File size: 9,984 Bytes
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 | import Mathlib
import OAI.Probability.Ballisticity.Estimates.GlobalProfile
namespace OAI
section
open MeasureTheory ProbabilityTheory TopologicalSpace
open scoped ENNReal NNReal Classical Topology
namespace DirectionalTransience
abbrev UpperRows {d : ℕ} (e : Direction d) := (ℕ×HorizontalSpace e) → Row d
noncomputable def upperEnvironment {d : ℕ} (e : Direction d) (ξ : UpperRows e) : Environment d :=
fun x => ξ ((signedHeight e x).toNat,horizontalProjection e x)
lemma upperEnvironment_continuous {d : ℕ} (e : Direction d) : Continuous (upperEnvironment e) := by
unfold upperEnvironment
fun_prop
lemma upperEnvironment_lift {d : ℕ} (e : Direction d) (ξ : UpperRows e)
(l : ℕ) (z : HorizontalSpace e) :
upperEnvironment e ξ (horizontalLift e l z)=ξ (l,z) := by
simp only [upperEnvironment,signedHeight_horizontalLift,horizontalProjection_lift,Int.toNat_natCast]
lemma continuous_wordWeight {d : ℕ} (x : Lattice d) (w : List (Direction d)) :
Continuous (fun ω : Environment d => wordWeight ω x w) := by
induction w generalizing x with
| nil => exact continuous_const
| cons u w ih =>
change Continuous (fun ω : Environment d => ((ω x).1 u:ℝ)*wordWeight ω (x+step u) w)
exact (NNReal.continuous_coe.comp ((continuous_apply u).comp (continuous_subtype_val.comp (continuous_apply x)))).mul (ih _)
lemma hitKernel_eq_tsum_words {d : ℕ} (ω : Environment d) (x : Lattice d)
(S T : Set (Lattice d)) (hST : Disjoint S T) (A : Set (Lattice d)) :
hitKernel S T (ω,x) A =
∑' w : {w : HitWord x S T // wordPath x w.val w.val.length∈A},
ENNReal.ofReal (wordWeight ω x w.val.val) := by
rw [←successfulWordLaw_endpoint ω x S T hST,
Measure.map_apply (measurable_of_countable _) (Set.to_countable A).measurableSet,
successfulWordLaw_apply]
rfl
lemma hitKernel_lowerSemicontinuous {d : ℕ} (x : Lattice d)
(S T : Set (Lattice d)) (hST : Disjoint S T) (A : Set (Lattice d)) :
LowerSemicontinuous (fun ω : Environment d => hitKernel S T (ω,x) A) := by
simp_rw [hitKernel_eq_tsum_words _ x S T hST A]
exact lowerSemicontinuous_tsum fun w =>
(ENNReal.continuous_ofReal.comp (continuous_wordWeight x w.val.val)).lowerSemicontinuous
noncomputable def upperHorizontalKernel {d : ℕ} (e : Direction d) (H : ℕ) (ξ : UpperRows e) :
Kernel (HorizontalSpace e) (HorizontalSpace e) :=
Kernel.mk (fun z => (variableHitKernel (realPosition (step e)) H (upperEnvironment e ξ,horizontalLift e 0 z)).map
(horizontalProjection e)) (measurable_of_countable _)
lemma upperHorizontalKernel_apply {d : ℕ} (e : Direction d) (H : ℕ) (ξ : UpperRows e)
(z : HorizontalSpace e) (A : Set (HorizontalSpace e)) :
upperHorizontalKernel e H ξ z A =
hitKernel (Strip (realPosition (step e)) (horizontalLift e 0 z) H)
(Upper (realPosition (step e)) (horizontalLift e 0 z) H)
(upperEnvironment e ξ,horizontalLift e 0 z) (horizontalProjection e ⁻¹' A) := by
rw [upperHorizontalKernel,Kernel.coe_mk,Measure.map_apply (measurable_of_countable _) (Set.to_countable A).measurableSet]
