| import OAI.Probability.FreeIsing.Energy.DiscrepancyEnergy |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped NNReal Topology |
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| namespace Problem367.DiscrepancyEnergy |
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| variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} |
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| theorem tendsto_integral_stopped_density [SFinite P] |
| {τ : ℕ → Ω → WithTop ℝ≥0} (hτ : ∀ n, Measurable (τ n)) |
| {E : ℝ → Ω → ℝ} {T : ℝ≥0} |
| (hE : Integrable (fun p : ℝ × Ω => E p.1 p.2) |
| ((volume.restrict (Ioc (0 : ℝ) T)).prod P)) |
| (hexit : ∀ᵐ ω ∂P, ∀ᶠ n in atTop, (T : WithTop ℝ≥0) ≤ τ n ω) : |
| Tendsto (fun n => ∫ s in (0 : ℝ)..(T : ℝ), |
| ∫ ω, if (s.toNNReal : WithTop ℝ≥0) ≤ τ n ω then E s ω else 0 ∂P) |
| atTop (𝓝 (∫ s in (0 : ℝ)..(T : ℝ), ∫ ω, E s ω ∂P)) := by |
| let Q := (volume.restrict (Ioc (0 : ℝ) (T : ℝ))).prod P |
| let A n := {p : ℝ × Ω | (p.1.toNNReal : WithTop ℝ≥0) ≤ τ n p.2} |
| have hA (n : ℕ) : MeasurableSet (A n) := |
| measurableSet_le ((measurable_real_toNNReal.comp measurable_fst).withTop_coe) |
| ((hτ n).comp measurable_snd) |
| have hEn (n : ℕ) : Integrable ((A n).indicator (fun p : ℝ × Ω => E p.1 p.2)) Q := |
| hE.indicator (hA n) |
| have hlim : Tendsto (fun n => ∫ p, (A n).indicator (fun p : ℝ × Ω => E p.1 p.2) p ∂Q) |
| atTop (𝓝 (∫ p : ℝ × Ω, E p.1 p.2 ∂Q)) := by |
| apply tendsto_integral_of_dominated_convergence (fun p : ℝ × Ω => ‖E p.1 p.2‖) |
| (fun n => (hEn n).aestronglyMeasurable) hE.norm |
| · intro n |
| apply Eventually.of_forall |
| intro p |
| by_cases hp : p ∈ A n <;> simp [hp] |
| · have hm : ∀ᵐ p ∂Q, p.1 ∈ Ioc (0 : ℝ) (T : ℝ) := |
| Measure.quasiMeasurePreserving_fst.ae (ae_restrict_mem measurableSet_Ioc) |
| have he : ∀ᵐ p ∂Q, ∀ᶠ n in atTop, (T : WithTop ℝ≥0) ≤ τ n p.2 := |
| Measure.quasiMeasurePreserving_snd.ae hexit |
| filter_upwards [hm, he] with p hp he |
| have hpT : p.1.toNNReal ≤ T := by |
| simpa only [Real.toNNReal_coe] using Real.toNNReal_mono hp.2 |
| apply tendsto_const_nhds.congr' |
| filter_upwards [he] with n hn |
| exact (Set.indicator_of_mem (show p ∈ A n from |
| (WithTop.coe_le_coe.mpr hpT).trans hn) (fun p : ℝ × Ω => E p.1 p.2)).symm |
| have heq (n : ℕ) : (∫ p, (A n).indicator (fun p : ℝ × Ω => E p.1 p.2) p ∂Q) = |
| ∫ s in (0 : ℝ)..(T : ℝ), |
| ∫ ω, if (s.toNNReal : WithTop ℝ≥0) ≤ τ n ω then E s ω else 0 ∂P := by |
| rw [integral_prod _ (hEn n), intervalIntegral.integral_of_le T.coe_nonneg] |
| simp only [Set.indicator, A, mem_ofPred_eq] |
