| import Mathlib |
| import OAI.Probability.Ballisticity.Estimates.RawPairLocality |
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| namespace OAI |
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| section |
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| section |
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| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| namespace DirectionalTransience |
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| def BelowHeight {d : ℕ} (ℓ : Vector d) (a : ℝ) : Set (Lattice d) := |
| {y | dot (realPosition y) ℓ < a} |
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| 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 _ |
|
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| 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 |
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| 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 |
|
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| 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 |
|
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| 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 |
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| 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 |
|
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| 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 |
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| end |
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| end |
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| end OAI |
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