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import OAI.Analysis.HardSphere.Kernels.ExposedPhysicalPair
namespace OAI
section
section
noncomputable section
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal NNReal Topology
namespace HardSphere.Bridge
open Wiener GaussianPacking Comparison
variable {E : Type*} [MeasurableSpace E]
lemma leg_all_continuous (t : ℝ) :
Continuous (fun p : (Point×Point)×Noise ↦ leg t p.1.1 p.1.2 p.2) := by
apply ContinuousMap.continuous_of_continuous_uncurry
change Continuous (fun p : ((Point×Point)×Noise)×Time ↦ fun j ↦
mean p.1.1.1 p.1.1.2 p.2 j+Real.sqrt (2*t)*p.1.2 j p.2)
unfold mean
fun_prop
def exposedBridgeKernel (t : ℝ) : Kernel (Point×Point) C(Time,Point) :=
((Kernel.deterministic id measurable_id).prod (Kernel.const (Point×Point) law)).map
(fun p ↦ leg t p.1.1 p.1.2 p.2)
instance exposedBridgeKernel_markov (t : ℝ) : IsMarkovKernel (exposedBridgeKernel t) := by
unfold exposedBridgeKernel
exact Kernel.IsMarkovKernel.map _ (leg_all_continuous t).measurable
lemma exposedBridgeKernel_apply (t : ℝ) (y z : Point) :
exposedBridgeKernel t (y,z)=law.map (leg t y z) := by
rw [exposedBridgeKernel,Kernel.map_apply _ (leg_all_continuous t).measurable,
Kernel.prod_apply,Kernel.deterministic_apply,Kernel.const_apply,Measure.dirac_prod,
Measure.map_map (leg_all_continuous t).measurable
(measurable_const.prodMk measurable_id : Measurable (Prod.mk (id (y,z))))]
rfl
def parameterBridgeKernel (t : ℝ) (y z : E→Point) (hy : Measurable y) (hz : Measurable z) :
Kernel E C(Time,Point) := (exposedBridgeKernel t).comap (fun e ↦ (y e,z e)) (hy.prodMk hz)
instance parameterBridgeKernel_markov (t : ℝ) (y z : E→Point) (hy : Measurable y) (hz : Measurable z) :
IsMarkovKernel (parameterBridgeKernel t y z hy hz) := inferInstanceAs
(IsMarkovKernel ((exposedBridgeKernel t).comap _ _))
lemma parameterBridgeKernel_apply (t : ℝ) (y z : E→Point) (hy : Measurable y) (hz : Measurable z) (e : E) :
parameterBridgeKernel t y z hy hz e=law.map (leg t (y e) (z e)) := exposedBridgeKernel_apply _ _ _
def parameterIndividualReference (t : ℝ) (y z : E→Point) (hy : Measurable y) (hz : Measurable z)
(S : Set (E×C(Time,Point))) (hS : MeasurableSet S) : Kernel E C(Time,Point) :=
parameterSuccessKernel (parameterBridgeKernel (2*t) y z hy hz) S hS
instance parameterIndividualReference_finite (t : ℝ) (y z : E→Point) (hy : Measurable y) (hz : Measurable z)
(S : Set (E×C(Time,Point))) (hS : MeasurableSet S) :
IsFiniteKernel (parameterIndividualReference t y z hy hz S hS) := inferInstanceAs
(IsFiniteKernel (parameterSuccessKernel _ _ _))
lemma parameterIndividualReference_apply (t : ℝ) (y z : E→Point) (hy : Measurable y) (hz : Measurable z)
(S : Set (E×C(Time,Point))) (hS : MeasurableSet S) (e : E) :
parameterIndividualReference t y z hy hz S hS e=individualReference t (y e) (z e) (Prod.mk e ⁻¹' S) := by
rw [parameterIndividualReference,parameterSuccessKernel_apply,parameterBridgeKernel_apply]
rfl
end HardSphere.Bridge
end
end
section
noncomputable section
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal NNReal Topology
namespace HardSphere.Bridge
