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67aee1d | 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 | import Mathlib
import OAI.Analysis.HardSphere.Bridges.BridgeResidual
import OAI.Analysis.HardSphere.Comparison.PiAC
namespace OAI
section
section
noncomputable section
open MeasureTheory MeasureTheory.Measure Set Filter ProbabilityTheory
open scoped ENNReal NNReal Topology BigOperators
namespace HardSphere
open Wiener
def localLiftBox {N : ℕ} (L : Length) (c : LiftedConfiguration N) : Set (LiftedConfiguration N) :=
Set.univ.pi (fun i ↦ Set.univ.pi (fun j ↦ Icc (c i j-L.val/2) (c i j+L.val/2)))
lemma localLiftBox_preserving {N : ℕ} (L : Length) (c : LiftedConfiguration N) :
MeasurePreserving (project N L) (volume.restrict (localLiftBox L c)) volume := by
have hi (i : Fin N) : MeasurePreserving (fun x : Fin 3 → ℝ ↦ fun j ↦ (x j : AddCircle L.val))
(volume.restrict (Set.univ.pi (fun j ↦ Icc (c i j-L.val/2) (c i j+L.val/2)))) volume := by
have hp := measurePreserving_pi (fun j : Fin 3 ↦ volume.restrict (Icc (c i j-L.val/2) (c i j+L.val/2)))
(fun _ : Fin 3 ↦ (volume : Measure (AddCircle L.val)))
(f:=fun _ x ↦ (x:AddCircle L.val)) (fun j ↦ ?_)
· change MeasurePreserving _ ((Measure.pi (fun _ : Fin 3 ↦ (volume : Measure ℝ))).restrict _) (Measure.pi (fun _ : Fin 3 ↦ (volume : Measure (AddCircle L.val))))
simpa only [Measure.restrict_pi_pi] using hp
· have hp := AddCircle.measurePreserving_mk L.val (c i j-L.val/2)
rw [restrict_Ioc_eq_restrict_Icc,show c i j-L.val/2+L.val=c i j+L.val/2 by ring] at hp
exact hp
have hp := measurePreserving_pi
(fun i ↦ volume.restrict (Set.univ.pi (fun j ↦ Icc (c i j-L.val/2) (c i j+L.val/2))))
(fun _ : Fin N ↦ (volume : Measure (Torus L))) hi
change MeasurePreserving (fun X : LiftedConfiguration N ↦ fun i j ↦ (X i j : AddCircle L.val)) ((Measure.pi (fun _ : Fin N ↦ (volume : Measure (Fin 3 → ℝ)))).restrict _) (Measure.pi (fun _ : Fin N ↦ (volume : Measure (Torus L))))
simpa only [project,localLiftBox,Measure.restrict_pi_pi] using hp
lemma ball_subset_localLiftBox {N : ℕ} (L : Length) (c : LiftedConfiguration N) {r : ℝ}
(hr : r≤L.val/2) : Metric.ball c r ⊆ localLiftBox L c := by
intro X hX i hi j hj
have hd : dist (X i j) (c i j)<r :=
(dist_le_pi_dist _ _ j).trans_lt ((dist_le_pi_dist _ _ i).trans_lt hX)
rw [Real.dist_eq,abs_lt] at hd
constructor <;> linarith
lemma gaussianCoordinates_ac (N : ℕ) : gaussianCoordinates N ≪ (volume : Measure (Direction N → ℝ)) :=
Measure.pi_ac _ _ (fun _ ↦ gaussianReal_absolutelyContinuous 0 (by norm_num))
lemma gaussianCoordinates_ac' (N : ℕ) : (volume : Measure (Direction N → ℝ)) ≪ gaussianCoordinates N :=
Measure.pi_ac _ _ (fun _ ↦ gaussianReal_absolutelyContinuous' 0 (by norm_num))
def flattenEquiv (N : ℕ) : LiftedConfiguration N ≃L[ℝ] (Direction N → ℝ) :=
(LinearEquiv.curry ℝ ℝ (Fin N) (Fin 3)).symm.toContinuousLinearEquiv
lemma flatten_quasi (N : ℕ) : QuasiMeasurePreserving (flattenEquiv N)
(volume : Measure (LiftedConfiguration N)) (volume : Measure (Direction N → ℝ)) := by
let := (flattenEquiv N).isAddHaarMeasure_map (volume : Measure (LiftedConfiguration N))
exact ⟨(flattenEquiv N).continuous.measurable,absolutelyContinuous_isAddHaarMeasure _ _⟩
lemma endpointNoise_quasi {N : ℕ} {t : ℝ} (ht : 0<t) (X : LiftedConfiguration N) :
QuasiMeasurePreserving (fun Y : LiftedConfiguration N ↦
fun j : Direction N ↦ (Y j.1 j.2-X j.1 j.2)/Real.sqrt (2*t))
(volume : Measure (LiftedConfiguration N)) (volume : Measure (Direction N → ℝ)) := by
have ht' : Real.sqrt (2*t) ≠ 0 := (Real.sqrt_pos.2 (by linarith)).ne'
have hs := (volume : Measure (LiftedConfiguration N)).quasiMeasurePreserving_smul (inv_ne_zero ht')
have ha := (measurePreserving_add_right (volume : Measure (LiftedConfiguration N)) (-X)).quasiMeasurePreserving
convert (flatten_quasi N).comp (hs.comp ha) using 1
funext Y j
change (Y j.1 j.2-X j.1 j.2)/Real.sqrt (2*t)=(Real.sqrt (2*t))⁻¹*(Y j.1 j.2+-X j.1 j.2)
ring
end HardSphere
end
end
end
end OAI
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