lean/OAI/Analysis/HardSphere/Conditioning/PhaseMeasure.lean

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