File size: 3,142 Bytes
9eeb9c7 | 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 | 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
|