| import OAI.Probability.FreeIsing.Factors.ContinuousRecovery |
| import OAI.Probability.FreeIsing.Martingales.ContinuousL2Limit |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory Filter Set |
| open scoped NNReal Topology |
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| namespace Problem367.ContinuousHorizonGluing |
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| variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} |
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| def glue (L : ℕ → ℝ≥0 → Ω → ℝ) (t : ℝ≥0) (ω : Ω) : ℝ := |
| limUnder atTop (fun n => L n t ω) |
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| theorem glue_isStronglyPredictable {F : Filtration ℝ≥0 mΩ} |
| {L : ℕ → ℝ≥0 → Ω → ℝ} (hL : ∀ n, IsStronglyPredictable F (L n)) : |
| IsStronglyPredictable F (glue L) := by |
| let : MeasurableSpace (ℝ≥0 × Ω) := F.predictable |
| exact StronglyMeasurable.limUnder hL |
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| theorem ae_all_eq_on_Iic {f g : ℝ≥0 → Ω → ℝ} (T : ℝ≥0) |
| (hf : ∀ᵐ ω ∂μ, ContinuousOn (fun t => f t ω) (Iic T)) |
| (hg : ∀ᵐ ω ∂μ, ContinuousOn (fun t => g t ω) (Iic T)) |
| (hfg : ∀ t ≤ T, f t =ᵐ[μ] g t) : |
| ∀ᵐ ω ∂μ, ∀ t ≤ T, f t ω = g t ω := by |
| let : Nonempty (Iic T) := ⟨⟨0, by simp⟩⟩ |
| have hf' : ∀ᵐ ω ∂μ, Continuous (fun t : Iic T => f t ω) := by |
| filter_upwards [hf] with ω hω |
| exact continuousOn_iff_continuous_domRestrict.mp hω |
| have hg' : ∀ᵐ ω ∂μ, Continuous (fun t : Iic T => g t ω) := by |
| filter_upwards [hg] with ω hω |
| exact continuousOn_iff_continuous_domRestrict.mp hω |
| have heq := ae_eq_of_continuous_of_evaluation_ae_eq hf' hg' |
| (fun t : Iic T => hfg t t.property) |
| filter_upwards [heq] with ω hω |
| intro t ht |
| exact congrFun hω ⟨t, ht⟩ |
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| theorem ae_all_glue_eq {L : ℕ → ℝ≥0 → Ω → ℝ} |
| (hc : ∀ n, ∀ᵐ ω ∂μ, ContinuousOn (fun t => L n t ω) (Iic (n : ℝ≥0))) |
| (hcompat : ∀ (n m : ℕ) (t : ℝ≥0), t ≤ (n : ℝ≥0) → t ≤ (m : ℝ≥0) → |
| L n t =ᵐ[μ] L m t) : |
| ∀ᵐ ω ∂μ, ∀ (n : ℕ) (t : ℝ≥0), t ≤ (n : ℝ≥0) → glue L t ω = L n t ω := by |
| have heq (n m : ℕ) : ∀ᵐ ω ∂μ, ∀ t ≤ min (n : ℝ≥0) (m : ℝ≥0), |
| L n t ω = L m t ω := by |
| apply ae_all_eq_on_Iic |
| · filter_upwards [hc n] with ω hω |
| exact hω.mono (Iic_subset_Iic.mpr (min_le_left _ _)) |
| · filter_upwards [hc m] with ω hω |
| exact hω.mono (Iic_subset_Iic.mpr (min_le_right _ _)) |
| · intro t ht |
| exact hcompat n m t (ht.trans (min_le_left _ _)) (ht.trans (min_le_right _ _)) |
| have hall : ∀ᵐ ω ∂μ, ∀ (n m : ℕ) (t : ℝ≥0), t ≤ min (n : ℝ≥0) (m : ℝ≥0) → |
| L n t ω = L m t ω := ae_all_iff.mpr (fun n => ae_all_iff.mpr (heq n)) |
| filter_upwards [hall] with ω hω |
| intro n t ht |
| apply Filter.Tendsto.limUnder_eq |
| apply tendsto_const_nhds.congr' |
| filter_upwards [eventually_ge_atTop n] with m hm |
| exact hω n m t (le_min ht (ht.trans (by exact_mod_cast hm))) |
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| theorem ae_continuous_glue {L : ℕ → ℝ≥0 → Ω → ℝ} |
| (hc : ∀ n, ∀ᵐ ω ∂μ, ContinuousOn (fun t => L n t ω) (Iic (n : ℝ≥0))) |
| (hcompat : ∀ (n m : ℕ) (t : ℝ≥0), t ≤ (n : ℝ≥0) → t ≤ (m : ℝ≥0) → |
| L n t =ᵐ[μ] L m t) : |
| ∀ᵐ ω ∂μ, Continuous (fun t => glue L t ω) := by |
| filter_upwards [ae_all_iff.mpr hc, ae_all_glue_eq hc hcompat] with ω hcω hω |
