| import OAI.Probability.FreeIsing.Brownian.DenseModification |
| import OAI.Probability.FreeIsing.Brownian.DyadicMoments |
| import OAI.Probability.FreeIsing.Brownian.Existence |
| import OAI.Probability.FreeIsing.Estimates.DyadicContinuity |
|
|
| namespace OAI |
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|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped Topology NNReal |
|
|
| namespace Problem367.BrownianContinuousExistence |
| open ContinuousModification BrownianDyadicMoments |
|
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| variable {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} {B : ℝ≥0 → Ω → ℝ} |
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| def version (B : ℝ≥0 → Ω → ℝ) (t : ℝ≥0) (ω : Ω) : ℝ := |
| limUnder atTop (fun n => B (Real.toNNReal (dyadicLeft t n)) ω) |
|
|
| lemma toNNReal_nat_grid (n k : ℕ) : |
| Real.toNNReal ((k : ℝ) / (2 : ℝ) ^ n) = time n k := by |
| apply NNReal.coe_injective |
| simp [time, Real.toNNReal_of_nonneg (by positivity : 0 ≤ (k : ℝ) / (2 : ℝ) ^ n)] |
|
|
| lemma ae_continuous_version (hB : IsPreBrownianReal B μ) : |
| ∀ᵐ ω ∂μ, Continuous (fun t => version B t ω) := by |
| filter_upwards [ae_eventually_increments hB] with ω hω |
| apply DyadicContinuity.continuous_nnreal_dyadicLimit_of_nat_grid |
| (fun t : ℝ => B t.toNNReal ω) (C := 1) (r := 9 / 10) |
| (by norm_num) (by norm_num) (by norm_num) |
| intro H |
| obtain ⟨N, hN⟩ := eventually_atTop.mp (hω H) |
| refine ⟨N, fun n hn k hk => ?_⟩ |
| have hh := hN n hn k hk |
| rw [show ((k : ℝ) + 1) / 2 ^ n = ((k + 1 : ℕ) : ℝ) / 2 ^ n by push_cast; rfl, |
| toNNReal_nat_grid, toNNReal_nat_grid] |
| simpa only [Real.dist_eq, abs_sub_comm, one_mul] using hh |
|
|
| omit [MeasurableSpace Ω] in |
| lemma version_grid (B : ℝ≥0 → Ω → ℝ) (k : ℤ) (hk : 0 ≤ k) (n : ℕ) (ω : Ω) : |
| version B (Real.toNNReal ((k : ℝ) / 2 ^ n)) ω = |
| B (Real.toNNReal ((k : ℝ) / 2 ^ n)) ω := by |
| have hg : 0 ≤ (k : ℝ) / (2 : ℝ) ^ n := div_nonneg (by exact_mod_cast hk) (by positivity) |
| unfold version |
| rw [Real.coe_toNNReal _ hg] |
| exact DyadicContinuity.dyadicLimit_grid (fun t : ℝ => B t.toNNReal ω) k n |
|
|
| theorem ae_eq_version (hB : IsPreBrownianReal B μ) (t : ℝ≥0) : |
| B t =ᵐ[μ] version B t := by |
| let s : ℕ → ℝ≥0 := fun n => (dyadicLeft t n).toNNReal |
| have hs : Tendsto s atTop (𝓝 t) := by |
| have hh := continuous_real_toNNReal.continuousAt.tendsto.comp (tendsto_dyadicLeft (t : ℝ)) |
| simpa only [Real.toNNReal_coe, Function.comp_def, s] using hh |
| apply preBrownian_ae_eq_of_approximating_sequence hB (ae_continuous_version hB) hs |
| intro n |
| apply Filter.Eventually.of_forall |
| intro ω |
| exact (version_grid B ⌊(2 : ℝ) ^ n * (t : ℝ)⌋ |
| (Int.floor_nonneg.mpr (mul_nonneg (by positivity) t.property)) n ω).symm |
|
|
| theorem isBrownianReal_version (hB : IsPreBrownianReal B μ) : |
| IsBrownianReal (version B) μ := |
| ⟨hB.congr (ae_eq_version hB), ae_continuous_version hB⟩ |
|
|
| theorem exists_brownianReal : |
| ∃ (ν : Measure (ℝ≥0 → ℝ)) (B : ℝ≥0 → (ℝ≥0 → ℝ) → ℝ), |
| IsProbabilityMeasure ν ∧ IsBrownianReal B ν := |
| ⟨BrownianExistence.preWienerMeasure, version (fun t ω => ω t), inferInstance, |
| isBrownianReal_version BrownianExistence.coordinate_isPreBrownianReal⟩ |
|
|
| end Problem367.BrownianContinuousExistence |
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| end |
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| end OAI |
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|