| have heq' : (∫ p : ℝ × Ω, E p.1 p.2 ∂Q) = |
| ∫ s in (0 : ℝ)..(T : ℝ), ∫ ω, E s ω ∂P := by |
| rw [integral_prod _ hE, intervalIntegral.integral_of_le T.coe_nonneg] |
| simpa only [heq, heq'] using hlim |
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| theorem tendsto_integral_stopped_square [IsFiniteMeasure P] |
| {F : Filtration ℝ≥0 mΩ} {X : ℝ≥0 → Ω → ℝ} |
| (hX : IsStronglyProgressive F X) |
| {τ : ℕ → Ω → WithTop ℝ≥0} (hτ : ∀ n, IsStoppingTime F (τ n)) |
| {T t : ℝ≥0} (ht : t ≤ T) {C : ℝ} |
| (hb : ∀ᵐ ω ∂P, ∀ s ≤ T, |X s ω| ≤ C) |
| (hexit : ∀ᵐ ω ∂P, ∀ᶠ n in atTop, (T : WithTop ℝ≥0) ≤ τ n ω) : |
| Tendsto (fun n => ∫ ω, (stoppedProcess X (τ n) t ω) ^ 2 ∂P) |
| atTop (𝓝 (∫ ω, (X t ω) ^ 2 ∂P)) := by |
| apply tendsto_integral_of_dominated_convergence (fun _ : Ω => C ^ 2) |
| · intro n |
| have hm : AEStronglyMeasurable (stoppedProcess X (τ n) t) P := |
| (((hX.stoppedProcess (hτ n)).stronglyAdapted t).mono (F.le t)).aestronglyMeasurable |
| exact hm.pow 2 |
| · exact integrable_const _ |
| · intro n |
| filter_upwards [hb] with ω hω |
| have hs : |stoppedProcess X (τ n) t ω| ≤ C := by |
| exact hω _ ((WithTop.untopA_le (min_le_left (t : WithTop ℝ≥0) (τ n ω))).trans ht) |
| rw [Real.norm_of_nonneg (sq_nonneg _)] |
| simpa only [sq_abs] using pow_le_pow_left₀ (abs_nonneg _) hs 2 |
| · filter_upwards [hexit] with ω hω |
| apply tendsto_const_nhds.congr' |
| filter_upwards [hω] with n hn |
| rw [stoppedProcess_eq_of_le ((WithTop.coe_le_coe.mpr ht).trans hn)] |
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| theorem secondMoment_integral_representation_of_stopped [IsFiniteMeasure P] |
| {F : Filtration ℝ≥0 mΩ} {X : ℝ≥0 → Ω → ℝ} |
| (hX : IsStronglyProgressive F X) |
| {τ : ℕ → Ω → WithTop ℝ≥0} (hτ : ∀ n, IsStoppingTime F (τ n)) |
| {T : ℝ≥0} {C : ℝ} |
| (hb : ∀ᵐ ω ∂P, ∀ s ≤ T, |X s ω| ≤ C) |
| (hexit : ∀ᵐ ω ∂P, ∀ᶠ n in atTop, (T : WithTop ℝ≥0) ≤ τ n ω) |
| {E : ℝ → Ω → ℝ} |
| (hE : Integrable (fun p : ℝ × Ω => E p.1 p.2) |
| ((volume.restrict (Ioc (0 : ℝ) T)).prod P)) |
| (hlocal : ∀ n, ∀ t ≤ T, |
| (∫ ω, (stoppedProcess X (τ n) t ω) ^ 2 ∂P) = |
| (∫ ω, (X 0 ω) ^ 2 ∂P) + ∫ s in (0 : ℝ)..(t : ℝ), |
| ∫ ω, if (s.toNNReal : WithTop ℝ≥0) ≤ τ n ω then E s ω else 0 ∂P) : |
| ∀ t ≤ T, (∫ ω, (X t ω) ^ 2 ∂P) = |
| (∫ ω, (X 0 ω) ^ 2 ∂P) + ∫ s in (0 : ℝ)..(t : ℝ), ∫ ω, E s ω ∂P := by |
| intro t ht |
| have hEt : Integrable (fun p : ℝ × Ω => E p.1 p.2) |
| ((volume.restrict (Ioc (0 : ℝ) (t : ℝ))).prod P) := |
| hE.mono_measure (Measure.prod_mono |
| (Measure.restrict_mono_set _ (Ioc_subset_Ioc_right (show (t : ℝ) ≤ T from ht))) le_rfl) |
