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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 | import OAI.Probability.InvariantIsing.Gaussian.MPDensityTransform
import Mathlib.Analysis.Calculus.ParametricIntegral
namespace OAI
/-! Differentiation of bounded positive-resolvent moments. -/
noncomputable section
open MeasureTheory ProbabilityTheory Set Filter
open scoped Topology
namespace InvariantIsing
lemma positiveResolventTest_hasDerivAt {t : ℝ} (ht : 0 < t) (x : ℝ) :
HasDerivAt (fun s => positiveResolventTest s x) (-(positiveResolventTest t x)^2) t := by
have hn : t+max x 0 ≠ 0 := (add_pos_of_pos_of_nonneg ht (le_max_right x 0)).ne'
convert ((hasDerivAt_id t).add_const (max x 0)).inv hn using 1
· funext s
simp only [positiveResolventTest,one_div,Pi.inv_apply,id_eq]
· simp only [positiveResolventTest,div_eq_mul_inv,inv_pow,one_mul,id_eq,neg_mul]
lemma positiveResolventPower_hasDerivAt {t : ℝ} (ht : 0 < t) (x : ℝ) (n : ℕ) :
HasDerivAt (fun s => (positiveResolventTest s x)^n)
(-(n : ℝ)*(positiveResolventTest t x)^(n+1)) t := by
induction n with
| zero => simpa only [pow_zero,Nat.cast_zero,neg_zero,zero_mul] using hasDerivAt_const t (1 : ℝ)
| succ n ih =>
convert ih.mul (positiveResolventTest_hasDerivAt ht x) using 1
· funext s
exact pow_succ _ _
· simp only [Nat.cast_succ,pow_succ]
ring
lemma positiveResolventPower_integrable {t : ℝ} (ht : 0 < t) (n : ℕ)
(μ : Measure ℝ) [IsFiniteMeasure μ] :
Integrable (fun x => (positiveResolventTest t x)^n) μ := by
apply Integrable.of_bound ((continuous_positiveResolventTest ht).pow n).aestronglyMeasurable ((1/t)^n)
exact ae_of_all _ fun x => by
change ‖(positiveResolventTest t x)^n‖ ≤ _
rw [Real.norm_eq_abs,abs_of_nonneg (pow_nonneg (positiveResolventTest_bound ht x).1 _)]
exact pow_le_pow_left₀ (positiveResolventTest_bound ht x).1 (positiveResolventTest_bound ht x).2 n
theorem positiveResolventPower_integral_hasDerivAt (μ : Measure ℝ) [IsFiniteMeasure μ]
{t : ℝ} (ht : 0 < t) (n : ℕ) :
HasDerivAt (fun s => ∫ x, (positiveResolventTest s x)^n ∂μ)
(-(n : ℝ)*(∫ x, (positiveResolventTest t x)^(n+1) ∂μ)) t := by
have hs : Ioi (t/2) ∈ 𝓝 t := Ioi_mem_nhds (by linarith)
have hm : ∀ᶠ s in 𝓝 t, AEStronglyMeasurable (fun x => (positiveResolventTest s x)^n) μ := by
filter_upwards [hs] with s hs'
exact ((continuous_positiveResolventTest (lt_trans (half_pos ht) hs')).pow n).aestronglyMeasurable
have hd : ∀ᵐ x ∂μ, ∀ s ∈ Ioi (t/2),
HasDerivAt (fun u => (positiveResolventTest u x)^n)
(-(n : ℝ)*(positiveResolventTest s x)^(n+1)) s :=
ae_of_all _ fun x s hs' => positiveResolventPower_hasDerivAt (lt_trans (half_pos ht) hs') x n
have hb : ∀ᵐ x ∂μ, ∀ s ∈ Ioi (t/2),
‖-(n : ℝ)*(positiveResolventTest s x)^(n+1)‖ ≤ (n : ℝ)*(2/t)^(n+1) := by
refine ae_of_all _ fun x s hs' => ?_
have hsp : 0 < s := lt_trans (half_pos ht) hs'
have hr : positiveResolventTest s x ≤ 2/t := by
calc
_ ≤ 1/s := (positiveResolventTest_bound hsp x).2
_ ≤ 1/(t/2) := one_div_le_one_div_of_le (half_pos ht) hs'.le
_ = 2/t := by ring
rw [norm_mul,norm_neg,Real.norm_of_nonneg (Nat.cast_nonneg n),
Real.norm_of_nonneg (pow_nonneg (positiveResolventTest_bound hsp x).1 _)]
exact mul_le_mul_of_nonneg_left
(pow_le_pow_left₀ (positiveResolventTest_bound hsp x).1 hr _) (Nat.cast_nonneg n)
have hh := (hasDerivAt_integral_of_dominated_loc_of_deriv_le hs hm
(positiveResolventPower_integrable ht n μ)
(((continuous_positiveResolventTest ht).pow (n+1)).const_mul (-(n : ℝ))).aestronglyMeasurable
hb (integrable_const _) hd).2
simpa only [integral_const_mul] using hh
end InvariantIsing
end
end OAI
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