| import OAI.Probability.FreeIsing.Martingales.ContinuousLimit |
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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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| variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ] |
| {F : Filtration ℝ≥0 mΩ} {M : ℕ → ℝ≥0 → Ω → ℝ} |
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| theorem exists_continuous_subsequence_of_terminal_L2_cauchy |
| (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)) (T : ℝ≥0) |
| (hC : CauchySeq (fun n => (h2 n T).toLp (M n T))) : |
| ∃ v : ℕ → ℕ, StrictMono v ∧ |
| ∀ᵐ ω ∂μ, ∃ f : C(Set.Iic T, ℝ), |
| ∀ t : Set.Iic T, Tendsto (fun n => M (v n) t ω) atTop (𝓝 (f t)) := by |
| let u : ℕ → Lp ℝ 2 μ := fun n => (h2 n T).toLp (M n T) |
| obtain ⟨v, hv, hb⟩ := Metric.exists_subseq_bounded_of_cauchySeq u hC |
| (fun n => (1 / 4 : ℝ) ^ n) (fun n => by positivity) |
| refine ⟨v, hv, ae_exists_continuous_limit_of_summable_energy |
| (fun n => hM (v n)) (fun n => h2 (v n)) (fun n => hc (v n)) T ?_⟩ |
| have hfinite : (∑' n : ℕ, (1 / 4 : ℝ≥0∞) ^ n) ≠ ∞ := |
| (tsum_geometric_lt_top.mpr (by norm_num)).ne |
| apply ne_top_of_le_ne_top hfinite |
| apply ENNReal.tsum_le_tsum |
| intro n |
| have hnorm : ‖u (v (n + 1)) - u (v n)‖ < (1 / 4 : ℝ) ^ n := by |
| simpa only [dist_eq_norm] using hb n (v (n + 1)) (hv.monotone (Nat.le_succ n)) |
| have heq : ‖u (v (n + 1)) - u (v n)‖ ^ 2 = |
| ∫ ω, (M (v (n + 1)) T ω - M (v n) T ω) ^ 2 ∂μ := by |
| rw [← real_inner_self_eq_norm_sq, L2.inner_def] |
| simp only [real_inner_self_eq_norm_sq, Real.norm_eq_abs, sq_abs] |
| apply integral_congr_ae |
| filter_upwards [Lp.coeFn_sub (u (v (n + 1))) (u (v n)), |
| (h2 (v (n + 1)) T).coeFn_toLp, (h2 (v n) T).coeFn_toLp] with ω hsub ha hb |
| simp only [Pi.sub_apply] at hsub |
| rw [hsub, ha, hb] |
| have henergy : (∫ ω, (M (v (n + 1)) T ω - M (v n) T ω) ^ 2 ∂μ) ≤ |
| ((1 / 4 : ℝ) ^ n) ^ 2 := by |
| rw [← heq] |
| nlinarith [norm_nonneg (u (v (n + 1)) - u (v n))] |
| have henergy' : ENNReal.ofReal |
| (∫ ω, (M (v (n + 1)) T ω - M (v n) T ω) ^ 2 ∂μ) ≤ |
| ((1 / 4 : ℝ≥0∞) ^ n) ^ 2 := by |
| have hh : ENNReal.ofReal (((1 / 4 : ℝ) ^ n) ^ 2) = |
| ((1 / 4 : ℝ≥0∞) ^ n) ^ 2 := by |
| rw [ENNReal.ofReal_pow (by positivity), ENNReal.ofReal_pow (by norm_num)] |
| congr 2 |
| rw [ENNReal.ofReal_div_of_pos (by norm_num : (0 : ℝ) < 4)] |
| norm_num |
| exact (ENNReal.ofReal_le_ofReal henergy).trans_eq hh |
| calc |
| _ ≤ ((1 / 4 : ℝ≥0∞) ^ n) ^ 2 / |
| ((((1 / 2 : ℝ≥0) ^ n : ℝ≥0) : ℝ≥0∞) ^ 2) := |
| ENNReal.div_le_div_right henergy' _ |
| _ = (1 / 4 : ℝ≥0∞) ^ n := by |
| have hh : ((((1 / 2 : ℝ≥0) ^ n : ℝ≥0) : ℝ≥0∞) ^ 2) = |
| ((1 / 2 : ℝ≥0∞) ^ n) ^ 2 := by |
| norm_num [ENNReal.coe_pow, ENNReal.coe_div (by norm_num : (2 : ℝ≥0) ≠ 0)] |
| rw [hh] |
| simp only [div_eq_mul_inv, ENNReal.inv_pow, ← mul_pow, ← pow_mul] |
| rw [Nat.mul_comm n 2, pow_mul] |
| congr 1 |
| apply (ENNReal.toReal_eq_toReal_iff' (by simp [ENNReal.mul_ne_top]) (by simp)).mp |
| norm_num |
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| end Problem367.MartingaleSquareMaximal |
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| end |
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| end OAI |
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