File size: 7,165 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 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 | import Mathlib
import OAI.Analysis.HardSphere.Operators.KilledTransitionAdd
import OAI.Analysis.HardSphere.Geometry.KilledAverage
import OAI.Analysis.HardSphere.Operators.KilledTransitionContinuousAe
namespace OAI
noncomputable section
open MeasureTheory Filter Set
open scoped ENNReal Topology BigOperators
open scoped BigOperators ComplexConjugate
open MeasureTheory ProbabilityTheory Filter Set
open scoped ENNReal NNReal Topology BigOperators
open MeasureTheory ProbabilityTheory Real Complex
open scoped ENNReal ComplexConjugate
namespace HardSphere
open Wiener ContinuousLinearMap
open scoped ComplexConjugate
lemma killedAverage_zero_of_not_allowed (N : ℕ) (L : Length) (a t : ℝ)
(g : Configuration N L→ℂ) {X : Configuration N L} (hX : X∉allowed N L a) :
killedAverage N L a t g X=0 := by
apply integral_eq_zero_of_ae
filter_upwards [coordinateWiener_start_zero N] with w hw
have hn : freeTrajectory N L t w X∉staysAllowed N L a := by
intro h
have h0 := h (Set.mem_range_self (0:Time))
rw [freeTrajectory_start N L t hw X] at h0
exact hX h0
exact Set.indicator_of_notMem hn _
def allowedProjection (N : ℕ) (L : Length) (a : ℝ) : ScalarL2 N L→L[ℂ] ScalarL2 N L :=
L2Multiplier.clm ((allowed N L a).indicator (fun _ ↦ (1:ℂ)))
((memLp_top_const (1:ℂ)).indicator (isOpen_allowed N L a).measurableSet)
lemma allowedProjection_ae (N : ℕ) (L : Length) (a : ℝ) (f : ScalarL2 N L) :
allowedProjection N L a f =ᵐ[volume] (allowed N L a).indicator f := by
filter_upwards [L2Multiplier.coe_clm ((allowed N L a).indicator (fun _ ↦ (1:ℂ)))
((memLp_top_const (1:ℂ)).indicator (isOpen_allowed N L a).measurableSet) f] with X hX
change (L2Multiplier.clm _ _ f) X=_
rw [hX]
by_cases h : X∈allowed N L a <;> simp [h]
lemma killedTransition_zero (N : ℕ) (L : Length) (a : ℝ) :
killedTransition N L a 0=allowedProjection N L a := by
apply ContinuousLinearMap.ext_on (s := Set.range (ContinuousMap.toLp 2 volume ℂ))
· exact (ContinuousMap.toLp_denseRange ℂ volume ℂ (by norm_num : (2:ℝ≥0∞)≠∞)).mono
(Submodule.subset_span)
· rintro _ ⟨g,rfl⟩
apply Lp.ext
filter_upwards [killedTransition_continuous_ae N L a 0 g,
allowedProjection_ae N L a (ContinuousMap.toLp 2 volume ℂ g),
ContinuousMap.coeFn_toLp (𝕜 := ℂ) (p := 2) volume g] with X h1 h2 h3
rw [h1,killedAverage_zero,h2]
by_cases h : X∈allowed N L a
· simpa only [Set.indicator_of_mem h] using h3.symm
· simp only [Set.indicator_of_notMem h]
lemma allowedProjection_killedTransition (N : ℕ) (L : Length) (a t : ℝ) :
(allowedProjection N L a).comp (killedTransition N L a t)=killedTransition N L a t := by
apply ContinuousLinearMap.ext_on (s := Set.range (ContinuousMap.toLp 2 volume ℂ))
· exact (ContinuousMap.toLp_denseRange ℂ volume ℂ (by norm_num : (2:ℝ≥0∞)≠∞)).mono
(Submodule.subset_span)
· rintro _ ⟨g,rfl⟩
apply Lp.ext
filter_upwards [killedTransition_continuous_ae N L a t g,
allowedProjection_ae N L a (killedTransition N L a t (ContinuousMap.toLp 2 volume ℂ g))]
with X h1 h2
change (allowedProjection N L a (killedTransition N L a t (ContinuousMap.toLp 2 volume ℂ g))) X=_
rw [h2]
by_cases h : X∈allowed N L a
· exact Set.indicator_of_mem h _
· rw [Set.indicator_of_notMem h,h1,killedAverage_zero_of_not_allowed N L a t g h]
