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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 | import Mathlib
import OAI.Probability.Ballisticity.Estimates.OutwardCanonical
namespace OAI
section
section
open MeasureTheory ProbabilityTheory Filter Function
open scoped ENNReal NNReal BigOperators Topology Classical
namespace DirectionalTransience
lemma wordCylinder_signed_up_disjoint_stay {d : ℕ} (e : Direction d)
(x y : Lattice d) (H : ℕ) :
Disjoint (wordCylinder y (List.replicate H e))
(StayUntil (Strip (realPosition (step e)) x H) H) := by
apply Set.disjoint_left.mpr
intro X hw hS
have hb := (hS 0 (Nat.zero_le H)).1
have ht := (hS H le_rfl).2
have hzero := hw 0 (by simp)
have hend := hw H (by simp)
rw [wordPath_replicate] at hend
rw [hzero,wordPath_zero] at hb
rw [hend,dot_realPosition_add,dot_realPosition_nsmul,dot_signed_direction e (step e),signedCoordinate_step_self,mul_one] at ht
linarith
lemma quenched_signed_strip_block {d : ℕ} (ω : Environment d) {κ : ℝ≥0}
(hκ : ∀ y e, κ ≤ (ω y).1 e) (e : Direction d) (x y : Lattice d) (H : ℕ) :
quenchedKernel (ω,y) (StayUntil (Strip (realPosition (step e)) x H) H) ≤
1-(κ:ℝ≥0∞)^H := by
have hsub := (wordCylinder_signed_up_disjoint_stay e x y H).subset_compl_left
calc
_ ≤ quenchedKernel (ω,y) (wordCylinder y (List.replicate H e))ᶜ := measure_mono hsub
_ = 1-ENNReal.ofReal (wordWeight ω y (List.replicate H e)) := by
rw [measure_compl (measurableSet_wordCylinder _ _) (measure_ne_top _ _),measure_univ,quenched_wordCylinder]
_ ≤ 1-(κ:ℝ≥0∞)^H := by
apply tsub_le_tsub_left
have hs := ENNReal.ofReal_le_ofReal (wordWeight_lower ω hκ y (List.replicate H e))
simpa only [List.length_replicate,ENNReal.ofReal_pow κ.coe_nonneg,ENNReal.ofReal_coe_nnreal] using hs
lemma quenched_signed_strip_geometric {d : ℕ} (ω : Environment d) {κ : ℝ≥0}
(hκ : ∀ y e, κ ≤ (ω y).1 e) (e : Direction d) (x y : Lattice d) (H k : ℕ) :
quenchedKernel (ω,y) (StayUntil (Strip (realPosition (step e)) x H) (k*H)) ≤
(1-(κ:ℝ≥0∞)^H)^k :=
quenched_stayUntil_block_bound ω _ H _ (fun y => quenched_signed_strip_block ω hκ e x y H) y k
lemma truncatedEndpointLaw_mass_error {d : ℕ} (ω : Environment d) {κ : ℝ≥0}
(hκ : ∀ y e, κ ≤ (ω y).1 e) (e : Direction d) (x : Lattice d)
{H : ℕ} (hH : 0 < H) (k : ℕ) :
hitKernel (Strip (realPosition (step e)) x H) (Upper (realPosition (step e)) x H) (ω,x) Set.univ ≤
truncatedEndpointLaw e H (k*H) ω x Set.univ+(1-(κ:ℝ≥0∞)^H)^k := by
rw [truncatedEndpointLaw_apply e hH,Set.inter_univ,hitKernel_total (disjoint_strip_upper _ _ _)]
exact (quenched_hit_truncation ω x _ _ _).trans
(add_le_add le_rfl (quenched_signed_strip_geometric ω hκ e x x H k))
end DirectionalTransience
end
end
end OAI
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