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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 67 68 69 70 71 72 73 74 75 76 77 78 79 80 | import OAI.Probability.InvariantIsing.Haar.HaarLogIntegration
import OAI.Probability.InvariantIsing.Haar.HaarHeatLogGradient
import OAI.Probability.InvariantIsing.Haar.HaarHeatTestLimit
namespace OAI
/-! Entropy dissipation along the actual positive polynomial heat flow. -/
noncomputable section
open Matrix MvPolynomial MeasureTheory Filter Set
open scoped Topology
namespace InvariantIsing
def haarHeatEntropy {N d : ℕ} (μ : Measure (SpecialOrthogonal N))
(p : haarPolynomialSpace N d) (t : ℝ) : ℝ :=
∫ U, haarPolynomialValue ((haarPolynomialHeat N d t p : haarPolynomialSpace N d) : MatrixPolynomial N) U*
Real.log (haarPolynomialValue ((haarPolynomialHeat N d t p : haarPolynomialSpace N d) : MatrixPolynomial N) U) ∂μ
theorem haarHeatEntropy_hasDerivAt {N d : ℕ} (μ : Measure (SpecialOrthogonal N))
[IsFiniteMeasure μ] [μ.IsMulLeftInvariant] (p : haarPolynomialSpace N d)
(ε : ℝ) (hε : 0 < ε)
(hp : ∀ U : SpecialOrthogonal N, ε ≤ haarPolynomialValue (p : MatrixPolynomial N) U)
{t : ℝ} (ht : 0 < t) :
HasDerivAt (haarHeatEntropy μ p)
(-(∫ U, haarPolynomialValue (haarPolynomialGamma
((haarPolynomialHeat N d t p : haarPolynomialSpace N d) : MatrixPolynomial N)
((haarPolynomialHeat N d t p : haarPolynomialSpace N d) : MatrixPolynomial N)) U/
haarPolynomialValue ((haarPolynomialHeat N d t p : haarPolynomialSpace N d) : MatrixPolynomial N) U ∂μ)) t := by
let q (s : ℝ) : MatrixPolynomial N := (haarPolynomialHeat N d s p : haarPolynomialSpace N d)
let u (s : ℝ) (U : SpecialOrthogonal N) := haarPolynomialValue (q s) U
let L (s : ℝ) (U : SpecialOrthogonal N) := haarPolynomialValue (haarPolynomialLaplacian N (q s)) U
let F' (s : ℝ) (U : SpecialOrthogonal N) := L s U*(Real.log (u s U)+1)
obtain ⟨P,hP⟩ := haarPolynomialValue_bound (p : MatrixPolynomial N)
have hpos (s : ℝ) (hs : 0 ≤ s) (U : SpecialOrthogonal N) : 0 < u s U :=
hε.trans_le ((haarPolynomialHeat_bounds p ε P hp (fun V => (le_abs_self _).trans (hP V)) hs U).1)
have hcont : ContinuousOn (fun z : ℝ × SpecialOrthogonal N => F' z.1 z.2)
(Icc (t/2) (t+1) ×ˢ (Set.univ : Set (SpecialOrthogonal N))) := by
apply ((continuous_haarPolynomialHeat_operator_value (haarPolynomialLaplacian N) p).continuousOn).mul
apply ContinuousOn.add _ continuousOn_const
apply (continuous_haarPolynomialHeat_value p).continuousOn.log
intro z hz
exact (hpos z.1 (by linarith [hz.1.1]) z.2).ne'
obtain ⟨B,hB⟩ := (isCompact_Icc.prod (isCompact_univ : IsCompact (Set.univ : Set (SpecialOrthogonal N)))).exists_bound_of_continuousOn hcont
have hnb : Ioo (t/2) (t+1) ∈ 𝓝 t := Ioo_mem_nhds (by linarith) (by linarith)
have hlog : Continuous (fun U : SpecialOrthogonal N => Real.log (u t U)) :=
(continuous_haarPolynomialValue _).log (fun U => (hpos t ht.le U).ne')
have hdi : HasDerivAt (haarHeatEntropy μ p) (∫ U, F' t U ∂μ) t := by
apply (hasDerivAt_integral_of_dominated_loc_of_deriv_le (μ := μ)
(s := Ioo (t/2) (t+1)) (F' := F') (bound := fun _ => B) hnb
(Eventually.of_forall fun s => (Real.continuous_mul_log.comp (continuous_haarPolynomialValue (q s))).measurable.aestronglyMeasurable)
(continuous_haar_integrable μ _ (Real.continuous_mul_log.comp (continuous_haarPolynomialValue (q t))))
(((continuous_haarPolynomialValue _).mul (hlog.add continuous_const)).measurable.aestronglyMeasurable)
(ae_of_all μ fun U s hs => hB (s,U) ⟨⟨hs.1.le,hs.2.le⟩,mem_univ U⟩)
(integrable_const B) ?_).2
apply ae_of_all
intro U s hs
have hu := haarPolynomialHeat_hasDerivAt p (U : Matrix (Fin N) (Fin N) ℝ) s
have hh := (Real.hasDerivAt_mul_log (hpos s (by linarith [hs.1]) U).ne').comp s hu
convert hh using 1
· rfl
· dsimp only [F',L,u,haarPolynomialValue,q]
ring
apply hdi.congr_deriv
have hmul (U : SpecialOrthogonal N) : F' t U = L t U*Real.log (u t U)+L t U := by
dsimp only [F']
ring
simp_rw [hmul]
have hi1 : Integrable (fun U => L t U*Real.log (u t U)) μ :=
continuous_haar_integrable μ _ ((continuous_haarPolynomialValue _).mul hlog)
have hi2 : Integrable (L t) μ := haarPolynomialValue_integrable _ μ
rw [integral_add hi1 hi2]
rw [integral_haar_laplacian_log μ (q t) (hpos t ht.le)]
change _+haarPolynomialMean μ (haarPolynomialLaplacian N (q t)) = _
rw [haarPolynomialMean_laplacian,add_zero]
end InvariantIsing
end
end OAI
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