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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 | import OAI.Geometry.RCD.Transport.StrongEntropyMeasure
namespace OAI
/-! # Strong entropy convexity on every bounded compressed optimal plan -/
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_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]
end RegularPointCoordinates.NoncollapsedRCD
end
end OAI
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