| 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 |
| |
