| import OAI.Probability.FreeIsing.Brownian.ArrayFiltrationIntegral |
| import OAI.Probability.FreeIsing.Stochastic.IntegralAssociativityLimit |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped NNReal Topology BigOperators |
|
|
| namespace Problem367 |
|
|
| namespace BrownianGrid |
|
|
| theorem uniformSum_sub_coeff {Ω : Type*} (B H K : ℝ≥0 → Ω → ℝ) |
| (T : ℝ≥0) (n : ℕ) (ω : Ω) : |
| uniformSum B (fun t ω => H t ω - K t ω) T n ω = |
| uniformSum B H T n ω - uniformSum B K T n ω := by |
| simp only [uniformSum, integralSum, sub_mul, Finset.sum_sub_distrib] |
|
|
| end BrownianGrid |
|
|
| namespace SimpleIntegral |
|
|
| theorem meanSquare_limit_sub {Ω α : Type*} [MeasurableSpace Ω] |
| {P : Measure Ω} {l : Filter α} {X Y : α → Ω → ℝ} {I J : Ω → ℝ} |
| (hX : ∀ a, MemLp (X a) 2 P) (hY : ∀ a, MemLp (Y a) 2 P) |
| (hI : MemLp I 2 P) (hJ : MemLp J 2 P) |
| (hx : Tendsto (fun a => ∫ ω, (X a ω - I ω) ^ 2 ∂P) l (𝓝 0)) |
| (hy : Tendsto (fun a => ∫ ω, (Y a ω - J ω) ^ 2 ∂P) l (𝓝 0)) : |
| Tendsto (fun a => ∫ ω, ((X a ω - Y a ω) - (I ω - J ω)) ^ 2 ∂P) |
| l (𝓝 0) := by |
| apply (tendsto_toLp_iff_meanSquare (fun a => (hX a).sub (hY a)) (hI.sub hJ)).mp |
| change Tendsto (fun a => (hX a).toLp (X a) - (hY a).toLp (Y a)) l |
| (𝓝 (hI.toLp I - hJ.toLp J)) |
| exact ((tendsto_toLp_iff_meanSquare hX hI).mpr hx).sub |
| ((tendsto_toLp_iff_meanSquare hY hJ).mpr hy) |
|
|
| end SimpleIntegral |
|
|
| namespace BrownianFiltrationGrid |
|
|
| variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} |
| {B : ℝ≥0 → Ω → ℝ} {F : Filtration ℝ≥0 mΩ} |
|
|
| theorem uniform_meanSquare_limit_isometry |
| (hB : IsPreBrownianReal B P) (hAd : StronglyAdapted F B) |
| (hind : ∀ s t, s ≤ t → Indep |
| (MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance) (F s) P) |
| (H : ℝ≥0 → Ω → ℝ) (hH : StronglyAdapted F H) |
| (hc : ∀ᵐ ω ∂P, Continuous (fun t => H t ω)) |
| {C : ℝ} (hC : 0 ≤ C) (hb : ∀ᵐ ω ∂P, ∀ t, |H t ω| ≤ C) |
| (T : ℝ≥0) {I : Ω → ℝ} (hI : MemLp I 2 P) |
| (happrox : Tendsto (fun n => ∫ ω, |
| (BrownianGrid.uniformSum B H T n ω - I ω) ^ 2 ∂P) atTop (𝓝 0)) : |
| (∫ ω, I ω ^ 2 ∂P) = |
| ∫ t in (0 : ℝ)..(T : ℝ), ∫ ω, H t.toNNReal ω ^ 2 ∂P := by |
| let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure |
| have hH2 (t : ℝ≥0) : MemLp (H t) 2 P := by |
| apply MemLp.of_bound ((hH t).mono (F.le t)).aestronglyMeasurable C |
| filter_upwards [hb] with ω hω |
| simpa only [Real.norm_eq_abs] using hω t |
| have hlim := (SimpleIntegral.tendsto_toLp_iff_meanSquare |
