| import Mathlib |
| import OAI.Probability.Ballisticity.Geometry.TubePrefixMonoRadius |
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| namespace OAI |
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| section |
| section |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Pointwise Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Pointwise Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical BoundedContinuousFunction |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical BoundedContinuousFunction |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| namespace DirectionalTransience |
|
|
| def JointTubeSuccess {d : ℕ} (ℓ : Vector d) (f : Direction d) (x : Lattice d) |
| (b : ℕ → ℝ) (H : ℕ) (z : ℝ) : Set (Path d) := |
| Hit (Strip ℓ x H) (Upper ℓ x H) ∩ |
| {X | (fun n => X n-x) ∉ MedianTubeFailure ℓ f b H z ∧ |
| (fun n => X n-x) ∉ MedianTubeFailure ℓ f (fun j => (j:ℝ)*b H/H) H z} |
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| noncomputable def jointTubeMass {d : ℕ} (ℓ : Vector d) (f : Direction d) |
| (b : ℕ → ℝ) (H : ℕ) (z : ℝ) (π : Measure (Lattice d)) (ω : Environment d) : ℝ≥0∞ := |
| ∫⁻ x, (quenchedKernel (ω,x)) (JointTubeSuccess ℓ f x b H z) ∂π |
|
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| lemma jointTubeMass_ne_top {d : ℕ} (ℓ : Vector d) (f : Direction d) |
| (b : ℕ → ℝ) (H : ℕ) (z : ℝ) (π : Measure (Lattice d)) [IsFiniteMeasure π] (ω : Environment d) : |
| jointTubeMass ℓ f b H z π ω ≠ ⊤ := by |
| apply ne_top_of_le_ne_top (measure_ne_top π Set.univ) |
| calc |
| _ ≤ ∫⁻ _x, (1:ℝ≥0∞) ∂π := lintegral_mono fun _ => prob_le_one |
| _ = π Set.univ := by simp |
|
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| lemma tubePrefix_subset_jointTubeSuccess {d : ℕ} (ℓ : Vector d) (f : Direction d) |
| (x : Lattice d) (b : ℕ → ℝ) (H : ℕ) (z a R : ℝ) |
| (hdet : ∀ j ≤ H, |b j-(j:ℝ)*b H/H| ≤ a) (hz : z ≤ R) (hza : z+a ≤ R) : |
| TubePrefix ℓ f x (b H/H) z H ⊆ JointTubeSuccess ℓ f x b H R := by |
| rintro X hX |
| obtain ⟨n,hn⟩ := Set.mem_iUnion.mp hX |
| refine ⟨Set.mem_iUnion.mpr ⟨n,hn.1⟩,?_,?_⟩ |
| · rintro ⟨j,hj,hbad⟩ |
| obtain ⟨k,hk,hr,hgood⟩ := hn.2 j hj |
| have he : recordIndexPosition ℓ j (fun m => X m-x) = X k-x := by |
| simp only [recordIndexPosition,recordOrZeroPrefix_time ℓ hr] |
| rw [he] at hbad |
| have hh : |signedCoordinate f (X k-x)-b j| ≤ R := by |
| calc |
| _ ≤ |signedCoordinate f (X k-x)-(j:ℝ)*b H/H|+|(j:ℝ)*b H/H-b j| := abs_sub_le _ _ _ |
| _ ≤ z+a := add_le_add (by simpa only [mul_div_assoc] using hgood) |
| (by simpa only [abs_sub_comm] using hdet j hj) |
| _ ≤ R := hza |
| exact (not_lt_of_ge hh) hbad |
| · rintro ⟨j,hj,hbad⟩ |
| obtain ⟨k,hk,hr,hgood⟩ := hn.2 j hj |
| have he : recordIndexPosition ℓ j (fun m => X m-x) = X k-x := by |
| simp only [recordIndexPosition,recordOrZeroPrefix_time ℓ hr] |
| rw [he] at hbad |
| have hh : |signedCoordinate f (X k-x)-(j:ℝ)*b H/H| ≤ R := |
| (by simpa only [mul_div_assoc] using hgood.trans hz) |
| exact (not_lt_of_ge hh) hbad |
|
|
| theorem short_block_survival {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν] |
| (hue : UniformElliptic ν) (e f : Direction d) (hef : e.1 ≠ f.1) |