open Wiener GaussianPacking Comparison
variable {E : Type*} [MeasurableSpace E]
[MeasurableSpace.CountableOrCountablyGenerated E C(Time,Point)]
lemma exposed_physical_auxiliary
(Q : Kernel E Point) [IsFiniteKernel Q] {t C : ℝ} (ht : 0<t) (hC : 0≤C)
(y z y' z' : E→Point) (hy : Measurable y) (hz : Measurable z)
(hy' : Measurable y') (hz' : Measurable z')
(S T : Set (E×C(Time,Point))) (hS : MeasurableSet S) (hT : MeasurableSet T)
(A : E→Set Point) (hA : ∀e,MeasurableSet (A e))
(hAf : ∀e,volume (A e)≠⊤) (hQ : ∀e,Q e=targetUniform (A e))
{a c : ℝ≥0∞} (ha : 0<a) (hc : 0<c) (hvol : ∀e,a≤volume (A e))
(hb : ∀e u,u∈A e → ‖u-(fun j ↦ (y e j+z e j)/2)‖≤C)
(hsuccess : ∀e,∀ᵐu ∂targetUniform (A e),c≤trialKernel t (y e) (z e) u (Prod.mk e ⁻¹' S))
(hother : ∀e,∀ᵐu ∂targetUniform (A e),0<trialKernel t (y' e) (z' e) u (Prod.mk e ⁻¹' T)) :
∃ κ : Kernel (E×C(Time,Point)) (Point×C(Time,Point)), IsFiniteKernel κ ∧
(parameterIndividualReference t y z hy hz S hS).compProd κ=
exposedPairedTrialKernel Q t y z y' z' hy hz hy' hz' S T hS hT ∧
(∀e f,κ (e,f) univ≤c⁻¹*(a⁻¹*(midpointLower t C)⁻¹)) ∧
(∀e,∫⁻f,κ (e,f) univ ∂parameterIndividualReference t y z hy hz S hS e=1) := by
have hA0 (e : E) : volume (A e)≠0 := (ha.trans_le (hvol e)).ne'
let P := parameterIndividualReference t y z hy hz S hS
let ρ := exposedPairedTrialKernel Q t y z y' z' hy hz hy' hz' S T hS hT
let _ : IsMarkovKernel P := by
constructor
intro e
change IsProbabilityMeasure (parameterIndividualReference t y z hy hz S hS e)
rw [parameterIndividualReference_apply]
exact individualReference_probability ht hC (y e) (z e) (hA e) (hA0 e) (hAf e) (hb e)
(Prod.mk e ⁻¹' S) (hS.preimage measurable_prodMk_left) hc (hsuccess e)
let _ : IsMarkovKernel ρ := by
constructor
intro e
change IsProbabilityMeasure (exposedPairedTrialKernel Q t y z y' z' hy hz hy' hz' S T hS hT e)
rw [exposedPairedTrialKernel_physical Q t y z y' z' hy hz hy' hz' S T hS hT A hQ]
exact pairedTrialLaw_probability t (y e) (z e) (y' e) (z' e) (A e) (hA0 e) (hAf e)
(Prod.mk e ⁻¹' S) (Prod.mk e ⁻¹' T)
(hS.preimage measurable_prodMk_left) (hT.preimage measurable_prodMk_left)
((hsuccess e).mono (fun _ hu ↦ hc.trans_le hu)) (hother e)
apply exists_parameter_auxiliary P ρ
· exact ENNReal.mul_ne_top (ENNReal.inv_ne_top.mpr hc.ne')
(ENNReal.mul_ne_top (ENNReal.inv_ne_top.mpr ha.ne')
(ENNReal.inv_ne_top.mpr (midpointLower_pos ht C).ne'))
· intro e
change (ρ e).fst≤_ • P e
dsimp [ρ,P]
rw [exposedPairedTrialKernel_physical Q t y z y' z' hy hz hy' hz' S T hS hT A hQ,
parameterIndividualReference_apply,
pairedTrialLaw_fst t (y e) (z e) (y' e) (z' e) (A e) (hA0 e) (hAf e)
(Prod.mk e ⁻¹' S) (Prod.mk e ⁻¹' T)
(hS.preimage measurable_prodMk_left) (hT.preimage measurable_prodMk_left) (hother e)]
apply (physical_trial_le_reference ht hC (y e) (z e) (hA e) (hAf e) (hb e)
(Prod.mk e ⁻¹' S) (hS.preimage measurable_prodMk_left) (hsuccess e)).trans
intro B
simp only [Measure.smul_apply,smul_eq_mul]
exact mul_le_mul' (mul_le_mul' le_rfl
(mul_le_mul' (ENNReal.inv_le_inv.mpr (hvol e)) le_rfl)) le_rfl
end HardSphere.Bridge
end
end
end
end OAI
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