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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 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 | import Mathlib
import OAI.Analysis.HardSphere.Geometry.GroundPathWeight
import OAI.Analysis.HardSphere.Bridges.TorusLocalBridge
import OAI.Analysis.HardSphere.Geometry.LocalAE
namespace OAI
noncomputable section
open MeasureTheory MeasureTheory.Measure Set Filter
open scoped ENNReal NNReal Topology BigOperators ComplexConjugate
namespace HardSphere
open Wiener
lemma IsGroundVector.ae_killed_zero_propagation {N : ℕ} {L : Length} {a t : ℝ}
{u : Jet N L} (hu : IsGroundVector N L a u) (ht : 0≤t) :
∀ᵐ ω ∂killedPathLaw N L a t, u.1 (ω 0)=0 → u.1 (ω 1)=0 := by
let Z : Set (Configuration N L) := {X | u.1 X=0}
have hZ : MeasurableSet Z := (Lp.stronglyMeasurable u.1).measurable (measurableSet_singleton 0)
let f : Configuration N L → ℂ := Z.indicator (fun _ ↦ 1)
have hf : MemLp f 2 volume := (memLp_const (1:ℂ)).indicator hZ
let F : ScalarL2 N L := hf.toLp f
have hF : F =ᵐ[volume] f := MemLp.coeFn_toLp hf
have hInner : inner ℂ F (modulusScalar u.1)=0 := by
rw [L2.inner_def]
apply integral_eq_zero_of_ae
filter_upwards [hF,coe_modulusScalar u.1] with X hX hM
simp only [RCLike.inner_apply,hX,hM]
by_cases hz : u.1 X=0
· simp [hz]
· simp [f,Z,hz]
have hi := integrable_killed_endpoints N L a t F (modulusScalar u.1) 0 1
have he : (∫ω, conj (F (ω 0))*modulusScalar u.1 (ω 1) ∂killedPathLaw N L a t)=0 := by
rw [←killedTransition_pairing,show modulusScalar u.1=(modulusJet u).1 from rfl,
hu.modulus.killed_eigenvector ht,inner_smul_right]
change _ * inner ℂ F (modulusScalar u.1)=0
rw [hInner,mul_zero]
let g : ConfigurationPath N L → ℝ := fun ω ↦ if u.1 (ω 0)=0 then ‖u.1 (ω 1)‖ else 0
have hg : (fun ω ↦ conj (F (ω 0))*modulusScalar u.1 (ω 1)) =ᵐ[killedPathLaw N L a t]
fun ω ↦ (g ω:ℂ) := by
filter_upwards [ae_restrict_of_ae ((freePathLaw_marginal N L t 0).quasiMeasurePreserving.ae_eq_comp hF),
ae_restrict_of_ae ((freePathLaw_marginal N L t 1).quasiMeasurePreserving.ae_eq_comp (coe_modulusScalar u.1))] with ω h0 h1
dsimp only [Function.comp_def] at h0 h1
rw [h0,h1]
by_cases hz : u.1 (ω 0)=0 <;> simp [f,Z,g,hz]
have hig : Integrable g (killedPathLaw N L a t) :=
(Complex.reCLM.integrable_comp (hi.congr hg)).congr (Eventually.of_forall fun _ ↦ rfl)
have heg : ∫ω,g ω ∂killedPathLaw N L a t=0 := by
have hh := congrArg Complex.re ((integral_congr_ae hg).symm.trans he)
exact (Complex.reCLM.integral_comp_comm (hi.congr hg)).trans hh
have hgn : ∀ᵐ ω ∂killedPathLaw N L a t, 0≤g ω := ae_of_all _ (fun ω ↦ by dsimp [g]; split_ifs <;> positivity)
have hz := (integral_eq_zero_iff_of_nonneg_ae hgn hig).mp heg
filter_upwards [hz] with ω hω h0
have hn : ‖u.1 (ω 1)‖=0 := by simpa [g,h0] using hω
exact norm_eq_zero.mp hn
def complexPhase (z : ℂ) : ℂ := z/(‖z‖:ℂ)
lemma complexPhase_eq_of_inner {z w : ℂ} (hz : z≠0) (hw : w≠0)
(h : conj z*w=(‖z‖*‖w‖:ℝ)) : complexPhase z=complexPhase w := by
have hnz : (‖z‖:ℂ)≠0 := Complex.ofReal_ne_zero.mpr (norm_ne_zero_iff.mpr hz)
have hnw : (‖w‖:ℂ)≠0 := Complex.ofReal_ne_zero.mpr (norm_ne_zero_iff.mpr hw)
apply (div_eq_div_iff hnz hnw).mpr
have hh := congrArg (fun u : ℂ ↦ z*u) h
rw [←mul_assoc,Complex.mul_conj',Complex.ofReal_mul] at hh
apply mul_left_cancel₀ hnz
calc
(‖z‖:ℂ)*(z*(‖w‖:ℂ))=z*((‖z‖:ℂ)*(‖w‖:ℂ)) := by ring
_ = (‖z‖:ℂ)^2*w := hh.symm
_ = (‖z‖:ℂ)*(w*(‖z‖:ℂ)) := by ring
