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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 81 82 83 84 85 86 87 88 89 90 | import OAI.Probability.InvariantIsing.Magnetic.MagneticContinuationEndpoint
import OAI.Probability.InvariantIsing.Fields.FieldGaussianIntegrability
namespace OAI
/-! Gaussian integration by parts for the bounded continuation test,
with the actual linearly growing scalar field as tilted potential. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
open scoped NNReal
namespace InvariantIsing
lemma magnetic_gaussian_tilted_test_ibp {F M A B : ℝ → ℝ}
(hF : Measurable F) (hG : HasLinearGrowth F)
(hM : Measurable M) (hA : Measurable A) (hB : Measurable B)
{K C D : ℝ} (bM : ∀ x, |M x| ≤ K) (bA : ∀ x, |A x| ≤ C)
(bB : ∀ x, |B x| ≤ D) (dF : ∀ x, HasDerivAt F (M x) x)
(dA : ∀ x, HasDerivAt A (B x) x) (r ζ z : ℝ) :
(∫ u, u * A (z + r * u)
∂(gaussianReal 0 1).tilted (fun u => ζ * F (z + r * u))) =
r * (∫ u, B (z + r * u) + ζ * A (z + r * u) * M (z + r * u)
∂(gaussianReal 0 1).tilted (fun u => ζ * F (z + r * u))) := by
have hC : 0 ≤ C := (abs_nonneg _).trans (bA 0)
let U := fun u => F (z + r * u)
let V := fun u => Real.exp (ζ * U u) * A (z + r * u)
let DV := fun u => r * (Real.exp (ζ * U u) *
(B (z + r * u) + ζ * A (z + r * u) * M (z + r * u)))
have hUm : Measurable U := hF.comp (by fun_prop)
have hUg : HasLinearGrowth U := (hG.add_left z).scale_argument r
have hw := integrable_exp_of_linearGrowth (gaussianReal 0 1)
(gaussianReal_exponentialNormMoments 0 1) hUm hUg ζ
have bgen (x : ℝ) : |B x + ζ * A x * M x| ≤ D + |ζ| * C * K := by
refine (abs_add_le _ _).trans (add_le_add (bB x) ?_)
rw [abs_mul, abs_mul]
exact mul_le_mul (mul_le_mul_of_nonneg_left (bA x) (abs_nonneg ζ))
(bM x) (abs_nonneg _) (mul_nonneg (abs_nonneg ζ) hC)
have hV : Integrable V (gaussianReal 0 1) :=
hw.mul_bdd (hA.comp (by fun_prop)).aestronglyMeasurable
(Filter.Eventually.of_forall fun u => by
simpa only [Real.norm_eq_abs] using bA (z + r * u))
have hDV : Integrable DV (gaussianReal 0 1) := by
apply Integrable.const_mul
exact hw.mul_bdd
((hB.comp (by fun_prop)).add
(((hA.comp (by fun_prop)).const_mul ζ).mul (hM.comp (by fun_prop)))).aestronglyMeasurable
(Filter.Eventually.of_forall fun u => by
simpa only [Real.norm_eq_abs] using bgen (z + r * u))
have hderiv (u : ℝ) : HasDerivAt V (DV u) u := by
have hs : HasDerivAt (fun x : ℝ => z + r * x) r u := by
convert ((hasDerivAt_id u).const_mul r).const_add z using 1 <;> first | rfl | simp
have hf : HasDerivAt (fun x => F (z + r * x)) (M (z + r * u) * r) u :=
(dF (z + r * u)).comp u hs
have ha : HasDerivAt (fun x => A (z + r * x)) (B (z + r * u) * r) u :=
(dA (z + r * u)).comp u hs
convert ((hf.const_mul ζ).exp).mul ha using 1
dsimp only [V, DV, U]
ring
have hlinear : HasLinearGrowth (fun u => u * A (z + r * u)) := by
refine ⟨0, C, le_rfl, hC, fun u => ?_⟩
rw [abs_mul, Real.norm_eq_abs, zero_add]
simpa only [mul_comm] using mul_le_mul_of_nonneg_left (bA (z + r * u)) (abs_nonneg u)
have hUV : Integrable (fun u => u * V u) (gaussianReal 0 1) := by
convert field_exp_mul_linear_integrable 1 0 ζ hUm hUg
(measurable_id.mul (hA.comp (by fun_prop))) hlinear using 1
funext u
dsimp only [V, Pi.mul_apply, Function.comp_apply, id_eq]
ring
have hibp := gaussian_integration_by_parts hderiv hV hDV hUV
simp only [integral_tilted_eq_div]
have hn : (∫ u, Real.exp (ζ * F (z + r * u)) * (u * A (z + r * u))
∂gaussianReal 0 1) =
r * ∫ u, Real.exp (ζ * F (z + r * u)) *
(B (z + r * u) + ζ * A (z + r * u) * M (z + r * u)) ∂gaussianReal 0 1 := by
rw [← integral_const_mul]
convert hibp using 1
congr 1
funext u
dsimp only [V, DV, U]
ring
rw [hn]
ring
end InvariantIsing
end
end OAI
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