| (htrans : DirectionallyTransient ν (realPosition (step e))) |
| {v : ℝ} (hv : 0 < v) : |
| ∃ C : ℝ, 0 < C ∧ ∃ t₀ : ℝ, 0 < t₀ ∧ ∀ t : ℝ, 0 < t → t ≤ t₀ → |
| ∃ δ : ℝ, 0 < δ ∧ ∃ R : ℝ, 0 < R ∧ |
| ∀ r : ℝ, R ≤ r → ∀ H : ℕ, 0 < H → |
| (H:ℝ) ≤ t*fluctuationScale (independentConditionedPairLaw ν (realPosition (step e))) |
| (commonIncrementProcess (realPosition (step e)) f 0) r → |
| let ℓ := realPosition (step e) |
| let hp := ne_of_gt (noDrop_positive_of_directionallyTransient ν ℓ htrans) |
| ∀ (a : ℤ) (π : Measure (Lattice d)), IsProbabilityMeasure π → |
| (∀ᵐ x ∂π, signedHeight e x = a) → |
| (environmentLaw ν).real {ω | (jointTubeMass ℓ f |
| (fun j => (recordMedian ν ℓ hp f j:ℝ)) H (v*r) π ω).toReal < δ} ≤ C*t^2 := by |
| let w := min (v/8) 1 |
| have hw : 0 < w := lt_min (by positivity) zero_lt_one |
| have hw20 : w ≤ 20 := (min_le_right _ _).trans (by norm_num) |
| obtain ⟨C,hC,t₁,ht₁,hsurv⟩ := short_block_slope_survival ν hue e f hef htrans hw hw20 |
| obtain ⟨D,hD,t₂,ht₂,hmedian⟩ := recordMedian_shortBlock ν hue e f hef htrans |
| let t₀ := min t₁ (min t₂ ((w/(2*D))^2)) |
| have ht₀ : 0 < t₀ := lt_min ht₁ (lt_min ht₂ (sq_pos_of_pos (by positivity))) |
| refine ⟨C,hC,t₀,ht₀,?_⟩ |
| intro t ht htt |
| obtain ⟨δ,hδ,R,hR,hsurv⟩ := hsurv t ht (htt.trans (min_le_left _ _)) |
| refine ⟨δ,hδ,R,hR,?_⟩ |
| intro r hr H hH hscale |
| let ℓ := realPosition (step e) |
| let hp := ne_of_gt (noDrop_positive_of_directionallyTransient ν ℓ htrans) |
| let b := fun j => (recordMedian ν ℓ hp f j:ℝ) |
| have hr0 : 0 < r := hR.trans_le hr |
| have hmid := hmedian t ht ((htt.trans (min_le_right _ _)).trans (min_le_left _ _)) r hr0 H hH hscale |
| have hroot : Real.sqrt t ≤ w/(2*D) := (Real.sqrt_le_iff).mpr |
| ⟨by positivity,(htt.trans (min_le_right _ _)).trans (min_le_right _ _)⟩ |
| have hcoef : D*Real.sqrt t ≤ w/2 := by |
| have hh := mul_le_mul_of_nonneg_left hroot hD.le |
| have he : D*(w/(2*D))=w/2 := by field_simp |
| linarith |
| have hdet (j : ℕ) (hj : j ≤ H) : |b j-(j:ℝ)*b H/H| ≤ w*r/2 := by |
| exact (hmid j hj).trans (by nlinarith [mul_le_mul_of_nonneg_right hcoef hr0.le]) |
| have hwv : 8*w ≤ v := by have := min_le_left (v/8) 1; dsimp [w]; linarith |
| have hz : 3*(w*r) ≤ v*r := by nlinarith |
| have hza : 3*(w*r)+w*r/2 ≤ v*r := by nlinarith |
| dsimp only |
| intro a π hπ hsupp |
| let := hπ |
| have hb := hsurv r hr H hH hscale a π hπ hsupp |
| have hle : environmentLaw ν {ω | (jointTubeMass ℓ f b H (v*r) π ω).toReal < δ} ≤ |
| environmentLaw ν {ω | (tubePrefixMass ℓ f (b H/H) (3*(w*r)) H π ω).toReal < δ} := by |
| apply measure_mono |
| intro ω hω |
| have hh : tubePrefixMass ℓ f (b H/H) (3*(w*r)) H π ω ≤ jointTubeMass ℓ f b H (v*r) π ω := by |
| apply lintegral_mono; intro x |
| exact measure_mono (tubePrefix_subset_jointTubeSuccess ℓ f x b H (3*(w*r)) (w*r/2) (v*r) hdet hz hza) |
| exact (ENNReal.toReal_mono (jointTubeMass_ne_top ℓ f b H (v*r) π ω) hh).trans_lt hω |
| exact (ENNReal.toReal_mono (measure_ne_top _ _) hle).trans hb |
|
|
| end DirectionalTransience |
|
|
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
|
|
| end |
| end |
|
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| end OAI |
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