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b317718 | 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 | import OAI.Probability.FreeIsing.Martingales.ContinuousLimit
namespace OAI
noncomputable section
open MeasureTheory Filter Set
open scoped NNReal ENNReal Topology
namespace Problem367.MartingaleSquareMaximal
variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ]
{F : Filtration ℝ≥0 mΩ} {M : ℕ → ℝ≥0 → Ω → ℝ}
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
end Problem367.MartingaleSquareMaximal
end
end OAI
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