| import Mathlib |
| import OAI.Probability.Ballisticity.Walk.GaussianPathFiniteLawSingleton |
| import OAI.Probability.Ballisticity.Renewal.PositiveOpenTest |
|
|
| namespace OAI |
|
|
| section |
|
|
| section |
|
|
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal Topology BoundedContinuousFunction |
| namespace DirectionalTransience |
|
|
| def pathWindowEvent (τ : unitInterval) (δ ρ B : ℝ) : Set C(unitInterval,ℝ) := |
| {g | Set.MapsTo g (Metric.closedBall τ δ) (Set.Ioo (-ρ) ρ) ∧ ‖g‖ < B} |
|
|
| lemma isOpen_pathWindowEvent (τ : unitInterval) (δ ρ B : ℝ) : |
| IsOpen (pathWindowEvent τ δ ρ B) := |
| (ContinuousMap.isOpen_setOfPred_mapsTo (isCompact_closedBall τ δ) isOpen_Ioo).inter |
| (isOpen_lt continuous_norm continuous_const) |
|
|
| lemma pathWindowEvent_compl_nonempty (τ : unitInterval) (δ ρ B : ℝ) (hδ : 0 ≤ δ) : |
| (pathWindowEvent τ δ ρ B)ᶜ.Nonempty := by |
| refine ⟨ContinuousMap.const _ (ρ+1),?_⟩ |
| intro h |
| have hh := h.1 (Metric.mem_closedBall_self hδ) |
| change -ρ < ρ+1 ∧ ρ+1 < ρ at hh |
| linarith [hh.2] |
|
|
| lemma endpoint_window_covered (τ : unitInterval) (ρ : ℝ) : |
| {g : C(unitInterval,ℝ) | |g τ| < ρ} ⊆ |
| ⋃ n : ℕ, pathWindowEvent τ (1/((n:ℝ)+1)) ρ ((n:ℝ)+1) := by |
| intro g hg |
| have he : 0 < ρ-|g τ| := sub_pos.mpr hg |
| obtain ⟨δ,hδ,hgδ⟩ := Metric.continuousAt_iff.mp g.continuous.continuousAt (ρ-|g τ|) he |
| have hn : ∀ᶠ n : ℕ in atTop, 1/((n:ℝ)+1) < δ := |
| tendsto_one_div_add_atTop_nhds_zero_nat.eventually_lt_const hδ |
| have hn' : ∀ᶠ n : ℕ in atTop, ‖g‖ < (n:ℝ)+1 := by |
| have hh := (tendsto_natCast_atTop_atTop (R := ℝ)).eventually (eventually_gt_atTop ‖g‖) |
| filter_upwards [hh] with n hn |
| linarith |
| obtain ⟨n,hn,hbn⟩ := (hn.and hn').exists |
| refine Set.mem_iUnion.mpr ⟨n,?_,hbn⟩ |
| intro t ht |
| have hd := hgδ (lt_of_le_of_lt ht hn) |
| have hgt : |g t| < ρ := by |
| rw [Real.dist_eq] at hd |
| have ha := abs_add_le (g t-g τ) (g τ) |
| rw [sub_add_cancel] at ha |
| linarith |
| exact abs_lt.mp hgt |
|
|
| theorem positive_path_window_test (μ : Measure C(unitInterval,ℝ)) [IsFiniteMeasure μ] |
| (τ : unitInterval) (ρ : ℝ) (hpos : 0 < μ {g | |g τ| < ρ}) : |
| ∃ δ B : ℝ, 0 < δ ∧ 0 < B ∧ ∃ F : C(unitInterval,ℝ) →ᵇ ℝ, |
| (∀ g, 0 ≤ F g ∧ F g ≤ 1) ∧ |
| (∀ g, F g ≠ 0 → ‖g‖ < B ∧ ∀ t : unitInterval, dist t τ ≤ δ → |g t| < ρ) ∧ |
| 0 < ∫ g, F g ∂μ := by |
| have hc := endpoint_window_covered τ ρ |
