File size: 2,125 Bytes
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 | import Mathlib
import OAI.Analysis.HardSphere.Bridges.BridgeStrongRegular
import OAI.Analysis.HardSphere.Bridges.BridgeBlocks
namespace OAI
section
section
noncomputable section
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal NNReal Topology
namespace HardSphere
open Wiener Bridge GaussianPacking
def labelStartLift {N : ℕ} {L : Length} (e : BridgeEndpoints N L) (i : Fin N) : Point :=
representative N L e.2 i
def labelEndLift {N : ℕ} {L : Length} (t : ℝ) (e : BridgeEndpoints N L) (i : Fin N) : Point :=
fun j ↦ labelStartLift e i j+Real.sqrt (2*t)*e.1 (i,j)
lemma labelBridgePath_lift {N : ℕ} {L : Length} (t : ℝ)
(e : BridgeEndpoints N L) (β : Fin N→Bridge.Noise) (i : Fin N) :
onePathAt i (labelBridgePath N L t (e,β))=
projectCurve L (leg t (labelStartLift e i) (labelEndLift t e i) (β i)) := by
ext s k j
have hp := congrFun (congrFun (project_representative N L e.2) i) j
change e.2 i j+((Real.sqrt (2*t)*(e.1 (i,j)*(s:ℝ)+β i j s):ℝ):AddCircle L.val)=
(((1-(s:ℝ))*labelStartLift e i j+(s:ℝ)*labelEndLift t e i j+Real.sqrt (2*t)*β i j s:ℝ):AddCircle L.val)
change ((labelStartLift e i j:ℝ):AddCircle L.val)=e.2 i j at hp
rw [←hp,←QuotientAddGroup.mk_add]
congr 1
unfold labelEndLift
ring
lemma labelBridgePath_one_law {N : ℕ} {L : Length} (t : ℝ)
(e : BridgeEndpoints N L) (i : Fin N) :
(Measure.pi fun _ : Fin N ↦ Bridge.law).map
(fun β ↦ onePathAt i (labelBridgePath N L t (e,β)))=
(Bridge.law.map (leg t (labelStartLift e i) (labelEndLift t e i))).map (projectCurve L) := by
simp_rw [labelBridgePath_lift]
have h := congrArg (fun μ : Measure Bridge.Noise ↦ μ.map
(projectCurve L ∘ leg t (labelStartLift e i) (labelEndLift t e i)))
(measurePreserving_eval (fun _ : Fin N ↦ Bridge.law) i).map_eq
rw [Measure.map_map ((projectCurve_continuous L).measurable.comp (leg_continuous t _ _).measurable) (measurable_pi_apply i)] at h
rw [Measure.map_map (projectCurve_continuous L).measurable (leg_continuous _ _ _).measurable]
exact h
end HardSphere
end
end
end
end OAI
|