| import Mathlib |
| import OAI.Probability.Ballisticity.Walk.FreshRowWeights |
|
|
| namespace OAI |
|
|
| section |
|
|
| section |
|
|
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| namespace DirectionalTransience |
|
|
| theorem buffer_stage_fresh_success_weights {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))) : |
| ∃ a c : ℝ, 0 < a ∧ 0 < c ∧ ∀ ε : ℝ, 0 < ε → |
| ∃ g R : ℝ, 0 < g ∧ 0 < R ∧ ∀ r : ℝ, R ≤ r → |
| let H := ⌊fluctuationScale (independentConditionedPairLaw ν (realPosition (step e))) |
| (commonIncrementProcess (realPosition (step e)) f 0) r⌋₊ |
| 0 < H ∧ ∀ (z₀ : ℝ) (C : Set (Lattice d × Lattice d)) (S T : Set (Lattice d)), |
| Disjoint S T → |
| (∀ x ∈ C, Strip (realPosition (step e)) x.1 H ⊆ T ∧ |
| Strip (realPosition (step e)) x.2 H ⊆ T) → |
| ∀ π : Environment d → SupportedPairMeasures C, |
| @Measurable _ _ (rowSigma S) _ π → |
| ∀ A : Set (Environment d), MeasurableSet[rowSigma S] A → |
| (∀ η ∈ A, IsProbabilityMeasure (π η).val ∧ |
| ∀ᵐ x ∂(π η).val, z₀+r ≤ signedCoordinate f (x.2-x.1)) → |
| ∀ w : Environment d → ℝ≥0∞, @Measurable _ _ (rowSigma S) _ w → |
| ENNReal.ofReal c*(∫⁻ ω in A, w ω ∂environmentLaw ν) ≤ |
| ∫⁻ ω in {ω | ω ∈ A ∧ BufferStageEvent (realPosition (step e)) f H z₀ r ε a g |
| (π ω).val ω}, w ω ∂environmentLaw ν := by |
| obtain ⟨a,c,ha,hc,hchance⟩ := buffer_stage_success_probability ν hue e f hef htrans |
| refine ⟨a,c,ha,hc,fun ε hε => ?_⟩ |
| obtain ⟨g,R,hg,hR,hgood⟩ := hchance ε hε |
| refine ⟨g,R,hg,hR,fun r hr => ?_⟩ |
| let H := ⌊fluctuationScale (independentConditionedPairLaw ν (realPosition (step e))) |
| (commonIncrementProcess (realPosition (step e)) f 0) r⌋₊ |
| obtain ⟨hH,hπgood⟩ := hgood r hr |
| refine ⟨hH,fun z₀ C S T hST hC π hπ A hA hvalid w hw => ?_⟩ |
| apply fresh_buffer_weighted_test ν (realPosition (step e)) f hH z₀ r ε a g C hST hC π hπ A hA w hw |
| intro η hη |
| obtain ⟨hprob,hgap⟩ := hvalid η hη |
| let := hprob |
| have hb := hπgood z₀ (π η).val hprob hgap |
| have he : {ω | BufferStageEvent (realPosition (step e)) f H z₀ r ε a g (π η).val ω} = |
| {ω | BufferStageSuccess (realPosition (step e)) f H z₀ r ε a g (π η).val ω} := by |
| ext ω |
| exact bufferStageEvent_iff e f hH z₀ r ε a hg.le (π η).val ω |
| rw [he] |
| exact (ENNReal.ofReal_le_iff_le_toReal (measure_ne_top _ _)).mpr hb |
| end DirectionalTransience |
|
|
| end |
|
|
| end |
|
|
| end OAI |
|
|
