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b317718 | 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 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 | import OAI.Probability.FreeIsing.Estimates.ContinuousHorizonGluing
import OAI.Probability.FreeIsing.Martingales.L2Closure
import OAI.Probability.FreeIsing.Martingales.PredictableL2Completion
import OAI.Probability.FreeIsing.Martingales.StrongMaximal
namespace OAI
noncomputable section
open MeasureTheory Filter Finset
open scoped NNReal Topology
namespace Problem367.MartingaleSpatialLimit
variable {Ω ι : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω}
def finiteSum (M : ι → ℝ≥0 → Ω → ℝ) (s : Finset ι) (t : ℝ≥0) (ω : Ω) : ℝ :=
∑ i ∈ s, M i t ω
theorem finiteSum_memLp {M : ι → ℝ≥0 → Ω → ℝ}
(h2 : ∀ i t, MemLp (M i t) 2 μ) (s : Finset ι) (t : ℝ≥0) :
MemLp (finiteSum M s t) 2 μ :=
memLp_finsetSum s (fun i _ => h2 i t)
theorem finiteSum_martingale {F : Filtration ℝ≥0 mΩ} {M : ι → ℝ≥0 → Ω → ℝ}
(hM : ∀ i, Martingale (M i) F μ) (s : Finset ι) :
Martingale (finiteSum M s) F μ := by
classical
induction s using Finset.induction_on with
| empty =>
have heq : finiteSum M ∅ = 0 := by funext t ω; simp [finiteSum]
rw [heq]
exact martingale_zero ℝ F μ
| @insert i s hi ih =>
have heq : finiteSum M (insert i s) = M i + finiteSum M s := by
funext t ω
simp only [finiteSum, sum_insert hi, Pi.add_apply]
rw [heq]
exact (hM i).add ih
theorem finiteSum_ae_continuous {M : ι → ℝ≥0 → Ω → ℝ}
(hc : ∀ i, ∀ᵐ ω ∂μ, Continuous (fun t => M i t ω)) (s : Finset ι) :
∀ᵐ ω ∂μ, Continuous (fun t => finiteSum M s t ω) := by
have hall : ∀ᵐ ω ∂μ, ∀ i ∈ s, Continuous (fun t => M i t ω) :=
(Filter.eventually_all_finite s.finite_toSet).mpr (fun i _ => hc i)
filter_upwards [hall] with ω hω
exact continuous_finsetSum s hω
theorem summable_energy_of_integer [IsFiniteMeasure μ]
{F : Filtration ℝ≥0 mΩ} {M : ι → ℝ≥0 → Ω → ℝ}
(hM : ∀ i, Martingale (M i) F μ) (h2 : ∀ i t, MemLp (M i t) 2 μ)
(henergy : ∀ N : ℕ, Summable (fun i => ∫ ω, M i (N : ℝ≥0) ω ^ 2 ∂μ))
(t : ℝ≥0) : Summable (fun i => ∫ ω, M i t ω ^ 2 ∂μ) := by
obtain ⟨N, hN⟩ := exists_nat_ge t
exact Summable.of_nonneg_of_le (fun i => integral_nonneg (fun ω => sq_nonneg _))
(fun i => MartingaleSquareMaximal.integral_sq_le (hM i) (h2 i) hN) (henergy N)
theorem summable_toLp {M : ι → ℝ≥0 → Ω → ℝ}
(h2 : ∀ i t, MemLp (M i t) 2 μ) (t : ℝ≥0)
(horth : Pairwise (fun i j => ∫ ω, M i t ω * M j t ω ∂μ = 0))
(henergy : Summable (fun i => ∫ ω, M i t ω ^ 2 ∂μ)) :
Summable (fun i => (h2 i t).toLp (M i t)) := by
apply PredictableL2.summable_of_orthogonal_sq
· intro i j hij
simpa only [PredictableL2.inner_toLp] using horth hij
· simpa only [PredictableL2.norm_toLp_sq] using henergy
theorem exists_continuous_martingale_sum [Countable ι] [IsFiniteMeasure μ]
{F : Filtration ℝ≥0 mΩ} {M : ι → ℝ≥0 → Ω → ℝ}
