import Mathlib import OAI.Probability.Ballisticity.Geometry.TubePrefixMonoRadius namespace OAI 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} 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 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 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 end OAI