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180bc03 | 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 93 94 95 96 97 98 | import Mathlib
import OAI.Probability.Ballisticity.Coupling.FreshBuffer
import OAI.Probability.Ballisticity.Estimates.BufferFirstFailure
namespace OAI
section
section
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal BigOperators Topology Classical Function
namespace DirectionalTransience
lemma weighted_countable_partition_lower {Ω I : Type*} [MeasurableSpace Ω] [Countable I]
(μ : Measure Ω) (A B : I → Set Ω) (hA : ∀ i, MeasurableSet (A i))
(hB : ∀ i, MeasurableSet (B i)) (hdis : Pairwise (Disjoint on A))
(hsub : ∀ i, B i ⊆ A i) (w : Ω → ℝ≥0∞) (c : ℝ≥0∞)
(hbound : ∀ i, c*(∫⁻ ω in A i, w ω ∂μ) ≤ ∫⁻ ω in B i, w ω ∂μ) :
c*(∫⁻ ω in ⋃ i, A i, w ω ∂μ) ≤ ∫⁻ ω in ⋃ i, B i, w ω ∂μ := by
rw [lintegral_iUnion hA hdis,lintegral_iUnion hB
(fun i j hij => (hdis hij).mono (hsub i) (hsub j)),←ENNReal.tsum_mul_left]
exact ENNReal.tsum_le_tsum hbound
lemma actualFreshBuffer_countable_branches {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))) :
∃ α c : ℝ, 0 < α ∧ 0 < c ∧ ∀ ε : ℝ, 0 < ε →
∃ g R : ℝ, 0 < g ∧ 0 < R ∧ ∀ {I : Type} [Countable I]
(a r : I → ℝ) (_ : ∀ i, R ≤ r i) (z₀ : ℝ)
(π : (i : I) → Environment d → SupportedPairMeasures (PairAtHeight (realPosition (step e)) (a i)))
(_ : ∀ i, @Measurable _ _ (rowSigma (BelowHeight (realPosition (step e)) (a i))) _ (π i))
(A : I → Set (Environment d))
(_ : ∀ i, MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) (a i))] (A i))
(_ : Pairwise (Disjoint on A))
(_ : ∀ i η, η ∈ A i → IsProbabilityMeasure (π i η).val ∧
∀ᵐ x ∂(π i η).val, z₀+r i ≤ signedCoordinate f (x.2-x.1))
(w : Environment d → ℝ≥0∞)
(_ : ∀ i, @Measurable _ _ (rowSigma (BelowHeight (realPosition (step e)) (a i))) _
((A i).indicator w)),
let H := fun i => ⌊fluctuationScale (independentConditionedPairLaw ν (realPosition (step e)))
(commonIncrementProcess (realPosition (step e)) f 0) (r i)⌋₊
ENNReal.ofReal c*(∫⁻ ω in ⋃ i, A i, w ω ∂environmentLaw ν) ≤
∫⁻ ω in ⋃ i, {ω | ω ∈ A i ∧ BufferStageEvent (realPosition (step e)) f (H i)
z₀ (r i) ε α g (π i ω).val ω}, w ω ∂environmentLaw ν := by
obtain ⟨α,c,hα,hc,hchance⟩ := buffer_stage_fresh_success_weights ν hue e f hef htrans
refine ⟨α,c,hα,hc,fun ε hε => ?_⟩
obtain ⟨g,R,hg,hR,hgood⟩ := hchance ε hε
refine ⟨g,R,hg,hR,?_⟩
intro I _ a r hr z₀ π hπ A hA hdis hvalid w hw
let ℓ := realPosition (step e)
let H := fun i => ⌊fluctuationScale (independentConditionedPairLaw ν ℓ)
(commonIncrementProcess ℓ f 0) (r i)⌋₊
have hH (i : I) : 0 < H i := (hgood (r i) (hr i)).1
have hS (i : I) : Disjoint (BelowHeight ℓ (a i)) (BelowHeight ℓ (a i))ᶜ := disjoint_compl_right
have hC (i : I) (x : Lattice d × Lattice d) (hx : x ∈ PairAtHeight ℓ (a i)) :
Strip ℓ x.1 (H i) ⊆ (BelowHeight ℓ (a i))ᶜ ∧ Strip ℓ x.2 (H i) ⊆ (BelowHeight ℓ (a i))ᶜ := by
constructor
· intro y hy
change ¬ dot (realPosition y) ℓ < a i
have hh := hy.1
rw [hx.1] at hh
exact not_lt_of_ge hh
· intro y hy
change ¬ dot (realPosition y) ℓ < a i
have hh := hy.1
rw [hx.2] at hh
exact not_lt_of_ge hh
let B := fun i => {ω | ω ∈ A i ∧ BufferStageEvent ℓ f (H i) z₀ (r i) ε α g (π i ω).val ω}
have hAm (i : I) : MeasurableSet (A i) := (rowSigma_le _) _ (hA i)
have hBm (i : I) : MeasurableSet (B i) := by
apply (hAm i).inter
have hp := (hπ i).mono (rowSigma_le _) le_rfl
have hm := (measurableSet_bufferStageEvent_rows ℓ f (hH i) z₀ (r i) ε α g
(PairAtHeight ℓ (a i)) Set.univ (fun x hx => ⟨Set.subset_univ _,Set.subset_univ _⟩)).preimage
(hp.prodMk (measurable_id.mono le_rfl (rowSigma_le Set.univ)))
exact hm
apply weighted_countable_partition_lower (environmentLaw ν) A B hAm hBm hdis
(fun i _ h => h.1) w (ENNReal.ofReal c)
intro i
have hh := (hgood (r i) (hr i)).2 z₀ (PairAtHeight ℓ (a i)) (BelowHeight ℓ (a i))
(BelowHeight ℓ (a i))ᶜ (hS i) (hC i) (π i) (hπ i) (A i) (hA i) (hvalid i)
((A i).indicator w) (hw i)
rw [lintegral_congr_ae (ae_restrict_mem (hAm i) |>.mono (fun ω hω => Set.indicator_of_mem hω _))] at hh
have he : (∫⁻ ω in B i, (A i).indicator w ω ∂environmentLaw ν) =
∫⁻ ω in B i, w ω ∂environmentLaw ν := by
apply lintegral_congr_ae
filter_upwards [ae_restrict_mem (hBm i)] with ω hω
exact Set.indicator_of_mem hω.1 _
exact he ▸ hh
end DirectionalTransience
end
end
end OAI
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