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9eeb9c7 | 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 | import Mathlib
import OAI.Analysis.HardSphere.Geometry.KilledObservable
import OAI.Analysis.HardSphere.PathBounds.VisitPacking
namespace OAI
section
section
noncomputable section
open MeasureTheory Set Filter
open scoped ENNReal NNReal Topology BigOperators
namespace HardSphere.FreeVisits
open HardSphere.BathSlice HardSphere.Lattice HardSphere.CellOrbital HardSphere.Wiener
lemma pathVisitedSites_card_measurable {S : Type*} {n : ℕ} (L : Length)
(v : S → Torus L) (A : Finset S) (i : Fin n) :
Measurable (fun ω : ConfigurationPath n L ↦ (pathVisitedSites L v A i ω).card) := by
classical
simp only [pathVisitedSites,Finset.card_filter]
apply Finset.measurable_sum
intro a _
exact Measurable.ite (pathVisitsBall_isClosed L 12 (v a) i).measurableSet
measurable_const measurable_const
def visitCount {n K : ℕ} (L : Length) (A : Finset (DiscreteTorus K)) (i : Fin n)
(t : ℝ) (X : Configuration n L) (w : CoordinatePaths n) : ℕ :=
(pathVisitedSites L (siteCenter L) A i (freeTrajectory n L t w X)).card
lemma visitCount_measurable {n K : ℕ} (L : Length) (A : Finset (DiscreteTorus K)) (i : Fin n)
(t : ℝ) (X : Configuration n L) : Measurable (visitCount L A i t X) :=
(pathVisitedSites_card_measurable L (siteCenter L) A i).comp
((freeTrajectory_continuous n L t).comp (continuous_id.prodMk continuous_const)).measurable
lemma one_segment_visit_tail {c : ℝ} {M : ℝ≥0∞}
(htail : ∀ R : ℝ, 0≤R → wiener {w | R≤‖w‖}≤M*ENNReal.ofReal (Real.exp (-c*R^2)))
{n K : ℕ} [NeZero K] (L : Length) (hL : L.val=(K:ℝ))
(A : Finset (DiscreteTorus K)) (i : Fin n) {t : ℝ} (ht : 0<t) (X : Configuration n L) :
coordinateWiener n {w | 29^3<visitCount L A i t X w} ≤
3*M*ENNReal.ofReal (Real.exp (-c/(2*t))) := by
let R : ℝ := (Real.sqrt (2*t))⁻¹
have hs : 0<Real.sqrt (2*t) := Real.sqrt_pos.mpr (by positivity)
have hR : 0≤R := inv_nonneg.mpr hs.le
have hsub : {w | 29^3<visitCount L A i t X w} ⊆
⋃ j : Fin 3, {w | R≤‖w (i,j)‖} := by
intro w hw
by_contra hnot
have hn : ∀ j, ‖w (i,j)‖<R := by simpa only [Set.mem_iUnion,Set.mem_ofPred_eq,not_exists,not_le] using hnot
have hb := free_visited_card_le_of_norm L hL A i t w X (fun j ↦ ?_)
· exact (not_lt_of_ge hb) hw
· calc
Real.sqrt (2*t)*‖w (i,j)‖ ≤ Real.sqrt (2*t)*R := mul_le_mul_of_nonneg_left (hn j).le hs.le
_ = 1 := mul_inv_cancel₀ (ne_of_gt hs)
have he (j : Fin 3) : coordinateWiener n {w | R≤‖w (i,j)‖} = wiener {w | R≤‖w‖} := by
exact (measurePreserving_eval (fun _ : Direction n ↦ wiener) (i,j)).measure_preimage
(measurableSet_le measurable_const measurable_norm).nullMeasurableSet
calc
