| import Mathlib.MeasureTheory.Function.L2Space |
| import Mathlib.MeasureTheory.Integral.Bochner.Basic |
| import Mathlib.Analysis.Calculus.MeanValue |
| import Mathlib.Tactic |
|
|
| namespace OAI |
|
|
| /-! The Hilbert-space variance estimate for geodesic averages. Applying this |
| to the Jacobi curvature matrix in the Hilbert space of square-integrable |
| fields separates the averaging argument from the geometric flow identities. -/ |
|
|
| namespace PositiveEinsteinFour.Averages |
|
|
| open MeasureTheory Filter |
|
|
| noncomputable section |
|
|
| variable {Ω E : Type*} [MeasurableSpace Ω] [NormedAddCommGroup E] |
| [InnerProductSpace ℝ E] [CompleteSpace E] |
| {μ : Measure Ω} [IsProbabilityMeasure μ] {f : Ω → E} |
|
|
| omit [CompleteSpace E] in |
| theorem integrable_sq_distance (hf : Integrable f μ) |
| (hf2 : Integrable (fun t => ‖f t‖ ^ 2) μ) (a : E) : |
| Integrable (fun t => ‖a - f t‖ ^ 2) μ := by |
| have h := ((integrable_const (‖a‖ ^ 2)).sub ((hf.const_inner a).const_mul 2)).add hf2 |
| convert h using 1 |
| ext t |
| exact norm_sub_sq_real a (f t) |
|
|
| theorem integral_sq_distance (hf : Integrable f μ) |
| (hf2 : Integrable (fun t => ‖f t‖ ^ 2) μ) (a : E) : |
| (∫ t, ‖a - f t‖ ^ 2 ∂μ) = |
| ‖a‖ ^ 2 - 2 * inner ℝ a (∫ t, f t ∂μ) + ∫ t, ‖f t‖ ^ 2 ∂μ := by |
| simp_rw [norm_sub_sq_real] |
| have hadd := integral_add ((integrable_const (‖a‖ ^ 2)).sub |
| ((hf.const_inner a).const_mul 2)) hf2 |
| have hsub := integral_sub (integrable_const (‖a‖ ^ 2)) |
| ((hf.const_inner a).const_mul 2) |
| simp only [Pi.sub_apply] at hadd hsub |
| rw [hadd, hsub, integral_const_mul, integral_inner hf a] |
| simp only [integral_const, probReal_univ, one_smul] |
|
|
| theorem integrable_mean_sq_distance (hf : Integrable f μ) |
| (hf2 : Integrable (fun t => ‖f t‖ ^ 2) μ) : |
| Integrable (fun s => ∫ t, ‖f s - f t‖ ^ 2 ∂μ) μ := by |
| have h := (hf2.sub ((hf.inner_const (∫ t, f t ∂μ)).const_mul 2)).add |
| (integrable_const (∫ t, ‖f t‖ ^ 2 ∂μ)) |
| convert h using 1 |
| ext s |
| exact integral_sq_distance hf hf2 (f s) |
|
|
| /-- The double-variance identity, with only the integrability needed by the |
| application. The two integrals need not be interchanged. -/ |
| theorem double_variance_identity (hf : Integrable f μ) |
| (hf2 : Integrable (fun t => ‖f t‖ ^ 2) μ) : |
| (∫ s, ∫ t, ‖f s - f t‖ ^ 2 ∂μ ∂μ) = |
| 2 * ((∫ t, ‖f t‖ ^ 2 ∂μ) - ‖∫ t, f t ∂μ‖ ^ 2) := by |
| simp_rw [integral_sq_distance hf hf2] |
| have hadd := integral_add (hf2.sub ((hf.inner_const (∫ t, f t ∂μ)).const_mul 2)) |
| (integrable_const (∫ t, ‖f t‖ ^ 2 ∂μ)) |
| have hsub := integral_sub hf2 ((hf.inner_const (∫ t, f t ∂μ)).const_mul 2) |
| simp only [Pi.sub_apply] at hadd hsub |
| rw [hadd, hsub, integral_const_mul] |
| have hinner : (∫ s, inner ℝ (f s) (∫ t, f t ∂μ) ∂μ) = ‖∫ t, f t ∂μ‖ ^ 2 := by |
| calc |
| _ = ∫ s, inner ℝ (∫ t, f t ∂μ) (f s) ∂μ := by |
| apply integral_congr_ae |
| exact Eventually.of_forall fun s => real_inner_comm _ _ |
