File size: 6,451 Bytes
b37e936 | 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 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 | import OAI.Geometry.RCD.Heat.UniformCenteredSecondVariation
import OAI.Geometry.RCD.Heat.CompactHeatActualPlanLimit
import OAI.Geometry.RCD.Metric.DominatedCompactProbability
import OAI.Geometry.RCD.Transport.CompressedFiniteUnique
namespace OAI
/-! # Local heat second variation on the prescribed compressed optimal plan -/
open Set Filter MeasureTheory Metric TopologicalSpace
open scoped Topology NNReal ENNReal
noncomputable section
namespace RegularPointCoordinates.NoncollapsedRCD
variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X]
[ProperSpace X] {K : ℝ} {n : ℕ} (hX : NoncollapsedRCD K n X)
(hn : 2 ≤ n) [SeparableSpace X]
[IsLocallyFiniteMeasure (hausdorff n X)] [(hausdorff n X).Regular]
attribute [local instance] pathBorel pathBorelSpace
theorem bounded_plan_heat_second_variation (p₀ p : X) (hK : K ≤ 0)
(π : Measure C(Time,X)) [IsProbabilityMeasure π]
(hgeo : ∀ᵐ γ ∂π, IsGeodesic γ)
(hopt : IsOptimalCoupling (π.map (fun γ => γ 0)) (π.map (fun γ => γ 1))
(π.map (fun γ => (γ 0,γ 1))))
{B : ℝ≥0∞} (hB : B ≠ ∞)
(hcomp : ∀ t : Time, π.map (fun γ => γ t) ≤ B • hausdorff n X)
{R₀ r : ℝ} (hR₀ : 0 ≤ R₀) (hr : 0 < r) (hsmall : 6*R₀ ≤ r)
(hs₀ : ∀ᵐ x ∂π.map (fun γ => γ 0), dist x p ≤ R₀)
(hs₁ : ∀ᵐ x ∂π.map (fun γ => γ 1), dist x p ≤ R₀)
(f : Lp ℝ 2 (hausdorff n X)) (A : ℝ≥0)
(hf : ∀ᵐ x ∂hausdorff n X, |f x| ≤ A) {s : ℝ} (hs : 0 < s) :
let m := hausdorff n X
let q := hX.quadratic
let H := kernelHeat m q (hX.canonicalHeatKernel_isContinuousHeatKernel hn p) s f
let F t := if ht : t ∈ Ioo (0:ℝ) 1 then
∫ x, H x ∂π.map (fun γ => γ (⟨t,ht.1.le,ht.2.le⟩ : Time)) else 0
let w := heatSobolev m q s hs f
let C := hX.centeredActionBound hn p₀
∃ G : ℝ → ℝ,
(∀ t ∈ Ioo (0:ℝ) 1, DifferentiableAt ℝ F t) ∧
(∀ a b : ℝ, 0 < a → b < 1 → ∃ L : ℝ≥0, LipschitzOnWith L (deriv F) (Icc a b)) ∧
(∀ (δ : ℝ) (hδ : 0 < δ),
let L : ℝ≥0 := ⟨Real.sqrt (Real.exp |K| * (16*C*(R₀+32*r)^2/(δ/2)^2)),Real.sqrt_nonneg _⟩
∀ (t : ℝ) (ht : t ∈ Ioo δ 1), ∃ (v : SobolevSpace m q) (V : X → ℝ)
(J : Lp ℝ 1 (m.restrict (closedBall p r))),
LipschitzWith (3*L) V ∧ HasCompactSupport V ∧
(SobolevSpace.toLp m q v =ᵐ[m] V) ∧
(∀ x, |V x| ≤ 2*r*(L:ℝ)) ∧
IsCanonicalHeatHessian m q w v p r (radialCutoff p (r/2)) J ∧
G t = (1/4:ℝ)*(∫ x in closedBall p r, J x ∂π.map
(fun γ => γ (⟨t,(hδ.trans ht.1).le,ht.2.le⟩ : Time)))) ∧
∀ a b : ℝ, 0 < a → a ≤ b → b < 1 →
IntervalIntegrable G volume a b ∧ (∫ t in a..b, G t) = deriv F b-deriv F a := by
intro m q H F w C
let S := closedBall p R₀
let : CompactSpace S := isCompact_iff_compactSpace.mp (isCompact_closedBall p R₀)
have hsupport (t : Time) (ht : ∀ᵐ x ∂π.map (fun γ => γ t), dist x p ≤ R₀) :
∀ᵐ x ∂π.map (fun γ => γ t), x ∈ S := by
