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9b6e014 | 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 78 79 80 81 | import OAI.Probability.Ballisticity.RowBridge
namespace OAI
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal BigOperators Topology
namespace DirectionalTransience
lemma uniformElliptic_directional_zero_one {d : ℕ} (hd : 3 ≤ d)
(ν : Measure (Row d)) [IsProbabilityMeasure ν] (hue : UniformElliptic ν)
(ℓ : Vector d) (hℓ : ℓ ≠ 0) :
annealedLaw ν (TransientPaths ℓ) = 0 ∨ annealedLaw ν (TransientPaths ℓ) = 1 := by
have h := DirectionalZeroOne.directional_zero_one d hd (RowBridge.rowLaw ν)
(RowBridge.strictEllipticity ν hue) ℓ hℓ
change DirectionalZeroOne.annealed (RowBridge.rowLaw ν) 0 (TransientPaths ℓ) = 0 ∨
DirectionalZeroOne.annealed (RowBridge.rowLaw ν) 0 (TransientPaths ℓ) = 1 at h
rwa [← RowBridge.annealedLaw_eq ν] at h
lemma dot_div_right {d : ℕ} (v ℓ : Vector d) (r : ℝ) :
dot v (fun i => ℓ i / r) = dot v ℓ / r := by
simp only [dot, mul_div_assoc, Finset.sum_div]
lemma dot_self_pos_of_ne_zero {d : ℕ} (ℓ : Vector d) (hℓ : ℓ ≠ 0) :
0 < dot ℓ ℓ := by
obtain ⟨i, hi⟩ := Function.ne_iff.mp hℓ
change ℓ i ≠ 0 at hi
exact (mul_self_pos.mpr hi).trans_le
(Finset.single_le_sum (fun j _ => mul_self_nonneg (ℓ j)) (Finset.mem_univ i))
lemma transientPaths_div_pos {d : ℕ} (ℓ : Vector d) {r : ℝ} (hr : 0 < r) :
TransientPaths (fun i => ℓ i / r) = TransientPaths ℓ := by
ext X
simp only [TransientPaths, Set.mem_ofPred_eq, dot_div_right]
exact tendsto_div_const_atTop_of_pos hr
lemma velocity_of_positive_escape {d : ℕ} (hd : 3 ≤ d)
(ν : Measure (Row d)) [IsProbabilityMeasure ν] (hue : UniformElliptic ν)
(ℓ : Vector d) (hℓ : ℓ ≠ 0) (hpos : 0 < annealedLaw ν (TransientPaths ℓ)) :
∃ v : Vector d, 0 < dot v ℓ ∧ HasVelocity ν v := by
have hprob : annealedLaw ν (TransientPaths ℓ) = 1 :=
(uniformElliptic_directional_zero_one hd ν hue ℓ hℓ).resolve_left hpos.ne'
have hs : 0 < dot ℓ ℓ := dot_self_pos_of_ne_zero ℓ hℓ
let r : ℝ := Real.sqrt (dot ℓ ℓ)
have hr : 0 < r := Real.sqrt_pos.mpr hs
let u : Vector d := fun i => ℓ i / r
have hu : dot u u = 1 := by
have he : dot u u = dot ℓ ℓ / (r * r) := by
simp only [u, dot, div_mul_div_comm, Finset.sum_div]
rw [he, show r * r = dot ℓ ℓ from Real.mul_self_sqrt hs.le, div_self hs.ne']
have htrans : DirectionallyTransient ν u := by
apply (mem_ae_iff_prob_eq_one (measurableSet_transientPaths u)).mpr
rw [transientPaths_div_pos ℓ hr]
exact hprob
obtain ⟨v, hvu, hv⟩ := directional_transience_implies_ballisticity_unit
(by omega : 2 ≤ d) ν hue u hu htrans
refine ⟨v, ?_, hv⟩
have hp : 0 < dot v ℓ / r := by simpa only [u, dot_div_right] using hvu
exact (div_pos_iff_of_pos_right hr).mp hp
lemma escape_probability_one_of_velocity {d : ℕ} (ν : Measure (Row d))
[IsProbabilityMeasure ν] (v u : Vector d) (hv : HasVelocity ν v)
(hvu : 0 < dot v u) : annealedLaw ν (TransientPaths u) = 1 := by
apply (mem_ae_iff_prob_eq_one (measurableSet_transientPaths u)).mp
filter_upwards [hv] with X hX
have hproj := velocityPaths_projection v u X hX
have hprod := hproj.pos_mul_atTop hvu
(tendsto_natCast_atTop_atTop : Tendsto (fun n : ℕ => (n : ℝ)) atTop atTop)
apply hprod.congr'
filter_upwards [eventually_gt_atTop (0 : ℕ)] with n hn
exact div_mul_cancel₀ _ (by exact_mod_cast hn.ne')
lemma unit_direction_ne_zero {d : ℕ} (u : Vector d) (hu : dot u u = 1) : u ≠ 0 := by
intro h
subst u
simp [dot] at hu
end DirectionalTransience
end OAI
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