| import Mathlib |
| import OAI.Probability.Ballisticity.Crossings.FirstLayerHitRecordIndexTime |
| import OAI.Probability.Ballisticity.Estimates.GaussianAbsTail |
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| namespace OAI |
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| section |
| section |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Pointwise Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Pointwise Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical BoundedContinuousFunction |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical BoundedContinuousFunction |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| namespace DirectionalTransience |
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| def RegularPath {d : ℕ} (ℓ : Vector d) (x : Lattice d) : Set (Path d) := |
| {X | X 0 = x ∧ (∀ n, ∃ f, X (n+1) = X n+step f) ∧ X ∈ TransientPaths ℓ} |
|
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| lemma measurableSet_regularPath {d : ℕ} (ℓ : Vector d) (x : Lattice d) : |
| MeasurableSet (RegularPath ℓ x) := by |
| apply (measurableSet_eq_fun (measurable_pi_apply 0) measurable_const).inter |
| apply MeasurableSet.inter _ (measurableSet_transientPaths ℓ) |
| change MeasurableSet {X : Path d | ∀ n, ∃ f, X (n+1) = X n+step f} |
| simp only [Set.ofPred_forall,Set.ofPred_exists] |
| exact MeasurableSet.iInter fun n => MeasurableSet.iUnion fun f => |
| measurableSet_eq_fun (by fun_prop : Measurable (fun X : Path d => X (n+1))) |
| (by fun_prop : Measurable (fun X : Path d => X n+step f)) |
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| lemma annealedFrom_regularPath {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν] |
| (ℓ : Vector d) (htrans : DirectionallyTransient ν ℓ) (x : Lattice d) : |
| ∀ᵐ X ∂annealedFrom ν x, X ∈ RegularPath ℓ x := by |
| apply annealedFrom_ae_of_quenched ν x _ (measurableSet_regularPath ℓ x) |
| filter_upwards [quenched_directionallyTransient_ae ν ℓ htrans] with ω hω |
| filter_upwards [quenched_initial_ae (ω,x),quenched_nearest_neighbor (ω,x),hω x] with X h0 hnn ht |
| exact ⟨h0,hnn,ht⟩ |
|
|
| lemma sharedConditioned_regularPath {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν] |
| (ℓ : Vector d) (htrans : DirectionallyTransient ν ℓ) (x y : Lattice d) : |
| ∀ᵐ Z ∂sharedConditionedPairLaw ν ℓ x y, |
| (Z.1 ∈ RegularPath ℓ x ∧ Z.1 ∈ NoDrop ℓ x) ∧ |
| (Z.2 ∈ RegularPath ℓ y ∧ Z.2 ∈ NoDrop ℓ y) := by |
| have hh := shared_pair_ae ν x y _ _ (measurableSet_regularPath ℓ x) (measurableSet_regularPath ℓ y) |
| (annealedFrom_regularPath ν ℓ htrans x) (annealedFrom_regularPath ν ℓ htrans y) |
| filter_upwards [(sharedConditionedPairLaw_absolutelyContinuous ν ℓ x y).ae_le hh, |
| sharedConditionedPairLaw_noDrop ν ℓ x y] with Z hZ hD |
| exact ⟨⟨hZ.1,hD.1⟩,hZ.2,hD.2⟩ |
|
|
| lemma conditionedFrom_regularPath {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν] |
| (ℓ : Vector d) (htrans : DirectionallyTransient ν ℓ) (x : Lattice d) : |
| ∀ᵐ X ∂conditionedFrom ν ℓ x, X ∈ RegularPath ℓ x ∧ X ∈ NoDrop ℓ x := by |
| have hac : conditionedFrom ν ℓ x ≪ annealedFrom ν x := |
| Measure.smul_absolutelyContinuous.trans Measure.absolutelyContinuous_restrict |
| have hD : ∀ᵐ X ∂conditionedFrom ν ℓ x, X ∈ NoDrop ℓ x := by |
| exact Measure.smul_absolutelyContinuous.ae_le (ae_restrict_mem (measurableSet_noDrop ℓ x)) |
| have hR : ∀ᵐ X ∂conditionedFrom ν ℓ x, X ∈ RegularPath ℓ x := |
| hac.ae_le (annealedFrom_regularPath ν ℓ htrans x) |
| exact hR.and hD |
|
|
| lemma open_lane_record_iff {d : ℕ} (e f : Direction d) (x : Lattice d) (X : Path d) |
| (hX : X ∈ RegularPath (realPosition (step e)) x) |
| (hD : X ∈ NoDrop (realPosition (step e)) x) {k : ℕ} (hk : 0 < k) |
| (θ r a : ℝ) (hr : 0 < r) : |
| X ∈ SurvivingLane {z | dot (realPosition x) (realPosition (step e)) ≤ dot (realPosition z) (realPosition (step e))} |
| (Upper (realPosition (step e)) x k) |
| {z | |signedCoordinate f z-(signedCoordinate f x+(k:ℝ)*θ)| < a*r} ↔ |
| |scaledRecordEndpoint (realPosition (step e)) f θ r k x X| < a := by |
| rw [survivingLane_record_iff e x X hX.1 hD hX.2.1 hX.2.2 hk] |
| change |signedCoordinate f (_+x)-(signedCoordinate f x+(k:ℝ)*θ)| < a*r ↔ _ |
| rw [signedCoordinate_add] |
| have he (z : Lattice d) : signedCoordinate f z+signedCoordinate f x-(signedCoordinate f x+(k:ℝ)*θ) = |
| signedCoordinate f z-(k:ℝ)*θ := by ring |
| rw [he] |
| simp only [scaledRecordEndpoint,abs_div,abs_of_pos hr] |
| exact (div_lt_iff₀ hr).symm |
|
|
| end DirectionalTransience |
|
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| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
|
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| end |
| end |
|
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| end OAI |
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