lemma complexPhase_mul_norm (z : ℂ) : complexPhase z*(‖z‖:ℂ)=z := by
by_cases hz : z=0
· simp [complexPhase,hz]
· exact div_mul_cancel₀ _ (Complex.ofReal_ne_zero.mpr (norm_ne_zero_iff.mpr hz))
lemma norm_complexPhase {z : ℂ} (hz : z≠0) : ‖complexPhase z‖=1 := by
rw [complexPhase,norm_div,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (norm_nonneg _)]
exact div_self (norm_ne_zero_iff.mpr hz)
lemma inner_eq_of_complexPhase_eq {z w : ℂ} (h : complexPhase z=complexPhase w) :
conj z*w=(‖z‖*‖w‖:ℝ) := by
by_cases hz : z=0
· simp [hz]
by_cases hw : w=0
· simp [hw]
have hez := complexPhase_mul_norm z
have hew := complexPhase_mul_norm w
conv_lhs => rw [←hez,←hew,←h,map_mul]
simp only [Complex.conj_ofReal]
calc
conj (complexPhase z)*(‖z‖:ℂ)*(complexPhase z*(‖w‖:ℂ))=
(complexPhase z*conj (complexPhase z))*((‖z‖:ℂ)*(‖w‖:ℂ)) := by ring
_ = _ := by rw [Complex.mul_conj',norm_complexPhase hz]; simp
lemma measurable_complexPhase : Measurable complexPhase := by unfold complexPhase; fun_prop
lemma IsGroundVector.ae_killed_complexPhase {N : ℕ} {L : Length} {a t : ℝ}
{u : Jet N L} (hu : IsGroundVector N L a u) (ht : 0≤t) :
∀ᵐ ω ∂killedPathLaw N L a t, complexPhase (u.1 (ω 0))=complexPhase (u.1 (ω 1)) := by
have hrev := (reverse_killedPath_preserving N L a t).quasiMeasurePreserving.ae
(hu.ae_killed_zero_propagation ht)
filter_upwards [hu.ae_killed_zero_propagation ht,hu.ae_killed_endpoints_phase ht,hrev] with ω h0 hphase h1
have hr : u.1 (ω 1)=0 → u.1 (ω 0)=0 := by simpa [reverseConfigurationPath] using h1
by_cases hz : u.1 (ω 0)=0
· rw [hz,h0 hz]
· exact complexPhase_eq_of_inner hz (mt hr hz) hphase
end HardSphere
namespace HardSphere
lemma IsGroundVector.phase_region {N : ℕ} {L : Length} {a : ℝ} {u : Jet N L}
(hu : IsGroundVector N L a u) {x : Configuration N L} (hx : x∈allowed N L a) :
∃z, x∈LocalAE.region volume (fun y ↦ complexPhase (u.1 y)=z) := by
have hm := measurable_complexPhase.comp (Lp.stronglyMeasurable u.1).measurable
have hpm : MeasurableSet {p : Configuration N L×Configuration N L |
complexPhase (u.1 p.1)=complexPhase (u.1 p.2)} :=
measurableSet_eq_fun (hm.comp measurable_fst) (hm.comp measurable_snd)
obtain ⟨U,hU,hxU,hsub,hp⟩ := killed_ae_torus_local_pairs L a (P:=fun X Y ↦ complexPhase (u.1 X)=complexPhase (u.1 Y)) hpm
(fun t ht ↦ hu.ae_killed_complexPhase ht.le) hx
obtain ⟨z,hz⟩ := LocalAE.ae_local_const_of_ae_pairs volume (hU.measure_ne_zero volume ⟨x,hxU⟩) hp
exact ⟨z,U,hU,hxU,hz⟩
def groundPhaseRepresentative {N : ℕ} {L : Length} (u : Jet N L) : Configuration N L → ℂ :=
LocalAE.representative volume (fun y ↦ complexPhase (u.1 y))
lemma IsGroundVector.ae_phaseRepresentative {N : ℕ} {L : Length} {a : ℝ} {u : Jet N L}
(hu : IsGroundVector N L a u) :
∀ᵐ x ∂volume.restrict (allowed N L a), complexPhase (u.1 x)=groundPhaseRepresentative u x :=
LocalAE.representative_ae volume (isOpen_allowed N L a) (fun _ hx ↦ hu.phase_region hx)
lemma IsGroundVector.phaseRepresentative_eq_on_connected {N : ℕ} {L : Length} {a : ℝ} {u : Jet N L}
(hu : IsGroundVector N L a u) {S : Set (Configuration N L)} (hS : IsPreconnected S)
(hsub : S⊆allowed N L a) {x y : Configuration N L} (hx : x∈S) (hy : y∈S) :
groundPhaseRepresentative u x=groundPhaseRepresentative u y :=
LocalAE.representative_eq_on_connected volume hS (fun _ hz ↦ hu.phase_region (hsub hz)) hx hy
end HardSphere
end
end OAI
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