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f205175 | 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 | import OAI.Probability.DilutedSpin.CompoundMoment
namespace OAI
section
namespace DilutedSpinGlass
open _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
open scoped NNReal ENNReal
variable {E F : Type} [NormedAddCommGroup E] [NormedSpace ℝ E]
[MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
[NormedAddCommGroup F] [NormedSpace ℝ F]
[MeasurableSpace F] [BorelSpace F] [SecondCountableTopology F]
lemma compoundPoisson_map_linear [CompleteSpace F] (r : ℝ≥0) (μ : Measure E)
[IsProbabilityMeasure μ] (A : E →L[ℝ] F) :
Measure.map A (compoundPoisson r μ) = compoundPoisson r (Measure.map A μ) := by
let : IsProbabilityMeasure (Measure.map A μ) :=
(Measure.isProbabilityMeasure_map_iff A.measurable.aemeasurable).mpr inferInstance
apply Measure.ext_of_charFunDual
funext L
rw [charFunDual_map,charFunDual_compoundPoisson,charFunDual_compoundPoisson,charFunDual_map]
omit [NormedSpace ℝ E] [SecondCountableTopology E] in
lemma integrable_lipschitz_of_id (μ : Measure E) [IsFiniteMeasure μ]
(hi : Integrable id μ) {C : ℝ≥0} {f : E → ℝ} (hf : LipschitzWith C f) : Integrable f μ := by
apply ((hi.norm.const_mul (C:ℝ)).add (integrable_const |f 0|)).mono' hf.continuous.measurable.aestronglyMeasurable
apply ae_of_all
intro x
have hh := hf.norm_sub_le x 0
simp only [Real.norm_eq_abs,sub_zero] at hh
calc
|f x| = |(f x-f 0)+f 0| := by rw [sub_add_cancel]
_ ≤ |f x-f 0|+|f 0| := abs_add_le _ _
_ ≤ (C:ℝ)*‖x‖+|f 0| := by linarith
lemma integral_norm_sub_compound [CompleteSpace E] (r : ℝ≥0) (μ : Measure (E×E))
[IsProbabilityMeasure μ] (hi : Integrable id μ) :
(∫ z, ‖z.1-z.2‖ ∂compoundPoisson r μ) ≤ r*(∫ z,‖z.1-z.2‖ ∂μ) := by
let A := ContinuousLinearMap.fst ℝ E E-ContinuousLinearMap.snd ℝ E E
have hm : Measure.map A (compoundPoisson r μ) = compoundPoisson r (Measure.map A μ) :=
compoundPoisson_map_linear r μ A
let : IsProbabilityMeasure (Measure.map A μ) :=
(Measure.isProbabilityMeasure_map_iff A.measurable.aemeasurable).mpr inferInstance
have hai : Integrable id (Measure.map A μ) :=
(integrable_map_measure aestronglyMeasurable_id A.measurable.aemeasurable).mpr (A.integrable_comp hi)
have hb := compoundPoisson_norm_le r (Measure.map A μ) hai
rw [← hm,integral_map A.measurable.aemeasurable (by fun_prop),
integral_map A.measurable.aemeasurable (by fun_prop)] at hb
exact hb
/-- A first-moment transport bound for the physical marked Poisson process.
It also justifies deletion of all repeated-new-site bonds at their actual cost. -/
lemma compoundPoisson_lipschitz_coupling [CompleteSpace E] (r : ℝ≥0) (μ : Measure (E×E))
[IsProbabilityMeasure μ] (hi : Integrable id μ) {C : ℝ≥0} {f : E → ℝ}
(hf : LipschitzWith C f) :
|(∫ x, f x ∂compoundPoisson r (Measure.map Prod.fst μ))-
(∫ x, f x ∂compoundPoisson r (Measure.map Prod.snd μ))| ≤
C*r*(∫ z,‖z.1-z.2‖ ∂μ) := by
let A := ContinuousLinearMap.fst ℝ E E
let B := ContinuousLinearMap.snd ℝ E E
have hmA := compoundPoisson_map_linear r μ A
have hmB := compoundPoisson_map_linear r μ B
have hc := compoundPoisson_integrable_id r μ hi
have hia : Integrable (fun z : E×E => f z.1) (compoundPoisson r μ) := by
have hp : Integrable id (Measure.map A (compoundPoisson r μ)) :=
(integrable_map_measure aestronglyMeasurable_id A.measurable.aemeasurable).mpr (A.integrable_comp hc)
exact (integrable_map_measure hf.continuous.measurable.aestronglyMeasurable A.measurable.aemeasurable).mp
(integrable_lipschitz_of_id _ hp hf)
have hib : Integrable (fun z : E×E => f z.2) (compoundPoisson r μ) := by
have hp : Integrable id (Measure.map B (compoundPoisson r μ)) :=
(integrable_map_measure aestronglyMeasurable_id B.measurable.aemeasurable).mpr (B.integrable_comp hc)
exact (integrable_map_measure hf.continuous.measurable.aestronglyMeasurable B.measurable.aemeasurable).mp
(integrable_lipschitz_of_id _ hp hf)
change |(∫ x, f x ∂compoundPoisson r (Measure.map A μ))-
(∫ x, f x ∂compoundPoisson r (Measure.map B μ))| ≤ _
rw [← hmA,← hmB,integral_map A.measurable.aemeasurable hf.continuous.measurable.aestronglyMeasurable,
integral_map B.measurable.aemeasurable hf.continuous.measurable.aestronglyMeasurable]
change |(∫ z : E×E, f z.1 ∂compoundPoisson r μ)-(∫ z, f z.2 ∂compoundPoisson r μ)|≤_
rw [← integral_sub hia hib]
calc
_ ≤ ∫ z, |f z.1-f z.2| ∂compoundPoisson r μ := abs_integral_le_integral_abs
_ ≤ ∫ z : E×E, (C:ℝ)*‖z.1-z.2‖ ∂compoundPoisson r μ := by
apply integral_mono (hia.sub hib).abs
(((A-B).integrable_comp hc).norm.const_mul (C:ℝ))
intro z
exact hf.norm_sub_le z.1 z.2
_ = (C:ℝ)*(∫ z,‖z.1-z.2‖ ∂compoundPoisson r μ) := integral_const_mul _ _
_ ≤ _ := by
simpa only [mul_assoc] using mul_le_mul_of_nonneg_left (integral_norm_sub_compound r μ hi) C.coe_nonneg
end DilutedSpinGlass
end
end OAI
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