rfl
lemma upperHorizontalKernel_lowerSemicontinuous {d : ℕ} (e : Direction d) (H : ℕ)
(z : HorizontalSpace e) (A : Set (HorizontalSpace e)) :
LowerSemicontinuous (fun ξ : UpperRows e => upperHorizontalKernel e H ξ z A) := by
simp_rw [upperHorizontalKernel_apply]
exact (hitKernel_lowerSemicontinuous _ _ _ (disjoint_strip_upper _ _ _) _).comp
(upperEnvironment_continuous e)
lemma upperHorizontalKernel_mass_le_one {d : ℕ} (e : Direction d) (H : ℕ) (ξ : UpperRows e)
(z : HorizontalSpace e) : upperHorizontalKernel e H ξ z Set.univ ≤ 1 := by
rw [upperHorizontalKernel_apply,Set.preimage_univ]
exact hitKernel_total_le_one (disjoint_strip_upper _ _ _) _
end DirectionalTransience
end
section
open MeasureTheory ProbabilityTheory
open scoped ENNReal NNReal Classical BigOperators
namespace DirectionalTransience
lemma wordPath_endpoint_sum {d : ℕ} (x : Lattice d) (w : List (Direction d)) :
wordPath x w w.length=x+(w.map step).sum := by
induction w generalizing x with
| nil => simp [wordPath]
| cons u w ih => simp [wordPath,ih,add_assoc]
lemma wordPath_replicate_min {d : ℕ} (x : Lattice d) (u : Direction d) (k n : ℕ) :
wordPath x (List.replicate k u) n=x+(min n k:ℕ) • step u := by
induction k generalizing x n with
| zero => simp [wordPath]
| succ k ih =>
cases n with
| zero => simp
| succ n =>
rw [List.replicate_succ,wordPath,ih,min_add_add_right,succ_nsmul]
abel
lemma exists_horizontal_word {d : ℕ} (e : Direction d) (z : HorizontalSpace e) :
∃ w : List (Direction d), (w.map step).sum=z.val ∧ ∀ u∈w, u.1≠e.1 := by
have hsingle (i : Fin d) : ∃ w : List (Direction d),
(w.map step).sum=Pi.single i (z.val i) ∧ ∀ u∈w, u.1≠e.1 := by
by_cases hi : i=e.1
· subst i
refine ⟨[],?_,by simp⟩
have hz : z.val e.1=0 := z.property
simp [hz]
· cases hz : z.val i with
| ofNat n =>
refine ⟨List.replicate n (i,true),?_,?_⟩
· simp only [List.map_replicate,List.sum_replicate]
funext j
by_cases hj : j=i
· subst j; simp [step]
· simp [step,hj]
· intro u hu
have he := List.eq_of_mem_replicate hu
simpa only [he] using hi
| negSucc n =>
refine ⟨List.replicate (n+1) (i,false),?_,?_⟩
· simp only [List.map_replicate,List.sum_replicate]
funext j
by_cases hj : j=i
· subst j; simp [step]; omega
· simp [step,hj]
· intro u hu
have he := List.eq_of_mem_replicate hu
simpa only [he] using hi
have hf (s : Finset (Fin d)) : ∃ w : List (Direction d),
(w.map step).sum=∑ i∈s, Pi.single i (z.val i) ∧ ∀ u∈w, u.1≠e.1 := by
induction s using Finset.induction_on with
| empty => exact ⟨[],by simp,by simp⟩
| @insert i s hi ih =>
obtain ⟨v,hv,hve⟩ := hsingle i
obtain ⟨w,hw,hwe⟩ := ih
refine ⟨v++w,by simp [hv,hw,hi],?_⟩
intro u hu
rcases List.mem_append.mp hu with hu|hu
· exact hve u hu
· exact hwe u hu
obtain ⟨w,hw,hwe⟩ := hf Finset.univ
refine ⟨w,hw.trans ?_,hwe⟩
funext j
simp
lemma wordPath_horizontal {d : ℕ} (e : Direction d) (x : Lattice d)
(w : List (Direction d)) (hw : ∀ u∈w, u.1≠e.1) (n : ℕ) :
signedHeight e (wordPath x w n)=signedHeight e x := by
induction w generalizing x n with
| nil => rfl
