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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 82 83 84 85 86 87 88 89 90 91 92 | import Mathlib
import OAI.Probability.Ballisticity.Stationary.StationaryArrayLaw
import OAI.Probability.Ballisticity.Stationary.ArrayDust
namespace OAI
section
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal Classical Topology BigOperators
namespace DirectionalTransience
lemma stationary_ae_integer_family {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) {T : Ω → Ω} (hT : MeasurePreserving T μ μ)
(P : ℤ → Ω → Prop) (hp : ∀ i, MeasurableSet {ω | P i ω})
(hs : ∀ i, ∀ᵐ ω ∂μ, P i (T ω) ↔ P (i+1) ω)
(h0 : ∀ᵐ ω ∂μ, P 0 ω) : ∀ᵐ ω ∂μ, ∀ i, P i ω := by
apply ae_all_iff.mpr
have hb (i : ℤ) : (∀ᵐ ω ∂μ, P i ω) ↔ ∀ᵐ ω ∂μ, P (i+1) ω := by
have he : (∀ᵐ ω ∂μ, P i (T ω)) ↔ ∀ᵐ ω ∂μ, P (i+1) ω := eventually_congr (hs i)
rw [←ae_map_iff hT.measurable.aemeasurable (hp i),hT.map_eq] at he
exact he
intro i
induction i using Int.induction_on with
| zero => exact h0
| succ n ih => exact (hb n).mp ih
| pred n ih =>
apply (hb (-(n:ℤ)-1)).mpr
simpa only [sub_add_cancel] using ih
structure ArrayGrowthSector {d : ℕ} (e : Direction d)
(μ : Measure (ActualEpisodeArray e)) where
event : Set (ActualEpisodeArray e)
measurable_event : MeasurableSet event
positive_event : 0<μ event
invariant : ∀ᵐ Y ∂μ, StationaryCompact.shift Y∈event ↔ Y∈event
symmetric : ∀ σ, ∀ᵐ Y ∂μ, arrayPerm e σ Y∈event ↔ Y∈event
ζ : ℝ
positive_ζ : 0<ζ
sequence : ℕ → ℕ
strictMono : StrictMono sequence
growth : ∀ᵐ Y ∂μ, Y∈event → ∀ᶠ k in atTop,
ζ*(sequence k:ℝ)<(arrayWindowCost e 0 (sequence k) Y).toReal
no_dust : ∀ᵐ Y ∂μ, Y∈event → ∀ i a, ¬arrayDust e i a Y
namespace StationaryArrayLaw
variable {d : ℕ} {ν : Measure (Row d)} [IsProbabilityMeasure ν] {e : Direction d}
theorem growth_sector (L : StationaryArrayLaw ν e) (hue : UniformElliptic ν)
(htrans : DirectionallyTransient ν (realPosition (step e))) :
Nonempty (ArrayGrowthSector e (L.law : Measure (ActualEpisodeArray e))) := by
let μ := (L.law : Measure (ActualEpisodeArray e))
obtain ⟨g,hgm,hgb,hgT,ζ,hζ,hE,ns,hns,hlim,hgrowth⟩ := stationary_positive_growth μ
L.preserving (arrayRealCost e) (arrayRealCost_measurable e) (arrayRealCost_nonneg e)
L.integrable_cost L.positive_cost
let E := {Y | ζ<g Y}
have hmE : MeasurableSet E := measurableSet_lt measurable_const hgm
have heT : ∀ᵐ Y ∂μ, StationaryCompact.shift Y∈E ↔ Y∈E := by
filter_upwards [hgT] with Y hY
change ζ<g (StationaryCompact.shift Y) ↔ ζ<g Y
rw [hY]
have heP (σ : Equiv.Perm ℕ) : ∀ᵐ Y ∂μ, arrayPerm e σ Y∈E ↔ Y∈E := by
filter_upwards [array_growth_symmetry e μ g ns hlim L.symmetric σ] with Y hY
change ζ<g (arrayPerm e σ Y) ↔ ζ<g Y
rw [hY]
have hc : ∀ᵐ Y ∂μ, ∀ j, (Y.1 j).2.2≠⊤ := L.finite_marks.mono fun Y hY j => (hY j).2.2.2.ne
have hd : ∀ᵐ Y ∂μ, Y∈E → ∀ ε : ℝ, 0<ε → ∃ m,
(expNeg (arrayWindowCost e 0 m Y)).toReal<ε := by
filter_upwards [hc,hgrowth] with Y hY hg hy
exact array_cost_decay_of_growth e Y hY ζ hζ ns hns (hg hy)
have hn0 := array_no_dust_on_cost_decay e ν hue htrans μ L.consistent L.zero_sampling
L.entropy_finite (L.finite_marks.mono fun Y hY j => (hY j).2.1) L.survival L.symmetric 0 E hmE heP hd
have hnd : ∀ᵐ Y ∂μ, Y∈E → ∀ i a, ¬arrayDust e i a Y := by
have hh := stationary_ae_integer_family μ L.preserving (fun i Y => Y∈E → ∀ a, ¬arrayDust e i a Y)
(fun i => hmE.imp (by
simpa only [Set.ofPred_forall, Set.compl_ofPred] using MeasurableSet.iInter fun a => (arrayDust_measurable e i a).compl))
(fun i => heT.mono fun Y hY => by
rw [hY]
rfl) hn0
filter_upwards [hh] with Y hY hy i
exact hY i hy
refine ⟨⟨E,hmE,hE,heT,heP,ζ,hζ,ns,hns,?_,hnd⟩⟩
filter_upwards [hc,hgrowth] with Y hY hg hy
simpa only [arrayWindowCost_toReal e Y hY] using hg hy
end StationaryArrayLaw
end DirectionalTransience
end
end OAI
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