| import Mathlib |
| import OAI.Analysis.CoulombIonization.RadialBounds.BarrierForgettingBarrier |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff |
| open Set Filter MeasureTheory Laplacian Metric ProbabilityTheory |
| open scoped Topology BigOperators |
| namespace CoulombBarrier |
| open CoulombAnalysis CoulombPDE CoulombAtom |
| variable {Ω D : Type*} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] |
| variable [MeasurableSpace D] (P : Measure Ω) [IsProbabilityMeasure P] |
| variable (X : Ω → D) (hX : Measurable X) |
|
|
| def conditionalField (u : Ω → TFSpace → ℝ) (d : D) (x : TFSpace) : ℝ := |
| ∫ sample, u sample x ∂condDistrib id X P d |
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| lemma conditionalField_measurable {u : Ω → TFSpace → ℝ} |
| (hu : Measurable (Function.uncurry u)) : |
| Measurable (Function.uncurry (conditionalField P X u)) := by |
| have hm : StronglyMeasurable (fun p : (D × TFSpace) × Ω => u p.2 p.1.2) := |
| (hu.comp (measurable_snd.prodMk measurable_fst.snd)).stronglyMeasurable |
| exact (hm.integral_kernel_prod_right' |
| (κ := (condDistrib id X P).comap Prod.fst measurable_fst)).measurable |
|
|
| lemma conditionalField_bound {u : Ω → TFSpace → ℝ} (hb : DeterministicLocalBound u) : |
| DeterministicLocalBound (conditionalField P X u) := by |
| intro K hK |
| obtain ⟨C,hC⟩ := hb K hK |
| refine ⟨C,fun d x hx => ?_⟩ |
| calc |
| _ ≤ ∫ _ : Ω, C ∂condDistrib id X P d := norm_integral_le_of_norm_le (integrable_const C) |
| (Eventually.of_forall fun sample => hC sample x hx) |
| _ = C := by simp |
|
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| include hX |
|
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| lemma conditionalField_continuous {u : Ω → TFSpace → ℝ} |
| (hu : Measurable (Function.uncurry u)) (hb : DeterministicLocalBound u) |
| (hc : ∀ᵐ sample ∂P, Continuous (u sample)) : |
| ∀ᵐ d ∂P.map X, Continuous (conditionalField P X u d) := by |
| filter_upwards [conditional_all_properties P hX hc] with d hd |
| exact average_continuous hu hb hd |
|
|
| theorem weak_nuclear_conditional {U : Set TFSpace} (Z : ℝ) |
| {u g : Ω → TFSpace → ℝ} |
| (hu : Measurable (Function.uncurry u)) (hub : DeterministicLocalBound u) |
| (huc : ∀ᵐ sample ∂P, Continuous (u sample)) |
| (hg : Measurable (Function.uncurry g)) (hgb : DeterministicLocalBound g) |
| {χ : TFSpace → ℝ} (hχ : Measurable χ) (hχb : ∀ x, ‖χ x‖ ≤ 1) (hχn : ∀ x, 0 ≤ χ x) |
| {R k : ℝ} (hR : 0 < R) (hk : 0 ≤ k) (hzero : ∀ x, ‖x‖ < R → χ x = 0) |
| (hw : ∀ᵐ sample ∂P, WeakNuclearLowerOn U Z (fun x => nuclearField Z x+u sample x) |
| (fun x => g sample x+χ x*reaction k (nuclearField Z x+u sample x))) : |
| ∀ᵐ d ∂P.map X, WeakNuclearLowerOn U Z |
| (fun x => nuclearField Z x+conditionalField P X u d x) |
| (fun x => conditionalField P X g d x+ |
| χ x*reaction k (nuclearField Z x+conditionalField P X u d x)) := by |
| filter_upwards [conditional_all_properties P hX huc,conditional_all_properties P hX hw] with d hc hw |
| exact weak_nuclear_average Z hu hub hc hg hgb hχ hχb hχn hR hk hzero hw |
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| end CoulombBarrier |
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| end |
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| end OAI |
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|
