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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 | import OAI.Probability.FreeIsing.Martingales.PredictableL2Completion
import OAI.Probability.FreeIsing.Martingales.TransformL2
import OAI.Probability.FreeIsing.Stochastic.SimpleIntegralAssociativity
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped Topology NNReal
namespace Problem367.SimpleIntegral
section Multiplier
variable {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω}
{K : Ω → ℝ} {C : ℝ}
abbrev bounded_mul_memLp (hK : AEStronglyMeasurable K μ)
(hbound : ∀ᵐ ω ∂μ, |K ω| ≤ C) {X : Ω → ℝ} (hX : MemLp X 2 μ) :
MemLp (fun ω => K ω * X ω) 2 μ :=
BrownianGrid.memLp_compose_of_bounded hX hK hbound
def boundedMulLinear (hK : AEStronglyMeasurable K μ)
(hbound : ∀ᵐ ω ∂μ, |K ω| ≤ C) : Lp ℝ 2 μ →ₗ[ℝ] Lp ℝ 2 μ where
toFun X := (bounded_mul_memLp hK hbound (Lp.memLp X)).toLp (fun ω => K ω * X ω)
map_add' X Y := by
apply Lp.ext
filter_upwards
[(bounded_mul_memLp hK hbound (Lp.memLp (X + Y))).coeFn_toLp,
Lp.coeFn_add X Y,
Lp.coeFn_add
((bounded_mul_memLp hK hbound (Lp.memLp X)).toLp (fun ω => K ω * X ω))
((bounded_mul_memLp hK hbound (Lp.memLp Y)).toLp (fun ω => K ω * Y ω)),
(bounded_mul_memLp hK hbound (Lp.memLp X)).coeFn_toLp,
(bounded_mul_memLp hK hbound (Lp.memLp Y)).coeFn_toLp] with ω hxy hsum hout hx hy
simp only [Pi.add_apply] at *
rw [hxy, hsum, hout, hx, hy, mul_add]
map_smul' r X := by
apply Lp.ext
filter_upwards
[(bounded_mul_memLp hK hbound (Lp.memLp (r • X))).coeFn_toLp,
Lp.coeFn_smul r X,
Lp.coeFn_smul r
((bounded_mul_memLp hK hbound (Lp.memLp X)).toLp (fun ω => K ω * X ω)),
(bounded_mul_memLp hK hbound (Lp.memLp X)).coeFn_toLp] with ω hrx hsm hout hx
simp only [Pi.smul_apply, smul_eq_mul, RingHom.id_apply] at *
rw [hrx, hsm, hout, hx]
ring
theorem norm_boundedMulLinear_le (hK : AEStronglyMeasurable K μ)
(hbound : ∀ᵐ ω ∂μ, |K ω| ≤ C) (X : Lp ℝ 2 μ) :
‖boundedMulLinear hK hbound X‖ ≤ C * ‖X‖ := by
apply Lp.norm_le_mul_norm_of_ae_le_mul
filter_upwards [hbound,
(bounded_mul_memLp hK hbound (Lp.memLp X)).coeFn_toLp] with ω hb he
change ‖(bounded_mul_memLp hK hbound (Lp.memLp X)).toLp
(fun ω => K ω * X ω) ω‖ ≤ C * ‖X ω‖
rw [he, norm_mul]
exact mul_le_mul_of_nonneg_right hb (norm_nonneg _)
def boundedMulL2 (hK : AEStronglyMeasurable K μ)
(hbound : ∀ᵐ ω ∂μ, |K ω| ≤ C) : Lp ℝ 2 μ →L[ℝ] Lp ℝ 2 μ :=