| (uniformSum_memLp hB hAd hind H hH hH2 T) hI).mpr happrox |
| have hiso := uniformSumLp_limit_isometry hB hAd hind H hH hH2 hc hC hb T hlim |
| calc |
| (∫ ω, I ω ^ 2 ∂P) = ∫ ω, (hI.toLp I ω) ^ 2 ∂P := by |
| apply integral_congr_ae |
| filter_upwards [hI.coeFn_toLp] with ω hω |
| rw [hω] |
| _ = _ := hiso |
|
|
| theorem uniform_meanSquare_difference_isometry |
| (hB : IsPreBrownianReal B P) (hAd : StronglyAdapted F B) |
| (hind : ∀ s t, s ≤ t → Indep |
| (MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance) (F s) P) |
| (H K : ℝ≥0 → Ω → ℝ) (hH : StronglyAdapted F H) (hK : StronglyAdapted F K) |
| (hcH : ∀ᵐ ω ∂P, Continuous (fun t => H t ω)) |
| (hcK : ∀ᵐ ω ∂P, Continuous (fun t => K t ω)) |
| {C D : ℝ} (hC : 0 ≤ C) (hD : 0 ≤ D) |
| (hbH : ∀ᵐ ω ∂P, ∀ t, |H t ω| ≤ C) |
| (hbK : ∀ᵐ ω ∂P, ∀ t, |K t ω| ≤ D) |
| (T : ℝ≥0) {I J : Ω → ℝ} (hI : MemLp I 2 P) (hJ : MemLp J 2 P) |
| (haI : Tendsto (fun n => ∫ ω, |
| (BrownianGrid.uniformSum B H T n ω - I ω) ^ 2 ∂P) atTop (𝓝 0)) |
| (haJ : Tendsto (fun n => ∫ ω, |
| (BrownianGrid.uniformSum B K T n ω - J ω) ^ 2 ∂P) atTop (𝓝 0)) : |
| (∫ ω, (I ω - J ω) ^ 2 ∂P) = |
| ∫ t in (0 : ℝ)..(T : ℝ), ∫ ω, (H t.toNNReal ω - K t.toNNReal ω) ^ 2 ∂P := by |
| let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure |
| have hH2 (t : ℝ≥0) : MemLp (H t) 2 P := by |
| apply MemLp.of_bound ((hH t).mono (F.le t)).aestronglyMeasurable C |
| filter_upwards [hbH] with ω hω |
| simpa only [Real.norm_eq_abs] using hω t |
| have hK2 (t : ℝ≥0) : MemLp (K t) 2 P := by |
| apply MemLp.of_bound ((hK t).mono (F.le t)).aestronglyMeasurable D |
| filter_upwards [hbK] with ω hω |
| simpa only [Real.norm_eq_abs] using hω t |
| refine uniform_meanSquare_limit_isometry hB hAd hind (fun t ω => H t ω - K t ω) |
| (fun t => (hH t).sub (hK t)) ?_ (add_nonneg hC hD) ?_ T (hI.sub hJ) ?_ |
| · filter_upwards [hcH, hcK] with ω hωH hωK |
| exact hωH.sub hωK |
| · filter_upwards [hbH, hbK] with ω hωH hωK |
| intro t |
| exact (abs_sub _ _).trans (add_le_add (hωH t) (hωK t)) |
| · simpa only [BrownianGrid.uniformSum_sub_coeff, Pi.sub_apply] using |
| SimpleIntegral.meanSquare_limit_sub |
| (uniformSum_memLp hB hAd hind H hH hH2 T) |
| (uniformSum_memLp hB hAd hind K hK hK2 T) hI hJ haI haJ |
|
|
| end BrownianFiltrationGrid |
|
|
| namespace BrownianArrayFiltration |
|
|
| variable {Ω ι : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} |
| {B : ι → ℝ≥0 → Ω → ℝ} {F : Filtration ℝ≥0 mΩ} |
|
|
| theorem uniform_meanSquare_limits_orthogonal [IsProbabilityMeasure P] |
