File size: 3,047 Bytes
159357f | 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 | 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
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
include hX
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
end CoulombBarrier
end
end OAI
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