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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 | import Mathlib
import OAI.Probability.Ballisticity.Estimates.ConfinementTemplates
namespace OAI
section
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal Classical Topology BigOperators
namespace DirectionalTransience
def StripExcursion {d : ℕ} (ℓ : Vector d) (H R : ℝ) : Set (Path d) :=
{X | ∃ n, X ∈ BoundedHeightPrefix ℓ H n ∧ ∃ u : Direction d, R < signedCoordinate u (X n)}
lemma measurableSet_stripExcursion {d : ℕ} (ℓ : Vector d) (H R : ℝ) :
MeasurableSet (StripExcursion ℓ H R) := by
simp only [StripExcursion, Set.ofPred_exists, Set.ofPred_and]
apply MeasurableSet.iUnion
intro n
refine (measurableSet_boundedHeightPrefix ℓ H n).inter (MeasurableSet.iUnion fun u => ?_)
exact measurableSet_lt measurable_const ((measurable_of_countable (signedCoordinate u)).comp (measurable_pi_apply n))
lemma stripExcursion_subset_ratio {d : ℕ} (ℓ : Vector d) (X : Path d)
(h0 : X 0 = 0) (hNN : ∀ n, ∃ e : Direction d, X (n+1) = X n+step e)
(b a D : ℝ) (hb : 0 ≤ b) (ha : 1 ≤ a) (hD : 0 ≤ D) (N : ℕ) :
X ∈ StripExcursion ℓ N (a*(2*b+2+D)*(N+1)^2) →
X ∈ ⋃ u : Direction d, RatioHit ℓ u b a N (1+N*(2*b+1+D)) := by
rintro ⟨n,hn,u,hu⟩
refine Set.mem_iUnion.mpr ⟨u,Set.mem_iUnion.mpr ⟨n,⟨⟨h0,fun j _ => hNN j⟩,hn⟩,?_⟩⟩
have hm := runningHeight_le ℓ X n N (Nat.cast_nonneg _) (fun j hj => (hn j hj).2)
have hm0 := runningHeight_nonneg ℓ X n
have ht : 0 ≤ 1+(N:ℝ)*(2*b+1+D) := by positivity
have hfactor : 1+(N:ℝ)*(2*b+1+D) ≤ (2*b+2+D)*(N+1) := by nlinarith
have hbnd : (1+(N:ℝ)*(2*b+1+D))*(a*(1+runningHeight ℓ X n)) ≤
a*(2*b+2+D)*(N+1)^2 := by
calc
_ ≤ (1+(N:ℝ)*(2*b+1+D))*(a*(1+N)) :=
mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_left (by linarith) (by linarith)) ht
_ ≤ ((2*b+2+D)*(N+1))*(a*(1+N)) :=
mul_le_mul_of_nonneg_right hfactor (by positivity)
_ = _ := by ring
change _ < tiltedCoordinate ℓ u b a (X n)/(a*(1+runningHeight ℓ X n))
apply (lt_div_iff₀ (mul_pos (by linarith) (by linarith))).mpr
have hz := mul_nonneg (mul_nonneg hb (show 0 ≤ a by linarith)) (hn n le_rfl).1
dsimp only [tiltedCoordinate]
linarith
theorem spatial_confinement {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν]
(hue : UniformElliptic ν) (e : Direction d)
(htrans : DirectionallyTransient ν (realPosition (step e))) :
∃ C ε : ℝ, 0 < C ∧ 0 < ε ∧ ε ≤ 1 ∧ ∀ N : ℕ,
(annealedLaw ν).real (StripExcursion (realPosition (step e)) N (C*(N+1)^2)) ≤
(2*d : ℕ)*(1-ε)^N := by
obtain ⟨b,a,D,ε,hb,ha,hD,hε,hε1,hA⟩ := strip_templates ν hue e htrans
refine ⟨a*(2*b+2+D),ε,mul_pos (by linarith) (by linarith),hε,hε1,?_⟩
intro N
obtain ⟨A,hAm,hAp,hAc⟩ := hA N
apply le_trans ?_ (ratio_hit_union_bound ν (realPosition (step e))
(unit_direction_coordinate_bound _ (signed_direction_unit e)) b a D N ε hb ha hD hε.le hε1 A hAm hAp hAc N)
apply ENNReal.toReal_mono (measure_ne_top _ _)
apply measure_mono_ae
filter_upwards [annealed_initial ν,annealed_nearest_neighbor ν] with X h0 hNN
exact stripExcursion_subset_ratio _ X h0 hNN b a D hb ha hD N
end DirectionalTransience
end
end OAI
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