| 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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