γ 0)) |
| have hfin := coupling_transportCost_ne_top hp.1 hp.2.1 p |
| (integrable_dist_sq_of_ae_bounded _ p hs₀) (integrable_dist_sq_of_ae_bounded _ p hs₁) |
| have he (s t : Time) : |
| (wassersteinSq (π.map (fun γ => γ s)) (π.map (fun γ => γ t))).toReal= |
| |(s:ℝ)-(t:ℝ)|^2* |
| (wassersteinSq (π.map (fun γ => γ 0)) (π.map (fun γ => γ 1))).toReal := by |
| rw [wassersteinSq_eq_cost (geodesic_plan_optimal_marginals π hg hp hfin s t), |
| geodesic_plan_cost π hg,ENNReal.toReal_mul,ENNReal.toReal_pow, |
| ENNReal.toReal_ofReal (abs_nonneg _),wassersteinSq_eq_cost hp] |
| have hsupport : ∀ᵐ x ∂π.map (fun γ => γ t), dist x p ≤ 3*R := by |
| apply (ae_map_iff (measurable_path_eval t).aemeasurable |
| (measurableSet_le (continuous_id.dist continuous_const).measurable measurable_const)).mpr |
| exact (optimal_geodesic_plan_bounded _ _ π hg hp p hR hs₀ hs₁).mono fun γ hγ => hγ.2 t |
| apply hX.bounded_entropy_geodesic_measure hn p _ _ _ (hcomp 0) (hcomp 1) (hcomp t) |
| hs₀ hs₁ hsupport t.property.1 t.property.2 |
| · rw [he] |
| simp only [show ((0 : Time) : ℝ) = 0 from rfl,sub_zero, |
| abs_of_nonneg t.property.1] |
| · rw [he] |
| simp only [show ((1 : Time) : ℝ) = 1 from rfl, |
| abs_of_nonpos (sub_nonpos.mpr t.property.2),neg_sub] |
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|
| end RegularPointCoordinates.NoncollapsedRCD |
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| end |
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| end OAI |
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