| (hB : ∀ a, IsPreBrownianReal (B a) P) |
| (hAd : ∀ a, StronglyAdapted F (B a)) |
| (hind : ∀ s t, s ≤ t → Indep |
| (MeasurableSpace.comap (fun ω a => B a t ω - B a s ω) inferInstance) (F s) P) |
| (hcross : ∀ s t, s ≤ t → Pairwise fun a b => |
| IndepFun (fun ω => B a t ω - B a s ω) (fun ω => B b t ω - B b s ω) P) |
| (H : ι → ℝ≥0 → Ω → ℝ) (hH : ∀ a, StronglyAdapted F (H a)) |
| (hH2 : ∀ a t, MemLp (H a t) 2 P) (T : ℝ≥0) |
| {I : ι → Ω → ℝ} (hI : ∀ a, MemLp (I a) 2 P) |
| (ha : ∀ a, Tendsto (fun n => ∫ ω, |
| (BrownianGrid.uniformSum (B a) (H a) T n ω - I a ω) ^ 2 ∂P) atTop (𝓝 0)) : |
| Pairwise fun a b => ∫ ω, I a ω * I b ω ∂P = 0 := by |
| have hi (a : ι) (s t : ℝ≥0) (hst : s ≤ t) := |
| coordinate_increment_indep_past (hind s t hst) a |
| have hS (a : ι) := BrownianFiltrationGrid.uniformSum_memLp |
| (hB a) (hAd a) (hi a) (H a) (hH a) (hH2 a) T |
| have hlim (a : ι) := (SimpleIntegral.tendsto_toLp_iff_meanSquare (hS a) (hI a)).mpr (ha a) |
| intro a b hab |
| have hliminner := (hlim a).inner (𝕜 := ℝ) (hlim b) |
| have hz (n : ℕ) : inner ℝ |
| ((hS a n).toLp (BrownianGrid.uniformSum (B a) (H a) T n)) |
| ((hS b n).toLp (BrownianGrid.uniformSum (B b) (H b) T n)) = 0 := by |
| rw [PredictableL2.inner_toLp] |
| exact grid_sums_orthogonal hB hAd hind hcross |
| (BrownianGrid.uniformGrid T n) (BrownianGrid.uniformGrid_mono T n) |
| (fun a k => H a (BrownianGrid.uniformGrid T n k)) |
| (fun a k => hH a _) (fun a k => hH2 a _) hab (Finset.range n) (Finset.range n) |
| have heq : inner ℝ ((hI a).toLp (I a)) ((hI b).toLp (I b)) = 0 := |
| tendsto_nhds_unique hliminner (by simpa only [hz] using |
| (tendsto_const_nhds : Tendsto (fun _ : ℕ => (0 : ℝ)) atTop (𝓝 0))) |
| exact (PredictableL2.inner_toLp (hI a) (hI b)).symm.trans heq |
|
|
| theorem uniform_meanSquare_differences_orthogonal [IsProbabilityMeasure P] |
| (hB : ∀ a, IsPreBrownianReal (B a) P) |
| (hAd : ∀ a, StronglyAdapted F (B a)) |
| (hind : ∀ s t, s ≤ t → Indep |
| (MeasurableSpace.comap (fun ω a => B a t ω - B a s ω) inferInstance) (F s) P) |
| (hcross : ∀ s t, s ≤ t → Pairwise fun a b => |
| IndepFun (fun ω => B a t ω - B a s ω) (fun ω => B b t ω - B b s ω) P) |
| (H K : ι → ℝ≥0 → Ω → ℝ) |
| (hH : ∀ a, StronglyAdapted F (H a)) (hK : ∀ a, StronglyAdapted F (K a)) |
| (hH2 : ∀ a t, MemLp (H a t) 2 P) (hK2 : ∀ a t, MemLp (K a t) 2 P) |
| (T : ℝ≥0) {I J : ι → Ω → ℝ} |
| (hI : ∀ a, MemLp (I a) 2 P) (hJ : ∀ a, MemLp (J a) 2 P) |
| (haI : ∀ a, Tendsto (fun n => ∫ ω, |