_ ≤ coordinateWiener n (⋃ j : Fin 3, {w | R≤‖w (i,j)‖}) := measure_mono hsub
_ ≤ ∑' j : Fin 3, coordinateWiener n {w | R≤‖w (i,j)‖} := measure_iUnion_le _
_ = 3 * wiener {w | R≤‖w‖} := by simp [he,tsum_fintype]
_ ≤ 3*(M*ENNReal.ofReal (Real.exp (-c*R^2))) := mul_le_mul_right (htail R hR) _
_ = _ := by
dsimp [R]
rw [inv_pow,Real.sq_sqrt (by positivity : 0≤2*t)]
simp only [div_eq_mul_inv,mul_assoc]
lemma visit_count_split {n K : ℕ} (L : Length) (A : Finset (DiscreteTorus K)) (i : Fin n)
{s t : ℝ} (hs : 0<s) (ht : 0<t) (w : CoordinatePaths n) (X : Configuration n L) :
visitCount L A i (s+t) X w ≤
visitCount L A i s X (fun p ↦ normalizedPrefix (splitTime s t hs ht) (w p)) +
visitCount L A i t
(freeTrajectory n L s (fun p ↦ normalizedPrefix (splitTime s t hs ht) (w p)) X 1)
(fun p ↦ normalizedSuffix (splitTime s t hs ht) (w p)) := by
have hle := visited_card_split_le L (siteCenter L) A i (freeTrajectory n L (s+t) w X)
(splitTime s t hs ht) (splitTime_pos s t hs ht) (splitTime_lt_one s t hs ht)
have hp := congrArg Prod.fst (split_fixedTrajectory n L s t hs ht w X)
have hq := congrArg Prod.snd (split_fixedTrajectory n L s t hs ht w X)
dsimp only [Prod.fst,Prod.snd] at hp hq
rw [hp,hq] at hle
exact hle
attribute [local irreducible] freeTrajectory pathVisitedSites
lemma visit_tail_add {n K : ℕ} (L : Length) (A : Finset (DiscreteTorus K)) (i : Fin n)
{s t : ℝ} (hs : 0<s) (ht : 0<t) (a b : ℕ) (p q : ℝ≥0∞)
(hsbound : ∀ X, coordinateWiener n {w | a<visitCount L A i s X w}≤p)
(htbound : ∀ X, coordinateWiener n {w | b<visitCount L A i t X w}≤q)
(X : Configuration n L) :
coordinateWiener n {w | a+b<visitCount L A i (s+t) X w}≤p+q := by
let u := splitTime s t hs ht
let split : CoordinatePaths n → CoordinatePaths n × CoordinatePaths n := fun w ↦
(fun p ↦ normalizedPrefix u (w p),fun p ↦ normalizedSuffix u (w p))
let E₁ : Set (CoordinatePaths n × CoordinatePaths n) :=
{p | a<visitCount L A i s X p.1}
let E₂ : Set (CoordinatePaths n × CoordinatePaths n) :=
{p | b<visitCount L A i t (freeTrajectory n L s p.1 X 1) p.2}
have hE₁ : MeasurableSet E₁ := measurableSet_lt measurable_const
((visitCount_measurable L A i s X).comp measurable_fst)
have hE₂ : MeasurableSet E₂ := by
apply measurableSet_lt measurable_const
exact (pathVisitedSites_card_measurable L (siteCenter L) A i).comp
(measurable_snd.comp (split_fixedTrajectory_measurable n L s t X))
have hsub : {w | a+b<visitCount L A i (s+t) X w} ⊆ split ⁻¹' (E₁∪E₂) := by
intro w hw
have hle := visit_count_split L A i hs ht w X
change a<visitCount L A i s X (split w).1 ∨
b<visitCount L A i t (freeTrajectory n L s (split w).1 X 1) (split w).2
change a+b<visitCount L A i (s+t) X w at hw
change visitCount L A i (s+t) X w ≤ visitCount L A i s X (split w).1 +