| have hn : ∃ n : ℕ, 0 < μ (pathWindowEvent τ (1/((n:ℝ)+1)) ρ ((n:ℝ)+1)) := by |
| by_contra! h |
| have hz : ∀ n : ℕ, μ (pathWindowEvent τ (1/((n:ℝ)+1)) ρ ((n:ℝ)+1)) = 0 := |
| fun n => le_antisymm (h n) zero_le |
| have hu := measure_iUnion_null hz |
| exact (ne_of_gt hpos) (measure_mono_null hc hu) |
| obtain ⟨n,hn⟩ := hn |
| have hB : 0 < (n:ℝ)+1 := by positivity |
| have hδ : 0 < 1/((n:ℝ)+1) := by positivity |
| obtain ⟨F,hF,hFs,hFp⟩ := positive_bounded_test_of_open μ |
| (pathWindowEvent τ (1/((n:ℝ)+1)) ρ ((n:ℝ)+1)) (isOpen_pathWindowEvent _ _ _ _) |
| (pathWindowEvent_compl_nonempty _ _ _ _ hδ.le) hn |
| refine ⟨_,_,hδ,hB,F,hF,?_,hFp⟩ |
| intro g hg |
| have hh := hFs g hg |
| refine ⟨hh.2,fun t ht => ?_⟩ |
| exact abs_lt.mpr (hh.1 ht) |
|
|
| end DirectionalTransience |
|
|
| end |
|
|
| section |
|
|
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal Topology BoundedContinuousFunction |
| namespace DirectionalTransience |
|
|
| lemma gaussianReal_centered_window_pos (v : ℝ≥0) {ρ : ℝ} (hρ : 0 < ρ) : |
| 0 < gaussianReal 0 v {z : ℝ | |z| < ρ} := by |
| by_cases hv : v = 0 |
| · rw [hv,gaussianReal_zero_var] |
| simp [hρ] |
| · apply pos_iff_ne_zero.mpr |
| intro hz |
| have hh := gaussianReal_absolutelyContinuous' 0 hv hz |
| have hs : {z : ℝ | |z| < ρ} = Set.Ioo (-ρ) ρ := by |
| ext z |
| change |z| < ρ ↔ -ρ < z ∧ z < ρ |
| exact abs_lt |
| rw [hs,Real.volume_Ioo,ENNReal.ofReal_eq_zero] at hh |
| linarith |
|
|
| theorem brownian_positive_common_time_window (W : ProbabilityMeasure C(unitInterval,ℝ)) (T : ℝ) |
| (hW : ∀ I : Finset unitInterval, (W : Measure C(unitInterval,ℝ)).map |
| (fun g : C(unitInterval,ℝ) => I.restrict g) = gaussianPathFiniteLaw T I) |
| (τ : unitInterval) {ρ : ℝ} (hρ : 0 < ρ) : |
| ∃ δ B : ℝ, 0 < δ ∧ 0 < B ∧ ∃ F : C(unitInterval,ℝ) →ᵇ ℝ, |
| (∀ g, 0 ≤ F g ∧ F g ≤ 1) ∧ |
| (∀ g, F g ≠ 0 → ‖g‖ < B ∧ ∀ t : unitInterval, dist t τ ≤ δ → |g t| < ρ) ∧ |
| 0 < ∫ g, F g ∂(W : Measure C(unitInterval,ℝ)) := by |
| apply positive_path_window_test |
| have hp := gaussianReal_centered_window_pos (Real.toNNReal (T*(τ:ℝ))) hρ |
| rw [← wiener_path_eval W T hW τ] at hp |
| simpa only [Measure.map_apply (continuous_eval_const τ).measurable |
| (isOpen_lt (continuous_abs) continuous_const).measurableSet,Set.preimage_ofPred_eq] using hp |
|
|
| end DirectionalTransience |
|
|
| end |
|
|
| end |
|
|
| end OAI |
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|
