| 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 |
| |