| (BrownianGrid.uniformSum (B a) (H a) T n ω - I a ω) ^ 2 ∂P) atTop (𝓝 0)) |
| (haJ : ∀ a, Tendsto (fun n => ∫ ω, |
| (BrownianGrid.uniformSum (B a) (K a) T n ω - J a ω) ^ 2 ∂P) atTop (𝓝 0)) : |
| Pairwise fun a b => ∫ ω, (I a ω - J a ω) * (I b ω - J b ω) ∂P = 0 := by |
| apply uniform_meanSquare_limits_orthogonal hB hAd hind hcross |
| (fun a t ω => H a t ω - K a t ω) |
| (fun a t => (hH a t).sub (hK a t)) (fun a t => (hH2 a t).sub (hK2 a t)) T |
| (fun a => (hI a).sub (hJ a)) |
| intro a |
| have hi (s t : ℝ≥0) (hst : s ≤ t) := |
| coordinate_increment_indep_past (hind s t hst) a |
| simpa only [BrownianGrid.uniformSum_sub_coeff, Pi.sub_apply] using |
| SimpleIntegral.meanSquare_limit_sub |
| (BrownianFiltrationGrid.uniformSum_memLp (hB a) (hAd a) hi (H a) (hH a) (hH2 a) T) |
| (BrownianFiltrationGrid.uniformSum_memLp (hB a) (hAd a) hi (K a) (hK a) (hK2 a) T) |
| (hI a) (hJ a) (haI a) (haJ a) |
|
|
| section RowComparison |
|
|
| variable [IsProbabilityMeasure P] |
| (hB : ∀ a, IsPreBrownianReal (B a) P) |
| (hAd : ∀ a, StronglyAdapted F (B a)) |
| (hind : ∀ s t, s ≤ t → Indep |
| (MeasurableSpace.comap (fun ω a => B a t ω - B a s ω) inferInstance) (F s) P) |
| (hcross : ∀ s t, s ≤ t → Pairwise fun a b => |
| IndepFun (fun ω => B a t ω - B a s ω) (fun ω => B b t ω - B b s ω) P) |
| (H K : ι → ℝ≥0 → Ω → ℝ) |
| (hH : ∀ a, StronglyAdapted F (H a)) (hK : ∀ a, StronglyAdapted F (K a)) |
| (hcH : ∀ a, ∀ᵐ ω ∂P, Continuous (fun t => H a t ω)) |
| (hcK : ∀ a, ∀ᵐ ω ∂P, Continuous (fun t => K a t ω)) |
| (C D : ι → ℝ) (hC : ∀ a, 0 ≤ C a) (hD : ∀ a, 0 ≤ D a) |
| (hbH : ∀ a, ∀ᵐ ω ∂P, ∀ t, |H a t ω| ≤ C a) |
| (hbK : ∀ a, ∀ᵐ ω ∂P, ∀ t, |K a t ω| ≤ D a) |
| (T : ℝ≥0) {I J : ι → Ω → ℝ} |
| (hI : ∀ a, MemLp (I a) 2 P) (hJ : ∀ a, MemLp (J a) 2 P) |
| (haI : ∀ a, Tendsto (fun n => ∫ ω, |
| (BrownianGrid.uniformSum (B a) (H a) T n ω - I a ω) ^ 2 ∂P) atTop (𝓝 0)) |
| (haJ : ∀ a, Tendsto (fun n => ∫ ω, |
| (BrownianGrid.uniformSum (B a) (K a) T n ω - J a ω) ^ 2 ∂P) atTop (𝓝 0)) |
|
|
| include hB hAd hind hcross hH hK hcH hcK hC hD hbH hbK hI hJ haI haJ in |
|
|
| theorem finite_row_difference_isometry (s : Finset ι) : |
| (∫ ω, ((∑ a ∈ s, I a ω) - ∑ a ∈ s, J a ω) ^ 2 ∂P) = |
| ∑ a ∈ s, ∫ t in (0 : ℝ)..(T : ℝ), |
| ∫ ω, (H a t.toNNReal ω - K a t.toNNReal ω) ^ 2 ∂P := by |
| have hH2 (a : ι) (t : ℝ≥0) : MemLp (H a t) 2 P := by |
| apply MemLp.of_bound ((hH a t).mono (F.le t)).aestronglyMeasurable (C a) |
| filter_upwards [hbH a] with ω hω |
| simpa only [Real.norm_eq_abs] using hω t |
