| import Mathlib | |
| import OAI.Analysis.HardSphere.Conditioning.ReverseConfigurationPath | |
| import OAI.Analysis.HardSphere.Geometry.KilledObservable | |
| 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 | |
| open scoped ComplexConjugate | |
| theorem killedTransition_add (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 ∘L killedTransition N L a t := by | |
| have hd := ContinuousMap.toLp_denseRange ℂ (volume : Measure (Configuration N L)) ℂ | |
| (show (2:ℝ≥0∞) ≠ ∞ by norm_num) | |
| apply ContinuousLinearMap.ext | |
| intro g | |
| change killedTransition N L a (s+t) g = killedTransition N L a s (killedTransition N L a t g) | |
| have hc : IsClosed {g : ScalarL2 N L | killedTransition N L a (s+t) g = | |
| killedTransition N L a s (killedTransition N L a t g)} := | |
| isClosed_eq (ContinuousLinearMap.continuous _) ((ContinuousLinearMap.continuous _).comp | |
| (ContinuousLinearMap.continuous _)) | |
| apply hc.closure_subset_iff.mpr ?_ (hd g) | |
| rintro _ ⟨g,rfl⟩ | |
| change killedTransition N L a (s+t) (ContinuousMap.toLp 2 volume ℂ g) = | |
| killedTransition N L a s (killedTransition N L a t (ContinuousMap.toLp 2 volume ℂ g)) | |
| have hgLp : MemLp (fun X ↦ g X) 2 (volume : Measure (Configuration N L)) := | |
| MemLp.of_bound g.continuous.measurable.aestronglyMeasurable ‖g‖ | |
| (Filter.Eventually.of_forall g.norm_coe_le_norm) | |
| have hge : ContinuousMap.toLp 2 volume ℂ g = hgLp.toLp g := by | |
| apply Lp.ext | |
| exact (ContinuousMap.coeFn_toLp (𝕜 := ℂ) (p := 2) volume g).trans hgLp.coeFn_toLp.symm | |
| rw [hge,killedTransition_bounded N L a (s+t) g.continuous.measurable hgLp (norm_nonneg _) g.norm_coe_le_norm, | |
| killedTransition_bounded N L a t g.continuous.measurable hgLp (norm_nonneg _) g.norm_coe_le_norm, | |
| killedTransition_bounded N L a s (killedAverage_measurable N L a t g.continuous.measurable) | |
| (killedAverage_memLp N L a t g.continuous.measurable (norm_nonneg _) g.norm_coe_le_norm) | |
| (norm_nonneg g) (killedAverage_bound N L a t (norm_nonneg _) g.norm_coe_le_norm)] | |
| apply MemLp.toLp_congr | |
| exact Filter.Eventually.of_forall (killedAverage_add N L a s t hs ht g.continuous.measurable | |
| (norm_nonneg _) g.norm_coe_le_norm) | |
| end HardSphere | |
| namespace HardSphere | |
| lemma killedTransition_nonneg (N : ℕ) (L : Length) (a t : ℝ) (ht : 0<t) : | |
| 0 ≤ killedTransition N L a t := by | |
| have hh : 0<t/2 := by positivity | |
| rw [show t=t/2+t/2 by ring,killedTransition_add N L a (t/2) (t/2) hh hh] | |
| have ha := killedTransition_selfAdjoint N L a (t/2) | |
| have hp := ContinuousLinearMap.isPositive_adjoint_comp_self (killedTransition N L a (t/2)) | |
| simpa only [show (killedTransition N L a (t/2)).adjoint=killedTransition N L a (t/2) from ha] using ContinuousLinearMap.nonneg_iff_isPositive.mpr hp | |
| end HardSphere | |
| end | |
| end OAI | |