simpa only [S,mem_closedBall] using ht
obtain ⟨μ,ρ₀,B₀,hprob₀,hμ,hm₀,hp₀,hC₀,hd₀⟩ :=
dominated_supported_probability_law m (π.map (fun γ => γ 0)) S measurableSet_closedBall
(hsupport 0 hs₀) hB (hcomp 0)
obtain ⟨ν,ρ₁,B₁,hprob₁,hν,hm₁,hp₁,hC₁,hd₁⟩ :=
dominated_supported_probability_law m (π.map (fun γ => γ 1)) S measurableSet_closedBall
(hsupport 1 hs₁) hB (hcomp 1)
let := hprob₀
let := hprob₁
obtain ⟨a₀⟩ := nonempty_of_isProbabilityMeasure μ
obtain ⟨b₀⟩ := nonempty_of_isProbabilityMeasure ν
let R := max 1 r
have hR : 1 ≤ R := le_max_left _ _
have hS₀ (y : S) : dist p y ≤ R₀ := by
simpa only [S,mem_closedBall,dist_comm] using y.property
have hS (y : S) : dist p y ≤ R := (hS₀ y).trans
((by linarith : R₀ ≤ r).trans (le_max_right _ _))
have he1 (k : ℕ) : heatStep k ≤ 1 := by
unfold heatStep
exact pow_le_one₀ (by norm_num) (by norm_num)
obtain ⟨D⟩ := hX.exists_compact_heat_scaling_data hn p hK a₀ b₀ μ ν
ρ₀ ρ₁ hm₀ hm₁ hp₀ hp₁ hC₀ hC₁ hd₀ hd₁ heatStep heatStep_pos he1 heatStep_tendsto
(zero_le_one.trans hR) hS hS
obtain ⟨π',j,B',_A,hj,hB',hg',hop',_hlen,hcomp',_hfor,_hback,hweak,_hcal⟩ :=
D.exists_actual_heat_limit_plan a₀ b₀ hK he1 heatStep_tendsto
ρ₀ ρ₁ hm₀ hm₁ hp₀ hp₁ hC₀ hC₁ hd₀ hd₁ hR hS hS
have hop : IsOptimalCoupling (μ.map Subtype.val) (ν.map Subtype.val)
(π.map (fun γ => (γ 0,γ 1))) := by rw [hμ,hν]; exact hopt
have heq : π=(π' : Measure C(Time,X)) :=
hX.compressed_plan_unique_finite hn p hR₀ π π' hgeo hg' hop hop' hB hB' hcomp hcomp'
(hμ.symm ▸ hs₀) (hν.symm ▸ hs₁)
let Q₀ (t : Ioo (0:ℝ) 1) : ProbabilityMeasure X :=
⟨π.map (fun γ => γ (⟨t,t.property.1.le,t.property.2.le⟩ : Time)),
inferInstance⟩
have hQ (t : Ioo (0:ℝ) 1) :
Tendsto (fun k => D.interpolationLaw (D.index (j k)) t) atTop (𝓝 (Q₀ t)) := by
have hqt : Q₀ t=π'.map (fun γ => γ (⟨t,t.property.1.le,t.property.2.le⟩ : Time)) := by
apply ProbabilityMeasure.toMeasure_injective
exact congrArg (fun ρ : Measure C(Time,X) => ρ.map
(fun γ => γ (⟨t,t.property.1.le,t.property.2.le⟩ : Time))) heq
rw [hqt]
exact (hweak t).1
have hbound := optimal_geodesic_plan_bounded (π.map (fun γ => γ 0))
(π.map (fun γ => γ 1)) π hgeo hopt p hR₀ hs₀ hs₁
have hstay (t : Ioo (0:ℝ) 1) : ∀ᵐ x ∂(Q₀ t : Measure X), x∈closedBall p (r/2) := by
apply (ae_map_iff (measurable_path_eval _).aemeasurable measurableSet_closedBall).mpr
filter_upwards [hbound] with γ hγ
exact (hγ.2 _).trans (by linarith)
obtain ⟨G,hder,hLip,hident,hint⟩ := D.centered_limit_mean_second_variation_uniform p₀
a₀ b₀ hK he1 heatStep_tendsto ρ₀ ρ₁ hm₀ hm₁ hp₀ hp₁ hC₀ hC₁ hd₀ hd₁ hR hS hS
j hj Q₀ hQ hr ((le_max_right 1 r).trans (by linarith)) hstay hR₀ hS₀ f A hf hs
refine ⟨G,hder,hLip,?_,hint⟩
intro δ hδ L t ht
obtain ⟨v,V,J,hV,hVc,hv,hVb,_hloc,hcan,hGi⟩ := hident δ hδ t ht
exact ⟨v,V,J,hV,hVc,hv,hVb,hcan,hGi⟩
end RegularPointCoordinates.NoncollapsedRCD
end
end OAI
|