| import OAI.Probability.FreeIsing.Martingales.PredictableMeshApproximation |
| import OAI.Probability.FreeIsing.Martingales.SquareMaximal |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped NNReal ENNReal Topology |
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| namespace Problem367.MartingaleSquareMaximal |
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| private def dyadicPoint (n k : ℕ) : ℝ≥0 := (k : ℝ≥0) / (2 ^ n : ℕ) |
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| private lemma dyadicPoint_mono (n : ℕ) : Monotone (dyadicPoint n) := by |
| intro i j hij |
| dsimp [dyadicPoint] |
| gcongr |
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| private lemma dyadicPoint_double (n k : ℕ) : |
| dyadicPoint (n + 1) (2 * k) = dyadicPoint n k := by |
| simp only [dyadicPoint, pow_succ, Nat.cast_mul, Nat.cast_ofNat] |
| field_simp |
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| private lemma dyadicPoint_eq_mesh (n k : ℕ) : |
| dyadicPoint n k = PredictableMesh.point (2 ^ n - 1) k := by |
| have hn : 1 ≤ 2 ^ n := Nat.one_le_pow n 2 (by norm_num) |
| simp only [dyadicPoint, PredictableMesh.point, Nat.sub_add_cancel hn] |
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| variable {Ω : Type*} [mΩ : MeasurableSpace Ω] |
| {μ : Measure Ω} [IsFiniteMeasure μ] |
| {F : Filtration ℝ≥0 mΩ} {M : ℝ≥0 → Ω → ℝ} |
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| theorem continuous_maximal_sq |
| (hM : Martingale M F μ) (h2 : ∀ t, MemLp (M t) 2 μ) |
| (hc : ∀ᵐ ω ∂μ, Continuous (fun t => M t ω)) |
| (ε T : ℝ≥0) : |
| (ε : ℝ≥0∞) ^ 2 * μ {ω | ∃ t ≤ T, (ε : ℝ) < |M t ω|} ≤ |
| ENNReal.ofReal (∫ ω, M T ω ^ 2 ∂μ) := by |
| let A : ℕ → Set Ω := fun n => |
| {ω | ∃ k : ℕ, dyadicPoint n k ≤ T ∧ (ε : ℝ) < |M (dyadicPoint n k) ω|} |
| have hAmono : Monotone A := by |
| apply monotone_nat_of_le_succ |
| intro n ω hω |
| obtain ⟨k, hk, hv⟩ := hω |
| exact ⟨2 * k, by simpa only [dyadicPoint_double] using hk, |
| by simpa only [dyadicPoint_double] using hv⟩ |
| have hAn (n : ℕ) : (ε : ℝ≥0∞) ^ 2 * μ (A n) ≤ |
| ENNReal.ofReal (∫ ω, M T ω ^ 2 ∂μ) := by |
| let N : ℕ := ⌊(2 ^ n : ℕ) * T⌋₊ |
| have hp : (0 : ℝ≥0) < (2 ^ n : ℕ) := by positivity |
| have hlast : dyadicPoint n N ≤ T := by |
| apply (div_le_iff₀ hp).2 |
| simpa only [N, mul_comm] using Nat.floor_le (show (0 : ℝ≥0) ≤ ((2 ^ n : ℕ) : ℝ≥0) * T from by positivity) |
| have hsub : A n ⊆ {ω | ∃ k ≤ N, (ε : ℝ) ≤ |M (dyadicPoint n k) ω|} := by |
| intro ω hω |
| obtain ⟨k, hk, hv⟩ := hω |
| refine ⟨k, ?_, hv.le⟩ |
| apply Nat.le_floor |
| simpa only [mul_comm] using (div_le_iff₀ hp).mp hk |
| exact (mul_le_mul_right (measure_mono hsub) ((ε : ℝ≥0∞) ^ 2)).trans |
| (sampled_maximal_sq hM h2 (dyadicPoint n) (dyadicPoint_mono n) ε N hlast) |
| have hUnion : (ε : ℝ≥0∞) ^ 2 * μ (⋃ n, A n) ≤ |
| ENNReal.ofReal (∫ ω, M T ω ^ 2 ∂μ) := by |
| rw [hAmono.measure_iUnion, ENNReal.mul_iSup] |
| exact iSup_le hAn |
| have hsub : {ω | ∃ t ≤ T, (ε : ℝ) < |M t ω|} ≤ᵐ[μ] ⋃ n, A n := by |
| filter_upwards [hc] with ω hcω |
| rintro ⟨t, ht, hv⟩ |
| have hp : Tendsto (fun n : ℕ => 2 ^ n - 1) atTop atTop := |
| (tendsto_sub_atTop_nat 1).comp |
| (tendsto_pow_atTop_atTop_of_one_lt (by norm_num : (1 : ℕ) < 2)) |
| have hlim : Tendsto (fun n : ℕ => |M (PredictableMesh.left (2 ^ n - 1) t) ω|) |
| atTop (𝓝 |M t ω|) := |
| ((hcω.tendsto t).comp ((PredictableMesh.tendsto_left t).comp hp)).abs |
| obtain ⟨n, hn⟩ := ((tendsto_order.1 hlim).1 (ε : ℝ) hv).exists |
| apply Set.mem_iUnion.mpr |
| refine ⟨n, ?_⟩ |
| let k := ⌈((2 ^ n - 1 + 1 : ℕ) : ℝ≥0) * t⌉₊ - 1 |
| have heq : dyadicPoint n k = PredictableMesh.left (2 ^ n - 1) t := |
| dyadicPoint_eq_mesh n k |
| refine ⟨k, ?_, ?_⟩ |
| · rw [heq] |
| exact (PredictableMesh.left_le _ t).trans ht |
| · rw [heq] |
| exact hn |
| exact (mul_le_mul_right (measure_mono_ae hsub) ((ε : ℝ≥0∞) ^ 2)).trans hUnion |
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| end Problem367.MartingaleSquareMaximal |
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| end |
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| end OAI |
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