File size: 5,226 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 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 97 98 99 100 101 102 103 | import OAI.Probability.InvariantIsing.Magnetic.MagneticContinuationTwoJet
import OAI.Probability.InvariantIsing.Magnetic.MagneticStationaryFamily
import OAI.Probability.InvariantIsing.Fields.FieldAffineAverageDerivative
namespace OAI
/-! Joint variance/spatial differentiation of the actual scalar
continuation. The covariance differential is left as integrals here;
Gaussian integration by parts then identifies its time generator. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron Filter Set
open scoped NNReal Topology
namespace InvariantIsing
def magneticGaussianAverageDifferential (ζ v : ℝ) (F M A B : ℝ → ℝ) (z : ℝ) :
(ℝ × ℝ) →L[ℝ] ℝ :=
let ν := (gaussianReal 0 1).tilted (fun u => ζ * F (z + Real.sqrt v * u))
let D := fun u => pairLinear
(M (z + Real.sqrt v * u) * ((1 / (2 * Real.sqrt v)) * u))
(M (z + Real.sqrt v * u))
(∫ u, pairLinear
(B (z + Real.sqrt v * u) * ((1 / (2 * Real.sqrt v)) * u))
(B (z + Real.sqrt v * u)) + (ζ * A (z + Real.sqrt v * u)) • D u ∂ν) -
(∫ u, A (z + Real.sqrt v * u) ∂ν) • (∫ u, ζ • D u ∂ν)
theorem magneticGaussianAverageTwo_hasFDerivAt (P : MagneticContinuationJet) (A : MagneticContinuationTwoJet)
(F : ℝ → ℝ) (hF : Measurable F) (dF : ∀ z, HasDerivAt F (P.value z) z)
(ζ : ℝ) {v : ℝ} (hv : 0 < v) (z : ℝ) :
HasFDerivAt (fun q : ℝ × ℝ => gaussianTiltAverage q.1 ζ F A.value q.2)
(magneticGaussianAverageDifferential ζ v F P.value A.value A.first z) (v, z) := by
obtain ⟨K, hK, bK⟩ := P.bValue
obtain ⟨L, _, bL⟩ := P.bFirst
obtain ⟨CA, hCA, bCA⟩ := A.bValue
obtain ⟨CB, hCB, bCB⟩ := A.bFirst
let I := Ioo (v / 2) (v + 1)
let S := (magneticStationaryFamily I F P hF K L bK bL dF).toFieldSmoothFamily
let α : (ℝ × ℝ) → ℝ → ℝ := fun q u => A.value (q.2 + Real.sqrt q.1 * u)
let Dα : (ℝ × ℝ) → ℝ → (ℝ × ℝ) →L[ℝ] ℝ := fun q u => pairLinear
(A.first (q.2 + Real.sqrt q.1 * u) * ((1 / (2 * Real.sqrt q.1)) * u))
(A.first (q.2 + Real.sqrt q.1 * u))
let R := 1 / (2 * Real.sqrt (v / 2))
have hR : 0 ≤ R := by dsimp only [R]; positivity
have hm : 0 < v / 2 := by linarith
have hbound (q : ℝ × ℝ) (hq : q.1 ∈ I) :
|1 / (2 * Real.sqrt q.1)| ≤ R := by
simpa only [abs_one] using abs_div_two_sqrt_le hm hq.1.le 1
have hdα (q : ℝ × ℝ) (hq : q.1 ∈ I) (u : ℝ) :
HasFDerivAt (fun r => α r u) (Dα q u) q := by
have hb (p : ℝ × ℝ) (_ : p.1 ∈ I) :
HasFDerivAt (fun r : ℝ × ℝ => A.value r.2) (pairLinear 0 (A.first p.2)) p := by
convert (A.dValue p.2).hasFDerivAt.comp p hasFDerivAt_snd using 1
· rfl
· apply ContinuousLinearMap.ext
intro w
simp [pairLinear, mul_comm]
simpa only [zero_add, one_mul] using
affine_gaussian_shift_hasFDerivAt hb 0 1 u hq (by
dsimp only [I] at hq
linarith [hq.1])
have hbα (q : ℝ × ℝ) (_ : q.1 ∈ I) (u : ℝ) : |α q u| ≤ CA * (1 + |u|) := by
exact (bCA _).trans (le_mul_of_one_le_right hCA (by linarith [abs_nonneg u]))
have hbDα (q : ℝ × ℝ) (hq : q.1 ∈ I) (u : ℝ) :
‖Dα q u‖ ≤ (CB * (R + 1)) * (1 + |u|) ^ 2 := by
refine (norm_pairLinear_le _ _).trans ?_
have hf : |A.first (q.2 + Real.sqrt q.1 * u) * ((1 / (2 * Real.sqrt q.1)) * u)| ≤
CB * R * |u| := by
rw [abs_mul, abs_mul]
exact (mul_le_mul (bCB _) (mul_le_mul_of_nonneg_right (hbound q hq) (abs_nonneg u))
(mul_nonneg (abs_nonneg _) (abs_nonneg _)) hCB).trans_eq (by ring)
have hg := bCB (q.2 + Real.sqrt q.1 * u)
have h1 : 1 ≤ (1 + |u|) ^ 2 := by nlinarith [abs_nonneg u]
have h2 : |u| ≤ (1 + |u|) ^ 2 := by nlinarith [sq_nonneg |u|, abs_nonneg u]
nlinarith [mul_le_mul_of_nonneg_left h1 hCB,
mul_le_mul_of_nonneg_left h2 (mul_nonneg hCB hR)]
have hp : (v, z) ∈ I ×ˢ univ := ⟨by dsimp only [I]; constructor <;> linarith, mem_univ z⟩
have hh := S.affine_average_hasFDerivAt isOpen_Ioo α Dα 0 1 ζ hm
(fun t ht => by simpa only [zero_add, one_mul] using ht.1.le)
(fun t ht => by simpa only [zero_add, one_mul] using ht.2.le)
hCA (mul_nonneg hCB (by linarith : 0 ≤ R + 1))
(fun q => A.mValue.comp (by fun_prop))
(fun q => (measurable_pairLinear
(A.mFirst.comp (by fun_prop) |>.mul (by fun_prop))
(A.mFirst.comp (by fun_prop))).aestronglyMeasurable)
hbα hbDα hdα hp.1
simpa only [α, Dα, S, magneticStationaryFamily, FieldSmoothFamily.affineShift,
FieldSmoothFamily.affineDifferential, fieldAmplitudeSlope, zero_add, one_mul,
gaussianTiltAverage, magneticGaussianAverageDifferential] using hh
theorem magneticGaussianAverage_hasFDerivAt (P A : MagneticContinuationJet)
(F : ℝ → ℝ) (hF : Measurable F) (dF : ∀ z, HasDerivAt F (P.value z) z)
(ζ : ℝ) {v : ℝ} (hv : 0 < v) (z : ℝ) :
HasFDerivAt (fun q : ℝ × ℝ => gaussianTiltAverage q.1 ζ F A.value q.2)
(magneticGaussianAverageDifferential ζ v F P.value A.value A.first z) (v, z) := by
exact magneticGaussianAverageTwo_hasFDerivAt P A.toTwoJet F hF dF ζ hv z
end InvariantIsing
end
end OAI
|