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7d9f027 | 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 | import Mathlib
import OAI.Probability.SKGap.Stability.FieldTonelli
import OAI.Probability.SKGap.Localization.ConditionalIntegralRate
namespace OAI
section
noncomputable section
namespace SKGap
open Matrix Real Set MeasureTheory ProbabilityTheory GaussianDensity
open scoped BigOperators Matrix.Norms.Frobenius ENNReal
lemma measurable_fieldIntegral {n : ℕ} (j t σ : ℝ)
(E : Set (((Fin n×Fin n) → ℝ) × (Fin n → ℝ))) (hE : MeasurableSet E) :
Measurable (fun p : (MatrixCoordinates (Fin n) → ℝ) × (Fin n → ℝ)=>
fieldIntegral j σ (plantedInteraction j (goeMatrix (j/(n:ℝ)) p.1))
(fun i=>t+p.2 i) {y | ((fun q=>goeMatrix (j/(n:ℝ)) p.1 q.1 q.2),y)∈E}) := by
let H : ((MatrixCoordinates (Fin n) → ℝ) × (Fin n → ℝ)) × (Fin n → ℝ) → ℝ≥0∞ :=
fun p=>ENNReal.ofReal (fieldMollifier j σ (plantedInteraction j (goeMatrix (j/(n:ℝ)) p.1.1))
(fun i=>t+p.1.2 i) p.2)* E.indicator (fun _=>1) ((fun q=>goeMatrix (j/(n:ℝ)) p.1.1 q.1 q.2),p.2)
have hsJ : Continuous (fun p : ((MatrixCoordinates (Fin n) → ℝ) × (Fin n → ℝ)) × (Fin n → ℝ) =>
plantedInteraction j (goeMatrix (j/(n:ℝ)) p.1.1)) :=
((goeMatrix_pi_lipschitz (j/(n:ℝ))).continuous.comp (continuous_fst.comp continuous_fst)).add continuous_const
have htime : Continuous (fun p : ((MatrixCoordinates (Fin n) → ℝ) × (Fin n → ℝ)) × (Fin n → ℝ) =>
(fun i=>t+p.1.2 i)) := by
apply continuous_pi
intro i
exact continuous_const.add ((continuous_apply i).comp (continuous_snd.comp continuous_fst))
have hmap : Continuous (fun p : ((MatrixCoordinates (Fin n) → ℝ) × (Fin n → ℝ)) × (Fin n → ℝ) =>
(plantedInteraction j (goeMatrix (j/(n:ℝ)) p.1.1),((fun i=>t+p.1.2 i),p.2))) :=
hsJ.prodMk (htime.prodMk continuous_snd)
have hφ := ((continuous_fieldMollifier j σ).comp hmap).measurable.ennreal_ofReal
have hI : Measurable (fun p : ((MatrixCoordinates (Fin n) → ℝ) × (Fin n → ℝ)) × (Fin n → ℝ) =>
E.indicator (fun _=>(1 : ℝ≥0∞)) ((fun q=>goeMatrix (j/(n:ℝ)) p.1.1 q.1 q.2),p.2)) := by
apply (measurable_const.indicator hE).comp
unfold goeMatrix
fun_prop
have hH : Measurable H := hφ.mul hI
have he (p : (MatrixCoordinates (Fin n) → ℝ) × (Fin n → ℝ)) :
fieldIntegral j σ (plantedInteraction j (goeMatrix (j/(n:ℝ)) p.1))
(fun i=>t+p.2 i) {y | ((fun q=>goeMatrix (j/(n:ℝ)) p.1 q.1 q.2),y)∈E}=
∫⁻ y,H (p,y) := by
have hs : MeasurableSet {y : Fin n → ℝ | ((fun q=>goeMatrix (j/(n:ℝ)) p.1 q.1 q.2),y)∈E} := hE.preimage (by fun_prop)
erw [fieldIntegral_indicator _ _ _ _ _ hs]
apply lintegral_congr
intro y
dsimp [H]
by_cases hy : ((fun q=>goeMatrix (j/(n:ℝ)) p.1 q.1 q.2),y)∈E <;> simp [hy]
simp_rw [he]
exact hH.lintegral_prod_right'
lemma measurableSet_badField_section {n : ℕ} (j A ε m : ℝ)
(W : Matrix (Fin n) (Fin n) ℝ) :
MeasurableSet {y : Field n | (W,y)∈badFieldSet n j A ε m} := by
have hm : Measurable (fun z : Fin n → ℝ=>((fun q : Fin n × Fin n=>W q.1 q.2),z)) :=
measurable_const.prodMk measurable_id
have hs := (measurableSet_badFieldFlat n j A ε m).preimage hm
convert hs using 1
ext y
rfl
lemma measurable_fieldIntegral_cylinder {n : ℕ} (j t σ : ℝ)
(E : Set (Field n)) (hE : MeasurableSet E) :
Measurable (fun p : (MatrixCoordinates (Fin n) → ℝ) × (Fin n → ℝ)=>
fieldIntegral j σ (plantedInteraction j (goeMatrix (j/(n:ℝ)) p.1))
(fun i=>t+p.2 i) E) := by
simpa only [mem_prod,mem_univ,true_and,ofPred_mem_eq] using
measurable_fieldIntegral j t σ (Set.univ ×ˢ E) (MeasurableSet.univ.prod hE)
lemma measureReal_union_detector {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsFiniteMeasure μ] (E D : Set Ω) (F : Ω → ℝ≥0∞)
(b p q : ℝ) (hbD : μ.real D ≤ p)
(hbF : μ.real {x | ENNReal.ofReal b ≤ F x} ≤ q)
(hdet : ∀ x ∈ E, x∉D → ENNReal.ofReal b ≤ F x) : μ.real E ≤ p+q := by
apply (measureReal_mono (s₂ := D ∪ {x | ENNReal.ofReal b ≤ F x}) ?_ (measure_ne_top _ _)).trans
· exact (measureReal_union_le _ _).trans (add_le_add hbD hbF)
· intro x hx
by_cases hd : x∈D
· exact Or.inl hd
· exact Or.inr (hdet x hx hd)
lemma markov_exponential {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)
{F : Ω → ℝ≥0∞} (hF : Measurable F) {a b d : ℝ}
(hb : ∫⁻ x,F x ∂μ ≤ ENNReal.ofReal (exp (-a*d))) :
μ.real {x | ENNReal.ofReal (exp (-b*d)) ≤ F x} ≤ exp (-(a-b)*d) := by
have hm := meas_ge_le_lintegral_div (μ := μ) (ε := ENNReal.ofReal (exp (-b*d))) hF.aemeasurable
(ENNReal.ofReal_pos.mpr (exp_pos _)).ne' ENNReal.ofReal_ne_top
have he : ENNReal.ofReal (exp (-a*d))/ENNReal.ofReal (exp (-b*d))=
ENNReal.ofReal (exp (-(a-b)*d)) := by
rw [← ENNReal.ofReal_div_of_pos (exp_pos _),← exp_sub]
congr 2
ring
have hx := hm.trans ((ENNReal.div_le_div_right hb _).trans_eq he)
exact (ENNReal.toReal_mono ENNReal.ofReal_ne_top hx).trans_eq (ENNReal.toReal_ofReal (exp_pos _).le)
end SKGap
end
end
end OAI
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