| import Mathlib |
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| namespace OAI |
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| open MeasureTheory ProbabilityTheory Filter |
| open scoped Topology NNReal |
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| namespace Problem367 |
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| 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 |
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| 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) |
|
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| 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)) |
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| 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⟩ |
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| end Problem367 |
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| end OAI |
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