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9b6e014 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 | 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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