File size: 3,332 Bytes
f61e547 | 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 | import Mathlib
namespace OAI
open MeasureTheory ProbabilityTheory Filter
open scoped Topology NNReal
namespace Problem367
theorem preBrownian_tendstoInMeasure {Ω : Type*} [MeasurableSpace Ω]
{P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B P)
{s : ℕ → ℝ≥0} {t : ℝ≥0} (hs : Tendsto s atTop (𝓝 t)) :
TendstoInMeasure P (fun n => B (s n)) atTop (B t) := by
let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
rw [tendstoInMeasure_iff_norm]
intro ε hε
have hmean (n : ℕ) : (∫ ω, B (s n) ω - B t ω ∂P) = 0 := by
rw [integral_sub (hB.integrable_eval _) (hB.integrable_eval _),
hB.integral_eval, hB.integral_eval, sub_self]
have hvar (n : ℕ) : variance (B (s n) - B t) P = dist (s n) t := by
rw [(hB.hasLaw_sub (s n) t).variance_eq, variance_id_gaussianReal]
rfl
have hbound (n : ℕ) : P {ω | ε ≤ ‖B (s n) ω - B t ω‖} ≤
ENNReal.ofReal (dist (s n) t / ε ^ 2) := by
simpa only [hvar, Pi.sub_apply, hmean, sub_zero, Real.norm_eq_abs] using
meas_ge_le_variance_div_sq (hB.hasLaw_sub (s n) t).hasGaussianLaw.memLp_two hε
have hlim : Tendsto (fun n => ENNReal.ofReal (dist (s n) t / ε ^ 2))
atTop (𝓝 0) := by
simpa [Function.comp_def] using ENNReal.continuous_ofReal.continuousAt.tendsto.comp
((hs.dist (tendsto_const_nhds : Tendsto (fun _ : ℕ => t) atTop (𝓝 t))).div_const
(ε ^ 2))
exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds hlim
(fun _ => bot_le) hbound
theorem preBrownian_ae_eq_of_approximating_sequence
{Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
{B Y : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B P)
(hY : ∀ᵐ ω ∂P, Continuous (fun t => Y t ω))
{s : ℕ → ℝ≥0} {t : ℝ≥0} (hs : Tendsto s atTop (𝓝 t))
(hseq : ∀ n, B (s n) =ᵐ[P] Y (s n)) : B t =ᵐ[P] Y t := by
let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
have hlim : ∀ᵐ ω ∂P, Tendsto (fun n => B (s n) ω) atTop (𝓝 (Y t ω)) := by
filter_upwards [hY, ae_all_iff.mpr hseq] with ω hcont heq
have ht := (hcont.tendsto t).comp hs
simpa only [Function.comp_def, fun n => heq n] using ht
exact tendstoInMeasure_ae_unique (preBrownian_tendstoInMeasure hB hs)
(tendstoInMeasure_of_tendsto_ae
(fun n => (hB.aemeasurable (s n)).aestronglyMeasurable) hlim)
theorem preBrownian_ae_eq_of_dense
{Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
{B Y : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B P)
(hY : ∀ᵐ ω ∂P, Continuous (fun t => Y t ω))
{D : Set ℝ≥0} (hD : Dense D) (heq : ∀ t ∈ D, B t =ᵐ[P] Y t) :
∀ t, B t =ᵐ[P] Y t := by
intro t
obtain ⟨s, hsD, hst⟩ := mem_closure_iff_seq_limit.mp (hD t)
exact preBrownian_ae_eq_of_approximating_sequence hB hY hst
(fun n => heq (s n) (hsD n))
theorem isBrownianReal_of_dense_agreement
{Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
{B Y : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B P)
(hY : ∀ᵐ ω ∂P, Continuous (fun t => Y t ω))
{D : Set ℝ≥0} (hD : Dense D) (heq : ∀ t ∈ D, B t =ᵐ[P] Y t) :
IsBrownianReal Y P :=
⟨hB.congr (preBrownian_ae_eq_of_dense hB hY hD heq), hY⟩
end Problem367
end OAI
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