File size: 3,186 Bytes
a89d282 | 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 | import OAI.Probability.InvariantIsing.Haar.HaarHeatMixing
namespace OAI
/-! Continuous tests of the polynomial heat flow and their Haar limits. -/
noncomputable section
open Matrix MvPolynomial MeasureTheory Filter Set
open scoped Topology
namespace InvariantIsing
lemma haarPolynomialHeat_test_bound {N d : ℕ} (p : haarPolynomialSpace N d)
(F : ℝ → ℝ) (hF : Continuous F) :
∃ B : ℝ, ∀ t : ℝ, 0 ≤ t → ∀ U : SpecialOrthogonal N,
‖F (haarPolynomialValue
((haarPolynomialHeat N d t p : haarPolynomialSpace N d) : MatrixPolynomial N) U)‖ ≤ B := by
obtain ⟨P,hP⟩ := haarPolynomialValue_bound (p : MatrixPolynomial N)
obtain ⟨B,hB⟩ := isCompact_Icc.exists_bound_of_continuousOn hF.continuousOn
refine ⟨B,?_⟩
intro t ht U
apply hB
exact haarPolynomialHeat_bounds p (-P) P
(fun V => (abs_le.mp (hP V)).1) (fun V => (abs_le.mp (hP V)).2) ht U
theorem haarPolynomialHeat_test_tendsto_mean {N d : ℕ} (hN : 3 ≤ N)
(μ : Measure (SpecialOrthogonal N)) [IsProbabilityMeasure μ] [μ.IsMulLeftInvariant]
(p : haarPolynomialSpace N d) (F : ℝ → ℝ) (hF : Continuous F) :
Tendsto (fun t : ℝ => ∫ U, F (haarPolynomialValue
((haarPolynomialHeat N d t p : haarPolynomialSpace N d) : MatrixPolynomial N) U) ∂μ)
atTop (𝓝 (F (∫ U, haarPolynomialValue (p : MatrixPolynomial N) U ∂μ))) := by
obtain ⟨B,hB⟩ := haarPolynomialHeat_test_bound p F hF
have h := tendsto_integral_filter_of_dominated_convergence (μ := μ) (l := atTop)
(F := fun t U => F (haarPolynomialValue
((haarPolynomialHeat N d t p : haarPolynomialSpace N d) : MatrixPolynomial N) U))
(f := fun _ => F (∫ U, haarPolynomialValue (p : MatrixPolynomial N) U ∂μ)) (fun _ => B)
simp only [integral_const,probReal_univ,smul_eq_mul,one_mul] at h
apply h
· exact Eventually.of_forall fun t =>
(hF.comp (continuous_haarPolynomialValue _)).measurable.aestronglyMeasurable
· filter_upwards [eventually_ge_atTop (0:ℝ)] with t ht
exact ae_of_all μ (hB t ht)
· exact integrable_const _
· exact ae_of_all μ fun U => hF.continuousAt.tendsto.comp (haarPolynomialHeat_tendsto_mean hN μ p U)
theorem continuousOn_haarPolynomialHeat_test_integral {N d : ℕ}
(μ : Measure (SpecialOrthogonal N)) [IsFiniteMeasure μ]
(p : haarPolynomialSpace N d) (F : ℝ → ℝ) (hF : Continuous F) :
ContinuousOn (fun t : ℝ => ∫ U, F (haarPolynomialValue
((haarPolynomialHeat N d t p : haarPolynomialSpace N d) : MatrixPolynomial N) U) ∂μ) (Ici 0) := by
obtain ⟨B,hB⟩ := haarPolynomialHeat_test_bound p F hF
intro t _
apply tendsto_integral_filter_of_dominated_convergence (fun _ => B)
· exact Eventually.of_forall fun s =>
(hF.comp (continuous_haarPolynomialValue _)).measurable.aestronglyMeasurable
· filter_upwards [self_mem_nhdsWithin] with s hs
exact ae_of_all μ (hB s hs)
· exact integrable_const _
· exact ae_of_all μ fun U => ((hF.comp
((continuous_haarPolynomialHeat_value p).comp (continuous_id.prodMk continuous_const))).continuousAt).tendsto.mono_left
nhdsWithin_le_nhds
end InvariantIsing
end
end OAI
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