| import Mathlib |
| import OAI.Probability.Ballisticity.Estimates.CurvePolicyPMF |
| import OAI.Probability.Ballisticity.Estimates.CurvePolicyLateral |
| import OAI.Probability.Ballisticity.Estimates.BindIntegral |
|
|
| namespace OAI |
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| section |
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| section |
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|
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal Topology Classical |
| namespace DirectionalTransience |
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| noncomputable def curvePolicyKernel {d : ℕ} (ℓ : Vector d) (f : Direction d) |
| (x : Lattice d) (b : ℕ → ℝ) (B : ℝ) {H : ℕ} (hH : 0 < H) |
| (E : Set (Lattice d)) (δ α : ℝ≥0∞) (dummy : Lattice d) : |
| Kernel (Environment d) (Lattice d) where |
| toFun := curvePolicy ℓ f x b B H E δ α dummy |
| measurable' := (curvePolicy_rows ℓ f x b B hH E δ α dummy).mono (rowSigma_le _) le_rfl |
|
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| lemma curvePolicyKernel_markov {d : ℕ} (ℓ : Vector d) (f : Direction d) |
| (x : Lattice d) (b : ℕ → ℝ) (B : ℝ) {H : ℕ} (hH : 0 < H) |
| (E : Set (Lattice d)) {δ α : ℝ≥0∞} (hδ : 0 < δ) (hα : 0 < α) (dummy : Lattice d) : |
| IsMarkovKernel (curvePolicyKernel ℓ f x b B hH E δ α dummy) := |
| ⟨curvePolicy_probability ℓ f x b B H E hδ hα dummy⟩ |
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| lemma curvePolicy_average_toMeasure {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν] |
| (ℓ : Vector d) (f : Direction d) (b : ℕ → ℝ) (B : ℝ) {H : ℕ} (hH : 0 < H) |
| (E : Set (Lattice d)) {δ α : ℝ≥0∞} (hδ : 0 < δ) (hα : 0 < α) (dummy : Lattice d) : |
| (curvePolicy_average ν ℓ f b B hH E hδ hα dummy).toMeasure = |
| (environmentLaw ν).bind (curvePolicy ℓ f 0 b B H E δ α dummy) := by |
| have := curvePolicy_average_probability ν ℓ f b B hH E hδ hα dummy |
| exact Measure.toPMF_toMeasure _ |
|
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| lemma curvePolicy_average_square {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν] |
| (e f : Direction d) (b : ℕ → ℝ) {B : ℝ} (hB : 0 ≤ B) {H : ℕ} (hH : 0 < H) |
| (E : Set (Lattice d)) {δ α : ℝ≥0∞} (hδ : 0 < δ) (hα : 0 < α) |
| (dummy : Lattice d) (hd : signedCoordinate f dummy = b H) : |
| (∫ u, (signedCoordinate f u-b H)^2 ∂(curvePolicy_average ν (realPosition (step e)) f b B hH E hδ hα dummy).toMeasure) = |
| ∫ ω, ∫ u, (signedCoordinate f u-b H)^2 ∂curvePolicy (realPosition (step e)) f 0 b B H E δ α dummy ω ∂environmentLaw ν := by |
| rw [curvePolicy_average_toMeasure] |
| exact integral_bind_nonneg_of_integrable _ _ |
| ((curvePolicy_rows _ f 0 b B hH E δ α dummy).mono (rowSigma_le _) le_rfl) |
| _ (measurable_of_countable _) (fun u => sq_nonneg _) (curvePolicy_square_integrable e f 0 b hB hH E hδ hα dummy hd) |
|
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| lemma curvePolicy_square_outer_integrable {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν] |
| (e f : Direction d) (x : Lattice d) (b : ℕ → ℝ) {B : ℝ} (hB : 0 ≤ B) |
| {H : ℕ} (hH : 0 < H) (E : Set (Lattice d)) {δ α : ℝ≥0∞} (hδ : 0 < δ) (hα : 0 < α) |
| (dummy : Lattice d) (hd : signedCoordinate f dummy = b H) : |
| Integrable (fun ω => ∫ u, (signedCoordinate f u-b H)^2 ∂curvePolicy (realPosition (step e)) f x b B H E δ α dummy ω) |
| (environmentLaw ν) := by |
| let Q := curvePolicyKernel (realPosition (step e)) f x b B hH E δ α dummy |
| have hm : Measurable (fun ω => ∫ u, (signedCoordinate f u-b H)^2 ∂Q ω) := |
| (measurable_of_countable _).stronglyMeasurable.integral_kernel.measurable |
| apply Integrable.of_bound hm.aestronglyMeasurable (B^2) |
| apply Eventually.of_forall |
| intro ω |
| have := curvePolicy_probability (realPosition (step e)) f x b B H E hδ hα dummy ω |
| have hi := curvePolicy_square_integrable e f x b hB hH E hδ hα dummy hd ω |
| have hh : (∫ u, (signedCoordinate f u-b H)^2 ∂curvePolicy (realPosition (step e)) f x b B H E δ α dummy ω) ≤ B^2 := by |
| have hb : ∀ᵐ u ∂curvePolicy (realPosition (step e)) f x b B H E δ α dummy ω, |
| (signedCoordinate f u-b H)^2 ≤ B^2 := by |
| filter_upwards [curvePolicy_lateral_bound e f x b hB hH E δ α dummy hd ω] with u hu |
| nlinarith [sq_abs (signedCoordinate f u-b H),sq_nonneg (B-|signedCoordinate f u-b H|),abs_nonneg (signedCoordinate f u-b H)] |
| simpa using integral_mono_ae hi (integrable_const (B^2)) hb |
| have h0 : 0 ≤ ∫ u, (signedCoordinate f u-b H)^2 ∂Q ω := integral_nonneg (fun u => sq_nonneg (signedCoordinate f u-b H)) |
| rw [Real.norm_eq_abs,abs_of_nonneg h0] |
| exact hh |
|
|
| end DirectionalTransience |
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| end |
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| end |
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| end OAI |
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|
