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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 | import OAI.Probability.InvariantIsing.Gaussian.GaussianTiltLinearGrowth
namespace OAI
/-! Completing the square under the finite scalar Gaussian law. This is
the shift identity used to control canonical increments uniformly in the
number and size of the cascade steps. -/
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace InvariantIsing
lemma gaussianPDFReal_exp_linear {v : ℝ≥0} (hv : v ≠ 0) (t x : ℝ) :
gaussianPDFReal 0 v x * Real.exp (t * x) =
Real.exp (t ^ 2 * v / 2) * gaussianPDFReal (t * v) v x := by
have hvR : (v : ℝ) ≠ 0 := by exact_mod_cast hv
have he : -(x ^ 2) / (2 * v) + t * x =
t ^ 2 * v / 2 + -(x - t * v) ^ 2 / (2 * v) := by
field_simp
ring
simp only [gaussianPDFReal, sub_zero]
rw [mul_assoc, ← Real.exp_add, he, Real.exp_add]
ring
theorem gaussian_exp_linear_shift (v : ℝ≥0) (t : ℝ) (F : ℝ → ℝ) (hF : Measurable F) :
(∫ x, Real.exp (t * x) * F x ∂gaussianReal 0 v) =
Real.exp (t ^ 2 * v / 2) *
∫ x, F (x + t * v) ∂gaussianReal 0 v := by
by_cases hv : v = 0
· subst v
simp
have hshift : (∫ x, F x ∂gaussianReal (t * v) v) =
∫ x, F (x + t * v) ∂gaussianReal 0 v := by
have he := integral_map (μ := gaussianReal 0 v)
(f := F) (by fun_prop : AEMeasurable (fun x : ℝ => x + t * v)
(gaussianReal 0 v)) hF.aestronglyMeasurable
simpa only [gaussianReal_map_add_const, zero_add] using he
rw [← hshift, integral_gaussianReal_eq_integral_smul hv,
integral_gaussianReal_eq_integral_smul hv, ← integral_const_mul]
apply integral_congr_ae
apply ae_of_all
intro x
simp only [smul_eq_mul]
rw [← mul_assoc, gaussianPDFReal_exp_linear hv]
ring
end InvariantIsing
end
end OAI
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