| 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 |
| |
