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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 99 100 101 102 103 104 105 106 | import Mathlib
import OAI.Probability.Ballisticity.Estimates.RawPairLocality
namespace OAI
section
section
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal BigOperators Topology Classical
namespace DirectionalTransience
def BelowHeight {d : ℕ} (ℓ : Vector d) (a : ℝ) : Set (Lattice d) :=
{y | dot (realPosition y) ℓ < a}
def PairAtHeight {d : ℕ} (ℓ : Vector d) (a : ℝ) : Set (Lattice d × Lattice d) :=
{x | dot (realPosition x.1) ℓ = a ∧ dot (realPosition x.2) ℓ = a}
noncomputable def rowHeightFiltration {d : ℕ} (ℓ : Vector d) (a : ℝ) :
Filtration ℕ (inferInstance : MeasurableSpace (Environment d)) where
seq n := rowSigma (BelowHeight ℓ (a+n))
mono' := by
intro n m hnm
apply rowSigma_mono
intro y hy
have hnmR : (n:ℝ) ≤ m := by exact_mod_cast hnm
dsimp [BelowHeight] at hy ⊢
linarith
le' n := rowSigma_le _
noncomputable def bufferTraversalMass {d : ℕ} (ℓ : Vector d) (f : Direction d) (a z : ℝ)
(π : Environment d → SupportedPairMeasures (PairAtHeight ℓ a)) (h : ℕ) (ω : Environment d) : ℝ≥0∞ :=
if h=0 then 1 else pairKernelMass ℓ f h z (π ω).val ω
lemma bufferTraversalMass_adapted {d : ℕ} (ℓ : Vector d) (f : Direction d) (a z : ℝ)
(π : Environment d → SupportedPairMeasures (PairAtHeight ℓ a))
(hπ : @Measurable _ _ (rowSigma (BelowHeight ℓ a)) _ π) :
Adapted (rowHeightFiltration ℓ a) (bufferTraversalMass ℓ f a z π) := by
intro h
by_cases hh : h=0
· simp only [hh]
exact measurable_const
· have hhpos : 0 < h := Nat.pos_of_ne_zero hh
have hle : rowSigma (BelowHeight ℓ a) ≤ rowSigma (BelowHeight ℓ (a+h)) := by
apply rowSigma_mono
intro y hy
dsimp [BelowHeight] at hy ⊢
have : (0:ℝ) ≤ h := Nat.cast_nonneg h
linarith
have hC (x : Lattice d × Lattice d) (hx : x ∈ PairAtHeight ℓ a) :
Strip ℓ x.1 h ⊆ BelowHeight ℓ (a+h) ∧ Strip ℓ x.2 h ⊆ BelowHeight ℓ (a+h) := by
constructor
· intro y hy
change dot (realPosition y) ℓ < a+h
simpa only [hx.1] using hy.2
· intro y hy
change dot (realPosition y) ℓ < a+h
simpa only [hx.2] using hy.2
have hp : @Measurable _ _ (rowSigma (BelowHeight ℓ (a+h))) _ π := hπ.mono hle le_rfl
have hm := (pairKernelMass_joint_rows ℓ f hhpos z (PairAtHeight ℓ a)
(BelowHeight ℓ (a+h)) hC).comp (hp.prodMk measurable_id)
change @Measurable _ _ (rowSigma (BelowHeight ℓ (a+h))) _ _
unfold bufferTraversalMass
simp only [ite_eq_right hh]
exact hm
noncomputable def bufferFirstFailure {d : ℕ} (ℓ : Vector d) (f : Direction d) (a z g : ℝ)
(π : Environment d → SupportedPairMeasures (PairAtHeight ℓ a)) (H : ℕ) : Environment d → ℕ :=
hittingBtwn (bufferTraversalMass ℓ f a z π) (Set.Iio (ENNReal.ofReal g)) 1 H
lemma bufferFirstFailure_stopping {d : ℕ} (ℓ : Vector d) (f : Direction d) (a z g : ℝ)
(π : Environment d → SupportedPairMeasures (PairAtHeight ℓ a))
(hπ : @Measurable _ _ (rowSigma (BelowHeight ℓ a)) _ π) (H : ℕ) :
IsStoppingTime (rowHeightFiltration ℓ a) (fun ω => (bufferFirstFailure ℓ f a z g π H ω : WithTop ℕ)) :=
(bufferTraversalMass_adapted ℓ f a z π hπ).isStoppingTime_hittingBtwn measurableSet_Iio
lemma bufferFirstFailure_bounds {d : ℕ} (ℓ : Vector d) (f : Direction d) (a z g : ℝ)
(π : Environment d → SupportedPairMeasures (PairAtHeight ℓ a)) {H : ℕ} (hH : 0 < H)
(ω : Environment d) : 1 ≤ bufferFirstFailure ℓ f a z g π H ω ∧
bufferFirstFailure ℓ f a z g π H ω ≤ H :=
hittingBtwn_mem_Icc hH ω
lemma before_bufferFirstFailure_pass {d : ℕ} (ℓ : Vector d) (f : Direction d) (a z g : ℝ)
(π : Environment d → SupportedPairMeasures (PairAtHeight ℓ a)) (H : ℕ)
(ω : Environment d) {j : ℕ} (hj : 0 < j) (hjlt : j < bufferFirstFailure ℓ f a z g π H ω) :
ENNReal.ofReal g ≤ pairKernelMass ℓ f j z (π ω).val ω := by
have hn := notMem_of_lt_hittingBtwn hjlt hj
simpa only [bufferTraversalMass,ite_eq_right (Nat.ne_of_gt hj),Set.mem_Iio,not_lt] using hn
lemma bufferFirstFailure_fails_of_lt {d : ℕ} (ℓ : Vector d) (f : Direction d) (a z g : ℝ)
(π : Environment d → SupportedPairMeasures (PairAtHeight ℓ a)) {H : ℕ} (hH : 0 < H)
(ω : Environment d) (hlt : bufferFirstFailure ℓ f a z g π H ω < H) :
pairKernelMass ℓ f (bufferFirstFailure ℓ f a z g π H ω) z (π ω).val ω < ENNReal.ofReal g := by
have hn := hittingBtwn_mem_set_of_hittingBtwn_lt hlt
change bufferTraversalMass ℓ f a z π (bufferFirstFailure ℓ f a z g π H ω) ω < ENNReal.ofReal g at hn
have hp := (bufferFirstFailure_bounds ℓ f a z g π hH ω).1
simpa only [bufferTraversalMass,ite_eq_right (by omega : bufferFirstFailure ℓ f a z g π H ω ≠ 0),Set.mem_Iio] using hn
end DirectionalTransience
end
end
end OAI
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