lemma killedTransition_allowedProjection (N : ℕ) (L : Length) (a t : ℝ) :
(killedTransition N L a t).comp (allowedProjection N L a)=killedTransition N L a t := by
have he := congrArg ContinuousLinearMap.adjoint (allowedProjection_killedTransition N L a t)
rw [adjoint_comp,← killedTransition_zero N L a,
(killedTransition_selfAdjoint N L a 0).adjoint_eq,
(killedTransition_selfAdjoint N L a t).adjoint_eq] at he
simpa only [killedTransition_zero] using he
lemma killedTransition_support (N : ℕ) (L : Length) (a t : ℝ) (f : ScalarL2 N L) :
∀ᵐ X ∂(volume : Measure (Configuration N L)), X∉allowed N L a → killedTransition N L a t f X=0 := by
have he := congrArg (fun A : ScalarL2 N L→L[ℂ] ScalarL2 N L ↦ A f)
(allowedProjection_killedTransition N L a t)
change allowedProjection N L a (killedTransition N L a t f)=killedTransition N L a t f at he
filter_upwards [allowedProjection_ae N L a (killedTransition N L a t f)] with X hX
rw [he] at hX
intro hn
rw [hX,Set.indicator_of_notMem hn]
lemma killedTransition_add_nonneg (N : ℕ) (L : Length) (a s t : ℝ) (hs : 0 ≤ s) (ht : 0 ≤ t) :
killedTransition N L a (s+t)=(killedTransition N L a s).comp (killedTransition N L a t) := by
rcases hs.eq_or_lt with hs | hs
· rw [← hs,zero_add,killedTransition_zero,allowedProjection_killedTransition]
rcases ht.eq_or_lt with ht | ht
· rw [← ht,add_zero,killedTransition_zero,killedTransition_allowedProjection]
exact killedTransition_add N L a s t hs ht
end HardSphere
namespace HardSphere
lemma exists_bounded_lift_of_tendsto {E F ι : Type*}
[NormedAddCommGroup E] [NormedAddCommGroup F]
[InnerProductSpace ℂ E] [InnerProductSpace ℂ F]
[CompleteSpace E] [CompleteSpace F]
{l : Filter ι} [l.NeBot] (A : E →L[ℂ] F) (f : ι → E)
{C : ℝ} (hb : ∀ i, ‖f i‖≤C) {u : F}
(hu : Tendsto (fun i ↦ A (f i)) l (𝓝 u)) :
∃ x : E, ‖x‖≤C ∧ A x=u := by
let φ : ι → WeakDual ℂ E := fun i ↦ (InnerProductSpace.toDual ℂ E (f i)).toWeakDual
have hφ : ∀ i, φ i∈WeakDual.toStrongDual ⁻¹' Metric.closedBall 0 C := by
intro i
simpa only [Set.mem_preimage,Metric.mem_closedBall,dist_zero_right,φ,
StrongDual.toStrongDual_toWeakDual,(InnerProductSpace.toDual ℂ E).norm_map] using hb i
obtain ⟨ψ,hψ,hcluster⟩ := (WeakDual.isCompact_closedBall (0 : StrongDual ℂ E) C).exists_mapClusterPt
(show Filter.map φ l≤𝓟 (WeakDual.toStrongDual ⁻¹' Metric.closedBall 0 C) from
Filter.le_principal_iff.mpr (Filter.eventually_map.mpr (Eventually.of_forall hφ)))
let x : E := (InnerProductSpace.toDual ℂ E).symm ψ.toStrongDual
refine ⟨x,?_,?_⟩
· change ‖(InnerProductSpace.toDual ℂ E).symm ψ.toStrongDual‖≤C
simpa only [Set.mem_preimage,Metric.mem_closedBall,dist_zero_right,
(InnerProductSpace.toDual ℂ E).symm.norm_map] using hψ
· apply ext_inner_right ℂ
intro v
have hv : Tendsto (fun d : WeakDual ℂ E ↦ d (A.adjoint v))
(Filter.map φ l) (𝓝 (inner ℂ u v)) := by
rw [tendsto_map'_iff]
have h := hu.inner (𝕜 := ℂ) (tendsto_const_nhds (x := v))
simpa only [Function.comp_def,φ,StrongDual.toWeakDual_apply,
InnerProductSpace.toDual_apply_apply,ContinuousLinearMap.adjoint_inner_right] using h
have hct := hcluster.map (WeakDual.eval_continuous (A.adjoint v)).continuousAt hv
have he : ψ (A.adjoint v)=inner ℂ u v := by
exact eq_of_nhds_neBot hct
simpa only [x,← WeakDual.toStrongDual_apply,← InnerProductSpace.toDual_symm_apply,
ContinuousLinearMap.adjoint_inner_right] using he
end HardSphere
end
end OAI
|