File size: 3,223 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 73 74 75 76 | 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
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped Topology NNReal
namespace Problem367.BrownianContinuousExistence
open ContinuousModification BrownianDyadicMoments
variable {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} {B : ℝ≥0 → Ω → ℝ}
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
end
end OAI
|