| apply continuous_iff_continuousAt.mpr |
| intro t |
| obtain ⟨n, hn⟩ := exists_nat_gt t |
| have hlocal : ContinuousOn (fun s => glue L s ω) (Iic (n : ℝ≥0)) := |
| (hcω n).congr (fun s hs => hω n s hs) |
| exact hlocal.continuousAt (Iic_mem_nhds hn) |
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| theorem exists_global_predictable_continuous_version |
| {F : Filtration ℝ≥0 mΩ} {X : ℝ≥0 → Ω → ℝ} |
| (hlocal : ∀ n : ℕ, ∃ L : ℝ≥0 → Ω → ℝ, |
| IsStronglyPredictable F L ∧ |
| (∀ᵐ ω ∂μ, ContinuousOn (fun t => L t ω) (Iic (n : ℝ≥0))) ∧ |
| ∀ t ≤ (n : ℝ≥0), L t =ᵐ[μ] X t) : |
| ∃ L : ℝ≥0 → Ω → ℝ, IsStronglyPredictable F L ∧ |
| (∀ᵐ ω ∂μ, Continuous (fun t => L t ω)) ∧ |
| ∀ t, L t =ᵐ[μ] X t := by |
| choose L hpred hc heq using hlocal |
| have hcompat (n m : ℕ) (t : ℝ≥0) (hn : t ≤ n) (hm : t ≤ m) : |
| L n t =ᵐ[μ] L m t := (heq n t hn).trans (heq m t hm).symm |
| refine ⟨glue L, glue_isStronglyPredictable hpred, ae_continuous_glue hc hcompat, ?_⟩ |
| intro t |
| obtain ⟨n, hn⟩ := exists_nat_ge t |
| have hglue : glue L t =ᵐ[μ] L n t := by |
| filter_upwards [ae_all_glue_eq hc hcompat] with ω hω |
| exact hω n t hn |
| exact hglue.trans (heq n t hn) |
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| end Problem367.ContinuousHorizonGluing |
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| namespace Problem367.MartingaleSquareMaximal |
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| theorem exists_global_predictable_continuous_L2_limit |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ] |
| {F : Filtration ℝ≥0 mΩ} {M : ℕ → ℝ≥0 → Ω → ℝ} |
| (hM : ∀ n, Martingale (M n) F μ) |
| (h2 : ∀ n t, MemLp (M n t) 2 μ) |
| (hc : ∀ n, ∀ᵐ ω ∂μ, Continuous (fun t => M n t ω)) |
| (hC : ∀ N : ℕ, CauchySeq (fun n => (h2 n (N : ℝ≥0)).toLp (M n N))) : |
| ∃ L : ℝ≥0 → Ω → ℝ, IsStronglyPredictable F L ∧ |
| (∀ᵐ ω ∂μ, Continuous (fun t => L t ω)) ∧ |
| ∀ t, ∃ hL2 : MemLp (L t) 2 μ, |
| Tendsto (fun n => (h2 n t).toLp (M n t)) atTop (𝓝 (hL2.toLp (L t))) := by |
| have hlocal (N : ℕ) := exists_predictable_continuous_L2_limit hM h2 hc |
| (N : ℝ≥0) (hC N) |
| choose L hpred hcont hlim using hlocal |
| have hcompat (n m : ℕ) (t : ℝ≥0) (hn : t ≤ n) (hm : t ≤ m) : |
| L n t =ᵐ[μ] L m t := by |
| obtain ⟨hn2, hnlim⟩ := hlim n t hn |
| obtain ⟨hm2, hmlim⟩ := hlim m t hm |
| have hclass := tendsto_nhds_unique hnlim hmlim |
| filter_upwards [hn2.coeFn_toLp, hm2.coeFn_toLp] with ω hnω hmω |
| rw [← hnω, ← hmω, hclass] |
| refine ⟨ContinuousHorizonGluing.glue L, |
| ContinuousHorizonGluing.glue_isStronglyPredictable hpred, |
| ContinuousHorizonGluing.ae_continuous_glue hcont hcompat, ?_⟩ |
| intro t |
| obtain ⟨n, hn⟩ := exists_nat_ge t |
| obtain ⟨hn2, hnlim⟩ := hlim n t hn |
| have heq : ContinuousHorizonGluing.glue L t =ᵐ[μ] L n t := by |
| filter_upwards [ContinuousHorizonGluing.ae_all_glue_eq hcont hcompat] with ω hω |
| exact hω n t hn |
| have hL2 : MemLp (ContinuousHorizonGluing.glue L t) 2 μ := hn2.ae_eq heq.symm |
| refine ⟨hL2, ?_⟩ |
| rw [hL2.toLp_congr hn2 heq] |
| exact hnlim |
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| end Problem367.MartingaleSquareMaximal |
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| end |
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| end OAI |
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