visitCount L A i t (freeTrajectory n L s (split w).1 X 1) (split w).2 at hle
omega
have hp₁ : ((coordinateWiener n).prod (coordinateWiener n)) E₁≤p := by
have he : E₁={w | a<visitCount L A i s X w} ×ˢ Set.univ := by ext v; simp [E₁]
rw [he,Measure.prod_prod,measure_univ,mul_one]
exact hsbound X
have hp₂ : ((coordinateWiener n).prod (coordinateWiener n)) E₂≤q := by
rw [Measure.prod_apply hE₂]
calc
_ ≤ ∫⁻ _w : CoordinatePaths n, q ∂coordinateWiener n := by
apply lintegral_mono
intro w
exact htbound (freeTrajectory n L s w X 1)
_ = q := by simp
calc
_ ≤ coordinateWiener n (split ⁻¹' (E₁∪E₂)) := measure_mono hsub
_ = ((coordinateWiener n).prod (coordinateWiener n)) (E₁∪E₂) :=
(coordinateSplit_preserving n u (splitTime_pos s t hs ht) (splitTime_lt_one s t hs ht)).measure_preimage
(hE₁.union hE₂).nullMeasurableSet
_ ≤ ((coordinateWiener n).prod (coordinateWiener n)) E₁ +
((coordinateWiener n).prod (coordinateWiener n)) E₂ := measure_union_le _ _
_ ≤ p+q := add_le_add hp₁ hp₂
lemma n_segment_visit_tail {c : ℝ} {M : ℝ≥0∞}
(htail : ∀ R : ℝ, 0≤R → wiener {w | R≤‖w‖}≤M*ENNReal.ofReal (Real.exp (-c*R^2)))
{n K : ℕ} [NeZero K] (L : Length) (hL : L.val=(K:ℝ))
(A : Finset (DiscreteTorus K)) (i : Fin n) {t : ℝ} (ht : 0<t) :
∀ k : ℕ, 0<k → ∀ X : Configuration n L,
coordinateWiener n {w | 29^3*k<visitCount L A i ((k:ℝ)*t) X w} ≤
(k:ℝ≥0∞)*(3*M*ENNReal.ofReal (Real.exp (-c/(2*t)))) := by
intro k hk
induction k, hk using Nat.le_induction with
| base =>
intro X
simpa only [Nat.succ_eq_add_one,Nat.zero_add,Nat.cast_one,one_mul,mul_one] using one_segment_visit_tail htail L hL A i ht X
| succ k hk ih =>
intro X
have hkt : 0<(k:ℝ)*t := mul_pos (by exact_mod_cast hk) ht
have h := visit_tail_add L A i hkt ht (29^3*k) (29^3)
((k:ℝ≥0∞)*(3*M*ENNReal.ofReal (Real.exp (-c/(2*t)))))
(3*M*ENNReal.ofReal (Real.exp (-c/(2*t)))) ih
(fun X ↦ one_segment_visit_tail htail L hL A i ht X) X
convert h using 1 <;> push_cast <;> ring_nf
lemma whole_time_visit_tail {c : ℝ} {M : ℝ≥0∞}
(htail : ∀ R : ℝ, 0≤R → wiener {w | R≤‖w‖}≤M*ENNReal.ofReal (Real.exp (-c*R^2)))
{n K : ℕ} [NeZero K] (L : Length) (hL : L.val=(K:ℝ))
(A : Finset (DiscreteTorus K)) (i : Fin n) {t : ℝ} (ht : 0<t)
(k : ℕ) (hk : 0<k) (X : Configuration n L) :
coordinateWiener n {w | 29^3*k<visitCount L A i t X w} ≤
(k:ℝ≥0∞)*(3*M*ENNReal.ofReal (Real.exp (-c*(k:ℝ)/(2*t)))) := by
have hk' : (0:ℝ)<k := by exact_mod_cast hk
have h := n_segment_visit_tail htail L hL A i (div_pos ht hk') k hk X
have he : (k:ℝ)*(t/k)=t := mul_div_cancel₀ t (ne_of_gt hk')
rw [he] at h
convert h using 1
congr 3
field_simp
end HardSphere.FreeVisits
end
end
end
end OAI
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