| 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 |
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| end OAI |
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