| have hK2 (a : ι) (t : ℝ≥0) : MemLp (K a t) 2 P := by |
| apply MemLp.of_bound ((hK a t).mono (F.le t)).aestronglyMeasurable (D a) |
| filter_upwards [hbK a] with ω hω |
| simpa only [Real.norm_eq_abs] using hω t |
| have horth := uniform_meanSquare_differences_orthogonal hB hAd hind hcross H K hH hK |
| hH2 hK2 T hI hJ haI haJ |
| simp_rw [← Finset.sum_sub_distrib] |
| rw [SimpleIntegral.integral_sq_sum_of_orthogonal s |
| (fun a ω => I a ω - J a ω) |
| (fun a _ b _ => ((hI a).sub (hJ a)).integrable_mul ((hI b).sub (hJ b))) |
| (fun a _ b _ hab => horth hab)] |
| apply Finset.sum_congr rfl |
| intro a ha |
| exact BrownianFiltrationGrid.uniform_meanSquare_difference_isometry |
| (hB a) (hAd a) (fun s t hst => coordinate_increment_indep_past (hind s t hst) a) |
| (H a) (K a) (hH a) (hK a) (hcH a) (hcK a) (hC a) (hD a) |
| (hbH a) (hbK a) T (hI a) (hJ a) (haI a) (haJ a) |
|
|
| include hB hAd hind hcross hH hK hcH hcK hC hD hbH hbK hI hJ haI haJ in |
|
|
| theorem infinite_row_difference_isometry |
| {Y Z : Ω → ℝ} (hY : MemLp Y 2 P) (hZ : MemLp Z 2 P) |
| (hsY : Tendsto (fun s : Finset ι => ∫ ω, |
| ((∑ a ∈ s, I a ω) - Y ω) ^ 2 ∂P) atTop (𝓝 0)) |
| (hsZ : Tendsto (fun s : Finset ι => ∫ ω, |
| ((∑ a ∈ s, J a ω) - Z ω) ^ 2 ∂P) atTop (𝓝 0)) |
| (henergy : Summable fun a => ∫ t in (0 : ℝ)..(T : ℝ), |
| ∫ ω, (H a t.toNNReal ω - K a t.toNNReal ω) ^ 2 ∂P) : |
| (∫ ω, (Y ω - Z ω) ^ 2 ∂P) = |
| ∑' a, ∫ t in (0 : ℝ)..(T : ℝ), |
| ∫ ω, (H a t.toNNReal ω - K a t.toNNReal ω) ^ 2 ∂P := by |
| have hIs (s : Finset ι) : MemLp (fun ω => ∑ a ∈ s, I a ω) 2 P := |
| memLp_finsetSum s (fun a _ => hI a) |
| have hJs (s : Finset ι) : MemLp (fun ω => ∑ a ∈ s, J a ω) 2 P := |
| memLp_finsetSum s (fun a _ => hJ a) |
| have hd := SimpleIntegral.meanSquare_limit_sub hIs hJs hY hZ hsY hsZ |
| have hlp := (SimpleIntegral.tendsto_toLp_iff_meanSquare |
| (fun s => (hIs s).sub (hJs s)) (hY.sub hZ)).mpr hd |
| have hnorm : Tendsto (fun s : Finset ι => ∫ ω, |
| ((∑ a ∈ s, I a ω) - ∑ a ∈ s, J a ω) ^ 2 ∂P) atTop |
| (𝓝 (∫ ω, (Y ω - Z ω) ^ 2 ∂P)) := by |
| simpa only [PredictableL2.norm_toLp_sq, Pi.sub_apply] using hlp.norm.pow 2 |
| have heq (s : Finset ι) := finite_row_difference_isometry hB hAd hind hcross |
| H K hH hK hcH hcK C D hC hD hbH hbK T hI hJ haI haJ s |
| apply tendsto_nhds_unique hnorm |
| simpa only [HasSum, SummationFilter.unconditional, heq] using henergy.hasSum |
| |
| end RowComparison |
| end BrownianArrayFiltration |
| end Problem367 |
| |
| end |
| |
| end OAI |
| |
