File size: 3,299 Bytes
f61e547 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 | import OAI.Probability.FreeIsing.Brownian.IntegralProcess
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal Topology
namespace Problem367.BrownianIntegralProcess
variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
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
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)))
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
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]
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
end Problem367.BrownianIntegralProcess
end
end OAI
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