| import OAI.Probability.FreeIsing.Brownian.IntegralProcess |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped NNReal Topology |
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| namespace Problem367.BrownianIntegralProcess |
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| variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} |
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| theorem exists_rapid_meanSquare_subsequence |
| {X : ℕ → Ω → ℝ} (hX : ∀ n, MemLp (X n) 2 P) |
| (hCauchy : CauchySeq (fun n => (hX n).toLp (X n))) : |
| ∃ k : ℕ → ℕ, StrictMono k ∧ |
| Summable (fun n => (4 : ℝ) ^ n * |
| ∫ ω, (X (k (n + 1)) ω - X (k n) ω) ^ 2 ∂P) := by |
| obtain ⟨k, hk, hstep⟩ := Metric.exists_subseq_bounded_of_cauchySeq |
| (fun n => (hX n).toLp (X n)) hCauchy |
| (fun n => (1 / 4 : ℝ) ^ n) (fun n => by positivity) |
| refine ⟨k, hk, ?_⟩ |
| apply Summable.of_nonneg_of_le |
| (fun n => mul_nonneg (by positivity) (integral_nonneg (fun ω => sq_nonneg _))) |
| (f := fun n => (1 / 4 : ℝ) ^ n) ?_ |
| (summable_geometric_of_lt_one (by norm_num) (by norm_num)) |
| intro n |
| have hnorm : ‖(hX (k (n + 1))).toLp (X (k (n + 1))) - |
| (hX (k n)).toLp (X (k n))‖ ≤ (1 / 4 : ℝ) ^ n := by |
| simpa only [dist_eq_norm] using |
| (hstep n (k (n + 1)) (hk.monotone (Nat.le_succ n))).le |
| have hsq := (sq_le_sq₀ (norm_nonneg _) (by positivity : 0 ≤ (1 / 4 : ℝ) ^ n)).2 hnorm |
| rw [BrownianGrid.norm_sq_toLp_sub] at hsq |
| calc |
| _ ≤ (4 : ℝ) ^ n * ((1 / 4 : ℝ) ^ n) ^ 2 := |
| mul_le_mul_of_nonneg_left hsq (by positivity) |
| _ = _ := by |
| simp only [pow_two, ← mul_pow] |
| norm_num |
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| omit [MeasurableSpace Ω] in |
| theorem continuous_stoppedUniformSum |
| {B H : ℝ≥0 → Ω → ℝ} {ω : Ω} |
| (hB : Continuous (fun t => B t ω)) (T : ℝ≥0) (n : ℕ) : |
| Continuous (fun t => stoppedUniformSum B H T n t ω) := by |
| unfold stoppedUniformSum |
| apply continuous_finsetSum |
| intro i hi |
| exact continuous_const.mul |
| ((hB.comp (continuous_id.min continuous_const)).sub |
| (hB.comp (continuous_id.min continuous_const))) |
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| theorem ae_continuous_stoppedUniformSum |
| {B H : ℝ≥0 → Ω → ℝ} |
| (hB : ∀ᵐ ω ∂P, Continuous (fun t => B t ω)) (T : ℝ≥0) (n : ℕ) : |
| ∀ᵐ ω ∂P, Continuous (fun t => stoppedUniformSum B H T n t ω) := by |
| filter_upwards [hB] with ω hω |
| exact continuous_stoppedUniformSum hω T n |
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| omit [MeasurableSpace Ω] in |
| @[simp] theorem stoppedUniformSum_zero (B H : ℝ≥0 → Ω → ℝ) (T : ℝ≥0) (n : ℕ) : |
| stoppedUniformSum B H T n 0 = 0 := by |
| funext ω |
| simp [stoppedUniformSum, StoppedSimpleIntegral.intervalSum] |
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| theorem initial_zero_of_stopped_uniform_limit |
| {B H M : ℝ≥0 → Ω → ℝ} (T : ℝ≥0) (hM : MemLp (M 0) 2 P) |
| (hlim : Tendsto (fun n => ∫ ω, |
| (stoppedUniformSum B H T n 0 ω - M 0 ω) ^ 2 ∂P) atTop (𝓝 0)) : |
| M 0 =ᵐ[P] 0 := by |
| have hlim' : Tendsto (fun _ : ℕ => ∫ ω, (M 0 ω) ^ 2 ∂P) atTop (𝓝 0) := by |
| simpa only [stoppedUniformSum_zero, Pi.zero_apply, zero_sub, neg_sq] using hlim |
| have hzero : (∫ ω, (M 0 ω) ^ 2 ∂P) = 0 := |
| tendsto_nhds_unique tendsto_const_nhds hlim' |
| apply ae_eq_of_integral_sq_sub_eq_zero |
| · simpa only [Pi.zero_apply, sub_zero] using hM.integrable_sq |
| · simpa only [Pi.zero_apply, sub_zero] using hzero |
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| end Problem367.BrownianIntegralProcess |
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| end |
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| end OAI |
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