| _ = inner ℝ (∫ t, f t ∂μ) (∫ t, f t ∂μ) := integral_inner hf _ |
| _ = _ := real_inner_self_eq_norm_sq _ |
| rw [hinner] |
| simp only [integral_const, probReal_univ, one_smul] |
| ring |
|
|
| /-- A stationary Hilbert-valued curve with a mean-square increment bound |
| retains this much squared norm after averaging. In the geodesic application, |
| `L` is the averaged squared norm of the directional derivative. -/ |
| theorem stationary_average_lower {ν : Measure ℝ} [IsProbabilityMeasure ν] |
| {A : ℝ → E} {q L : ℝ} |
| (hA : Integrable A ν) (hA2 : Integrable (fun t => ‖A t‖ ^ 2) ν) |
| (ht : Integrable (fun t : ℝ => t) ν) |
| (ht2 : Integrable (fun t : ℝ => t ^ 2) ν) |
| (hstationary : ∀ᵐ t ∂ν, ‖A t‖ ^ 2 = q) |
| (hincrement : ∀ s t, ‖A s - A t‖ ^ 2 ≤ L * (s - t) ^ 2) : |
| q - L * ((∫ t, t ^ 2 ∂ν) - (∫ t, t ∂ν) ^ 2) ≤ ‖∫ t, A t ∂ν‖ ^ 2 := by |
| have ht2' : Integrable (fun t : ℝ => ‖t‖ ^ 2) ν := by simpa using ht2 |
| have hpoint (s : ℝ) : (∫ t, ‖A s - A t‖ ^ 2 ∂ν) ≤ |
| L * ∫ t, ‖s - t‖ ^ 2 ∂ν := by |
| calc |
| _ ≤ ∫ t, L * ‖s - t‖ ^ 2 ∂ν := by |
| apply integral_mono (integrable_sq_distance hA hA2 (A s)) |
| ((integrable_sq_distance ht ht2' s).const_mul L) |
| intro t |
| simpa using hincrement s t |
| _ = _ := integral_const_mul _ _ |
| have hbound := integral_mono (integrable_mean_sq_distance hA hA2) |
| ((integrable_mean_sq_distance ht ht2').const_mul L) hpoint |
| rw [double_variance_identity hA hA2, integral_const_mul, |
| double_variance_identity ht ht2'] at hbound |
| have hq : (∫ t, ‖A t‖ ^ 2 ∂ν) = q := by |
| rw [integral_congr_ae hstationary] |
| simp |
| rw [hq] at hbound |
| simp only [Real.norm_eq_abs, sq_abs] at hbound |
| linarith |
|
|
| omit [CompleteSpace E] in |
| /-- An invariant averaged derivative bound controls increments of the |
| Hilbert-valued curvature curve. This is also valid for an `L²` Hilbert space. -/ |
| theorem increment_bound_of_derivative {A d : ℝ → E} {L : ℝ} (hL : 0 ≤ L) |
| (hd : ∀ t, HasDerivAt A (d t) t) (hbound : ∀ t, ‖d t‖ ^ 2 ≤ L) (s t : ℝ) : |
| ‖A s - A t‖ ^ 2 ≤ L * (s - t) ^ 2 := by |
| have hnorm (z : ℝ) : ‖d z‖ ≤ Real.sqrt L := by |
| nlinarith [hbound z, Real.sqrt_nonneg L, Real.sq_sqrt hL, norm_nonneg (d z)] |
| have hforward (a b : ℝ) (hab : a ≤ b) : |
| ‖A b - A a‖ ≤ Real.sqrt L * (b - a) := by |
| exact norm_image_sub_le_of_norm_deriv_le_segment' |
| (fun z _ => (hd z).hasDerivWithinAt) (fun z _ => hnorm z) b ⟨hab, le_rfl⟩ |
| have hdist : ‖A s - A t‖ ≤ Real.sqrt L * |s - t| := by |
| rcases le_total t s with hts | hst |
| · simpa only [abs_of_nonneg (sub_nonneg.mpr hts)] using hforward t s hts |
| · rw [norm_sub_rev, abs_of_nonpos (sub_nonpos.mpr hst), neg_sub] |
| exact hforward s t hst |
| have hsq := mul_le_mul hdist hdist (norm_nonneg _) (by positivity) |
| simpa only [← pow_two, mul_pow, Real.sq_sqrt hL, sq_abs] using hsq |
| |
| end |
| |
| end PositiveEinsteinFour.Averages |
| |
| end OAI |
| |