(hM : ∀ i, Martingale (M i) F μ)
(h2 : ∀ i t, MemLp (M i t) 2 μ)
(hc : ∀ i, ∀ᵐ ω ∂μ, Continuous (fun t => M i t ω))
(horth : ∀ t, Pairwise (fun i j => ∫ ω, M i t ω * M j t ω ∂μ = 0))
(henergy : ∀ N : ℕ, Summable (fun i => ∫ ω, M i (N : ℝ≥0) ω ^ 2 ∂μ)) :
∃ (L : ℝ≥0 → Ω → ℝ) (hL2 : ∀ t, MemLp (L t) 2 μ),
IsStronglyPredictable F L ∧ Martingale L F μ ∧
(∀ᵐ ω ∂μ, Continuous (fun t => L t ω)) ∧
(∀ t, HasSum (fun i => (h2 i t).toLp (M i t)) ((hL2 t).toLp (L t))) ∧
(∀ T : ℝ≥0, Tendsto (fun s : Finset ι =>
∫ ω, MartingaleStrongMaximal.maximalSquare (finiteSum M s - L) T ω ∂μ)
atTop (𝓝 0)) := by
classical
obtain ⟨s, hs⟩ := Filter.exists_seq_tendsto (atTop : Filter (Finset ι))
have hsum (t : ℝ≥0) : Summable (fun i => (h2 i t).toLp (M i t)) :=
summable_toLp h2 t (horth t) (summable_energy_of_integer hM h2 henergy t)
have heq (n : ℕ) (t : ℝ≥0) :
(finiteSum_memLp h2 (s n) t).toLp (finiteSum M (s n) t) =
∑ i ∈ s n, (h2 i t).toLp (M i t) :=
PredictableL2.toLp_finsetSum (fun i => M i t) (fun i => h2 i t) (s n)
have hseq (t : ℝ≥0) : Tendsto (fun n =>
(finiteSum_memLp h2 (s n) t).toLp (finiteSum M (s n) t)) atTop
(𝓝 (∑' i, (h2 i t).toLp (M i t))) := by
simp_rw [heq]
exact (hsum t).hasSum.comp hs
obtain ⟨L, hpred, hLc, hlim⟩ :=
MartingaleSquareMaximal.exists_global_predictable_continuous_L2_limit
(fun n => finiteSum_martingale hM (s n))
(fun n t => finiteSum_memLp h2 (s n) t)
(fun n => finiteSum_ae_continuous hc (s n))
(fun N => (hseq (N : ℝ≥0)).cauchySeq)
choose hL2 hLlim using hlim
have hLM : Martingale L F μ := MartingaleSquareMaximal.martingale_of_L2_limits
(fun n => finiteSum_martingale hM (s n))
(fun n t => finiteSum_memLp h2 (s n) t)
hpred.stronglyAdapted hL2 hLlim
have hHasSum (t : ℝ≥0) : HasSum (fun i => (h2 i t).toLp (M i t))
((hL2 t).toLp (L t)) := by
have heq := tendsto_nhds_unique (hseq t) (hLlim t)
rw [← heq]
exact (hsum t).hasSum
refine ⟨L, hL2, hpred, hLM, hLc, hHasSum, ?_⟩
intro T
have hterminal := PredictableL2.tendsto_integral_sq_error_of_hasSum
(fun i => M i T) (fun i => h2 i T) (hL2 T) (hHasSum T)
apply squeeze_zero
(fun q => integral_nonneg (fun ω => ENNReal.toReal_nonneg))
(fun q => ?_)
(show Tendsto (fun q : Finset ι => 4 * ∫ ω,
((∑ i ∈ q, M i T ω) - L T ω) ^ 2 ∂μ) atTop (𝓝 0) from by
simpa only [mul_zero] using hterminal.const_mul 4)
have hbound := MartingaleStrongMaximal.integral_maximalSquare_le
((finiteSum_martingale hM q).sub hLM)
(fun t => (finiteSum_memLp h2 q t).sub (hL2 t))
(show ∀ᵐ ω ∂μ, Continuous (fun t => (finiteSum M q - L) t ω) from by
filter_upwards [finiteSum_ae_continuous hc q, hLc] with ω hq hl
exact hq.sub hl) T
simpa only [Pi.sub_apply, finiteSum, MartingaleStrongMaximal.maximalSquare] using hbound
end Problem367.MartingaleSpatialLimit
end
end OAI
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