| have hxt : ∀ᵐ ω ∂P, ∀ᶠ n in atTop, (t : WithTop ℝ≥0) ≤ τ n ω := by |
| filter_upwards [hexit] with ω hω |
| filter_upwards [hω] with n hn |
| exact (WithTop.coe_le_coe.mpr ht).trans hn |
| have hl := tendsto_integral_stopped_square hX hτ ht hb hexit |
| have hr := (tendsto_integral_stopped_density (fun n => (hτ n).measurable') hEt hxt).const_add |
| (∫ ω, (X 0 ω) ^ 2 ∂P) |
| have heq : (fun n => ∫ ω, (stoppedProcess X (τ n) t ω) ^ 2 ∂P) = |
| (fun n => (∫ ω, (X 0 ω) ^ 2 ∂P) + ∫ s in (0 : ℝ)..(t : ℝ), |
| ∫ ω, if (s.toNNReal : WithTop ℝ≥0) ≤ τ n ω then E s ω else 0 ∂P) := |
| funext (fun n => hlocal n t ht) |
| rw [heq] at hl |
| exact tendsto_nhds_unique hl hr |
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| theorem secondMoment_analysis_of_stopped [IsFiniteMeasure P] |
| {F : Filtration ℝ≥0 mΩ} {X : ℝ≥0 → Ω → ℝ} |
| (hX : IsStronglyProgressive F X) |
| {τ : ℕ → Ω → WithTop ℝ≥0} (hτ : ∀ n, IsStoppingTime F (τ n)) |
| {T : ℝ≥0} {C : ℝ} |
| (hb : ∀ᵐ ω ∂P, ∀ s ≤ T, |X s ω| ≤ C) |
| (hexit : ∀ᵐ ω ∂P, ∀ᶠ n in atTop, (T : WithTop ℝ≥0) ≤ τ n ω) |
| {E : ℝ → Ω → ℝ} |
| (hE : Integrable (fun p : ℝ × Ω => E p.1 p.2) |
| ((volume.restrict (Ioc (0 : ℝ) T)).prod P)) |
| (hlocal : ∀ n, ∀ t ≤ T, |
| (∫ ω, (stoppedProcess X (τ n) t ω) ^ 2 ∂P) = |
| (∫ ω, (X 0 ω) ^ 2 ∂P) + ∫ s in (0 : ℝ)..(t : ℝ), |
| ∫ ω, if (s.toNNReal : WithTop ℝ≥0) ≤ τ n ω then E s ω else 0 ∂P) : |
| AbsolutelyContinuousOnInterval (fun t : ℝ => ∫ ω, (X t.toNNReal ω) ^ 2 ∂P) 0 T ∧ |
| (∀ᵐ t ∂volume.restrict (Icc (0 : ℝ) T), |
| deriv (fun s : ℝ => ∫ ω, (X s.toNNReal ω) ^ 2 ∂P) t = ∫ ω, E t ω ∂P) := by |
| have hrep : ∀ t ∈ Icc (0 : ℝ) (T : ℝ), |
| (∫ ω, (X t.toNNReal ω) ^ 2 ∂P) = |
| (∫ ω, (X 0 ω) ^ 2 ∂P) + ∫ s in (0 : ℝ)..t, ∫ ω, E s ω ∂P := by |
| intro t ht |
| have htT : t.toNNReal ≤ T := by |
| simpa only [Real.toNNReal_coe] using Real.toNNReal_mono ht.2 |
| simpa only [Real.coe_toNNReal t ht.1] using |
| secondMoment_integral_representation_of_stopped hX hτ hb hexit hE hlocal t.toNNReal htT |
| refine ⟨?_, ae_deriv_of_integral_representation T.coe_nonneg |
| (intervalIntegrable_mean T.coe_nonneg hE) hrep⟩ |
| apply absolutelyContinuousOnInterval_of_integral_representation |
| (intervalIntegrable_mean T.coe_nonneg hE) |
| simpa only [uIcc_of_le T.coe_nonneg] using hrep |
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| end Problem367.DiscrepancyEnergy |
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| end |
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| end OAI |
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