| import OAI.Probability.FreeIsing.Martingales.ContinuousMaximal |
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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} |
| {M : ℕ → ℝ≥0 → Ω → ℝ} |
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| theorem ae_exists_continuous_limit_of_summable_crossings |
| (hc : ∀ n, ∀ᵐ ω ∂μ, Continuous (fun t => M n t ω)) (T : ℝ≥0) |
| (hsum : (∑' n : ℕ, μ {ω | ∃ t ≤ T, |
| ((1 / 2 : ℝ) ^ n) < |M (n + 1) t ω - M n t ω|}) ≠ ∞) : |
| ∀ᵐ ω ∂μ, ∃ f : C(Set.Iic T, ℝ), |
| ∀ t : Set.Iic T, Tendsto (fun n => M n t ω) atTop (𝓝 (f t)) := by |
| have hbc := ae_eventually_notMem hsum |
| filter_upwards [ae_all_iff.mpr hc, hbc] with ω hcω hω |
| have hcompact : IsCompact (Set.Iic T) := by |
| convert (isCompact_Icc (a := (0 : ℝ≥0)) (b := T)) using 1 |
| ext t |
| simp only [Set.mem_Iic, Set.mem_Icc] |
| exact ⟨fun ht => ⟨zero_le, ht⟩, And.right⟩ |
| let : CompactSpace (Set.Iic T) := isCompact_iff_compactSpace.mp hcompact |
| let f : ℕ → C(Set.Iic T, ℝ) := fun n => |
| ⟨fun t => M n t ω, (hcω n).comp continuous_subtype_val⟩ |
| have hgeom : Summable (fun n : ℕ => (1 / 2 : ℝ) ^ n) := |
| summable_geometric_of_norm_lt_one (by norm_num) |
| have hdist : Summable (fun n : ℕ => dist (f n) (f (n + 1))) := by |
| apply hgeom.of_norm_bounded_eventually_nat |
| filter_upwards [hω] with n hn |
| rw [Real.norm_eq_abs, abs_of_nonneg dist_nonneg] |
| apply (ContinuousMap.dist_le (by positivity : (0 : ℝ) ≤ (1 / 2 : ℝ) ^ n)).2 |
| intro t |
| have hv : |M (n + 1) t ω - M n t ω| ≤ (1 / 2 : ℝ) ^ n := by |
| by_contra! h |
| exact hn ⟨t, t.property, h⟩ |
| simpa only [f, ContinuousMap.coe_mk, Real.dist_eq, abs_sub_comm] using hv |
| obtain ⟨g, hg⟩ := cauchySeq_tendsto_of_complete (cauchySeq_of_summable_dist hdist) |
| refine ⟨g, fun t => ?_⟩ |
| exact ((continuous_eval_const t).tendsto g).comp hg |
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| theorem ae_exists_continuous_limit_of_summable_energy [IsFiniteMeasure μ] |
| {F : Filtration ℝ≥0 mΩ} |
| (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) |
| (hsum : (∑' n : ℕ, |
| ENNReal.ofReal (∫ ω, (M (n + 1) T ω - M n T ω) ^ 2 ∂μ) / |
| (((1 / 2 : ℝ≥0) ^ n : ℝ≥0) : ℝ≥0∞) ^ 2) ≠ ∞) : |
| ∀ᵐ ω ∂μ, ∃ f : C(Set.Iic T, ℝ), |
| ∀ t : Set.Iic T, Tendsto (fun n => M n t ω) atTop (𝓝 (f t)) := by |
| apply ae_exists_continuous_limit_of_summable_crossings hc T |
| apply ne_top_of_le_ne_top hsum |
| apply ENNReal.tsum_le_tsum |
| intro n |
| have hcont : ∀ᵐ ω ∂μ, Continuous |
| (fun t => (M (n + 1) - M n) t ω) := by |
| filter_upwards [hc (n + 1), hc n] with ω ha hb |
| exact ha.sub hb |
| have hb := continuous_maximal_sq ((hM (n + 1)).sub (hM n)) |
| (fun t => (h2 (n + 1) t).sub (h2 n t)) hcont ((1 / 2 : ℝ≥0) ^ n) T |
| apply (ENNReal.le_div_iff_mul_le (Or.inl (by positivity)) |
| (Or.inl (by simp))).2 |
| simpa only [NNReal.coe_pow, NNReal.coe_div, NNReal.coe_one, NNReal.coe_ofNat, |
| Pi.sub_apply, mul_comm] using hb |
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| end Problem367.MartingaleSquareMaximal |
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| end |
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| end OAI |
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