| import OAI.Probability.FirstPassage.ShortCrossingProbability |
| import OAI.Probability.FirstPassage.RecordScales |
|
|
| namespace OAI |
|
|
| namespace PlanarFPP |
| open Set Filter MeasureTheory |
| open scoped Topology |
|
|
| noncomputable def firstDyadicExponent (p : ℕ → ℝ) (q : ℝ) : ℕ := |
| by |
| classical |
| exact if h : ∃ j, p j ≤ q then Nat.find h else 0 |
|
|
| theorem firstDyadicExponent_spec (p : ℕ → ℝ) (q : ℝ) (h : ∃ j, p j ≤ q) : |
| p (firstDyadicExponent p q) ≤ q := by |
| simpa only [firstDyadicExponent,dite_eq_left h] using Nat.find_spec h |
|
|
| theorem firstDyadicExponent_le (p : ℕ → ℝ) (q : ℝ) (j : ℕ) (hj : p j ≤ q) : |
| firstDyadicExponent p q ≤ j := by |
| have h : ∃ j, p j ≤ q := ⟨j,hj⟩ |
| simpa only [firstDyadicExponent,dite_eq_left h] using Nat.find_min' h hj |
|
|
| theorem firstDyadicExponent_min (p : ℕ → ℝ) (q : ℝ) (j : ℕ) |
| (hj : j < firstDyadicExponent p q) : q < p j := by |
| by_contra hh |
| exact (not_le_of_gt hj) (firstDyadicExponent_le p q j (le_of_not_gt hh)) |
|
|
| variable {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) (τ : Edge → Ω → ℝ) |
|
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| noncomputable def sourceDyadicCrossingProbability (μ : Plane → ℝ) (b : ℝ) (k j : ℕ) : ℝ := |
| P.real {ω | (fun e => τ e ω) ∈ crossingEvent b (μ (direction b)) (sourceCrossingK μ b) |
| ((2 : ℝ)^k) ((2 : ℤ)^j) 0} |
|
|
| noncomputable def characteristicExponent (μ : Plane → ℝ) (b q : ℝ) (k : ℕ) : ℕ := |
| firstDyadicExponent (sourceDyadicCrossingProbability P τ μ b k) q |
|
|
| noncomputable def characteristicLength (μ : Plane → ℝ) (b q : ℝ) (k : ℕ) : ℤ := |
| (2 : ℤ)^(characteristicExponent P τ μ b q k) |
|
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| variable {P τ} |
|
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| theorem characteristicLength_pos (μ : Plane → ℝ) (b q : ℝ) (k : ℕ) : |
| 0 < characteristicLength P τ μ b q k := by unfold characteristicLength; positivity |
|
|
| theorem exponential_characteristic_lengths_exist (h : ExponentialEnvironment P τ) |
| (μ : Plane → ℝ) (hμ : TimeConstantNorm μ) (ht : IsTimeConstant P τ μ) |
| (b q : ℝ) (hb : |b| ≤ 1) (hq : 0 < q) : |
| ∀ᶠ k : ℕ in atTop, |
| characteristicExponent P τ μ b q k ≤ 3*k ∧ |
| P.real {ω | (fun e => τ e ω) ∈ crossingEvent b (μ (direction b)) (sourceCrossingK μ b) |
| ((2 : ℝ)^k) (characteristicLength P τ μ b q k) 0} ≤ q := by |
| filter_upwards [(exponential_source_tilt h μ hμ ht b hb).eventually (gt_mem_nhds hq)] with k hk |
| have hh : sourceDyadicCrossingProbability P τ μ b k (3*k) ≤ q := hk.le |
| exact ⟨firstDyadicExponent_le _ _ _ hh,firstDyadicExponent_spec _ _ ⟨3*k,hh⟩⟩ |
|
|
| theorem characteristicLength_minimal (μ : Plane → ℝ) (b q : ℝ) (k j : ℕ) |
| (hj : (2 : ℤ)^j < characteristicLength P τ μ b q k) : |
| q < sourceDyadicCrossingProbability P τ μ b k j := by |
| have hj' : j < characteristicExponent P τ μ b q k := by |
| by_contra hh |
| have he := pow_le_pow_right₀ (by norm_num : (1 : ℤ) ≤ 2) (le_of_not_gt hh) |
| exact (not_le_of_gt hj) he |
| exact firstDyadicExponent_min _ _ _ hj' |
|
|
| theorem exponential_characteristic_lengths_ratio (h : ExponentialEnvironment P τ) |
| (μ : Plane → ℝ) (hμ : TimeConstantNorm μ) (ht : IsTimeConstant P τ μ) |
| (b q : ℝ) (hb : |b| ≤ 1) (hq : 0 < q) (hq1 : q < 1) : |
| Tendsto (fun k : ℕ => (characteristicLength P τ μ b q k : ℝ)/(2 : ℝ)^k) atTop atTop := by |
| apply tendsto_atTop.2 |
| intro B |
| let C := max 1 B |
| have hC : 0 < C := lt_of_lt_of_le (by norm_num) (le_max_left _ _) |
| have hK : 0 < sourceCrossingK μ b := by unfold sourceCrossingK; positivity |
| have hs := exponential_short_crossing_probability h μ hμ ht b (sourceCrossingK μ b) C hb hK hC q hq1 |
| have h2 := tendsto_pow_atTop_atTop_of_one_lt (by norm_num : (1 : ℝ)<2) |
| filter_upwards [h2.eventually hs,exponential_characteristic_lengths_exist h μ hμ ht b q hb hq] with k hk he |
| have hL : 1 ≤ characteristicLength P τ μ b q k := characteristicLength_pos μ b q k |
| have hgt : C*(2 : ℝ)^k < (characteristicLength P τ μ b q k : ℝ) := by |
| by_contra hn |
| exact (not_lt_of_ge he.2) (hk _ hL (le_of_not_gt hn)) |
| have hr : C < (characteristicLength P τ μ b q k : ℝ)/(2 : ℝ)^k := |
| (lt_div_iff₀ (by positivity)).mpr hgt |
| exact (le_max_right _ _).trans hr.le |
|
|
| theorem exponential_characteristic_record_scales (h : ExponentialEnvironment P τ) |
| (μ : Plane → ℝ) (hμ : TimeConstantNorm μ) (ht : IsTimeConstant P τ μ) |
| (b q : ℝ) (hb : |b| ≤ 1) (hq : 0 < q) : |
| ∃ k0 : ℕ, ∀ N, ∃ k, N < k ∧ k0 ≤ k ∧ |
| ∀ j, j ≤ k-k0 → (1/16 : ℝ)^j*(characteristicLength P τ μ b q k : ℝ) ≤ |
| (characteristicLength P τ μ b q (k-j) : ℝ) := by |
| obtain ⟨k0,hk0⟩ := eventually_atTop.mp (exponential_characteristic_lengths_exist h μ hμ ht b q hb hq) |
| refine ⟨k0,record_scales_from (fun k => (characteristicLength P τ μ b q k : ℝ)) k0 (fun k _ => by exact_mod_cast characteristicLength_pos μ b q k) ?_⟩ |
| intro k hk |
| have hh := pow_le_pow_right₀ (by norm_num : (1 : ℤ) ≤ 2) (hk0 k hk).1 |
| have he : characteristicLength P τ μ b q k ≤ (2 : ℤ)^(3*k) := hh |
| exact_mod_cast he |
|
|
| end PlanarFPP |
|
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| end OAI |
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|