| cons u w ih =>
cases n with
| zero => rfl
| succ n =>
rw [wordPath,ih (x+step u) (fun g hg => hw g (by simp [hg]))]
have hu := hw u (by simp)
rw [signedHeight_add]
simp [signedHeight, step,Ne.symm hu]
lemma positive_wordWeight {d : ℕ} (ω : Environment d)
(hω : ∀ x u, 0<(ω x).val u) (x : Lattice d) (w : List (Direction d)) :
0<wordWeight ω x w := by
induction w generalizing x with
| nil => norm_num [wordWeight]
| cons u w ih =>
exact mul_pos (NNReal.coe_pos.mpr (hω x u)) (ih _)
lemma upperHorizontalKernel_pos {d : ℕ} (e : Direction d) (H : ℕ) (hH : 0<H)
(ξ : UpperRows e) (hξ : ∀ p u, 0<(ξ p).val u) (x y : HorizontalSpace e) :
0<upperHorizontalKernel e H ξ x {y} := by
obtain ⟨v,hv,hve⟩ := exists_horizontal_word e (y-x)
let w := v++List.replicate H e
have hend : wordPath (horizontalLift e 0 x) v v.length=horizontalLift e 0 y := by
rw [wordPath_endpoint_sum,hv]
simp only [horizontalLift,zero_zsmul,add_zero,AddSubgroup.coe_sub]
abel
have hy : wordPath (horizontalLift e 0 x) w w.length=horizontalLift e H y := by
simp only [w,List.length_append,List.length_replicate]
rw [wordPath_append_suffix,hend,wordPath_replicate_min,Nat.min_self]
simp [horizontalLift]
have hh : wordPath (horizontalLift e 0 x) w∈HitAt
(Strip (realPosition (step e)) (horizontalLift e 0 x) H)
(Upper (realPosition (step e)) (horizontalLift e 0 x) H) w.length := by
constructor
· rw [coordinate_mem_upper_iff,hy,signedHeight_horizontalLift,signedHeight_horizontalLift]
omega
· intro n hn
rw [coordinate_mem_strip_iff,signedHeight_horizontalLift]
by_cases hnv : n≤v.length
· have he := wordPath_append_prefix (horizontalLift e 0 x) v (List.replicate H e) hnv
rw [he,wordPath_horizontal e _ v hve,signedHeight_horizontalLift]
omega
· have hn' : n=v.length+(n-v.length) := by omega
rw [hn',wordPath_append_suffix,hend,wordPath_replicate_min,signedHeight_add,
signedHeight_horizontalLift]
have hmin : min (n-v.length) H=n-v.length := by
have hh : n<v.length+H := by simpa only [w,List.length_append,List.length_replicate] using hn
omega
rw [hmin]
have he : signedHeight e ((n-v.length) • step e)=(n-v.length:ℕ) := by
exact_mod_cast signedHeight_zsmul_step e (n-v.length:ℕ)
rw [he]
have hh : n<v.length+H := by simpa only [w,List.length_append,List.length_replicate] using hn
omega
rw [upperHorizontalKernel_apply,hitKernel_eq_tsum_words _ _ _ _ (disjoint_strip_upper _ _ _) _]
let a : {w : HitWord (horizontalLift e 0 x)
(Strip (realPosition (step e)) (horizontalLift e 0 x) H)
(Upper (realPosition (step e)) (horizontalLift e 0 x) H) //
wordPath (horizontalLift e 0 x) w.val w.val.length∈horizontalProjection e ⁻¹' {y}} :=
⟨⟨w,hh⟩,by change horizontalProjection e (wordPath _ w w.length)=y; rw [hy,horizontalProjection_lift]⟩
exact lt_of_lt_of_le (ENNReal.ofReal_pos.mpr (positive_wordWeight (upperEnvironment e ξ)
(fun z u => hξ _ u) (horizontalLift e 0 x) w))
(ENNReal.le_tsum (f := fun term => ENNReal.ofReal
(wordWeight (upperEnvironment e ξ) (horizontalLift e 0 x) term.val.val)) a)
end DirectionalTransience
end
end OAI
|