(boundedMulLinear hK hbound).mkContinuous C (norm_boundedMulLinear_le hK hbound)
theorem boundedMulL2_toLp (hK : AEStronglyMeasurable K μ)
(hbound : ∀ᵐ ω ∂μ, |K ω| ≤ C) {X : Ω → ℝ} (hX : MemLp X 2 μ) :
boundedMulL2 hK hbound (hX.toLp X) =
(bounded_mul_memLp hK hbound hX).toLp (fun ω => K ω * X ω) := by
apply Lp.ext
filter_upwards
[(bounded_mul_memLp hK hbound (Lp.memLp (hX.toLp X))).coeFn_toLp,
hX.coeFn_toLp, (bounded_mul_memLp hK hbound hX).coeFn_toLp] with ω hleft hx hright
change (bounded_mul_memLp hK hbound (Lp.memLp (hX.toLp X))).toLp
(fun ω => K ω * hX.toLp X ω) ω = _
rw [hleft, hx, hright]
theorem hasSum_bounded_mul {ι : Type*}
(hK : AEStronglyMeasurable K μ) (hbound : ∀ᵐ ω ∂μ, |K ω| ≤ C)
(X : ι → Ω → ℝ) (hX : ∀ i, MemLp (X i) 2 μ)
{Y : Ω → ℝ} (hY : MemLp Y 2 μ)
(hsum : HasSum (fun i => (hX i).toLp (X i)) (hY.toLp Y)) :
HasSum (fun i => (bounded_mul_memLp hK hbound (hX i)).toLp
(fun ω => K ω * X i ω))
((bounded_mul_memLp hK hbound hY).toLp (fun ω => K ω * Y ω)) := by
simpa only [boundedMulL2_toLp] using hsum.mapL (boundedMulL2 hK hbound)
end Multiplier
section Limits
variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω}
theorem tendsto_toLp_iff_meanSquare {α : Type*} {l : Filter α}
{X : α → Ω → ℝ} (hX : ∀ a, MemLp (X a) 2 μ)
{Y : Ω → ℝ} (hY : MemLp Y 2 μ) :
Tendsto (fun a => (hX a).toLp (X a)) l (𝓝 (hY.toLp Y)) ↔
Tendsto (fun a => ∫ ω, (X a ω - Y ω) ^ 2 ∂μ) l (𝓝 0) := by
have heq (a : α) :
‖(hX a).toLp (X a) - hY.toLp Y‖ ^ 2 =
∫ ω, (X a ω - Y ω) ^ 2 ∂μ := by
rw [← MemLp.toLp_sub (hX a) hY]
exact PredictableL2.norm_toLp_sq ((hX a).sub hY)
rw [tendsto_iff_norm_sub_tendsto_zero]
constructor
· intro h
simpa only [heq, zero_pow (by decide : (2 : ℕ) ≠ 0)] using h.pow 2
· intro h
have hs := Real.continuous_sqrt.continuousAt.tendsto.comp h
simpa only [Function.comp_def, ← heq, Real.sqrt_sq (norm_nonneg _),
Real.sqrt_zero] using hs
theorem meanSquare_limit_of_close {α : Type*} {l : Filter α}
{X Y : α → Ω → ℝ} (hX : ∀ a, MemLp (X a) 2 μ)
(hY : ∀ a, MemLp (Y a) 2 μ) {Z : Ω → ℝ} (hZ : MemLp Z 2 μ)
(hclose : Tendsto (fun a => ∫ ω, (X a ω - Y a ω) ^ 2 ∂μ) l (𝓝 0))
(hlim : Tendsto (fun a => ∫ ω, (X a ω - Z ω) ^ 2 ∂μ) l (𝓝 0)) :
Tendsto (fun a => ∫ ω, (Y a ω - Z ω) ^ 2 ∂μ) l (𝓝 0) := by
apply (tendsto_toLp_iff_meanSquare hY hZ).mp
apply ((tendsto_toLp_iff_meanSquare hX hZ).mpr hlim).congr_dist
have heq (a : α) :
dist ((hX a).toLp (X a)) ((hY a).toLp (Y a)) ^ 2 =
∫ ω, (X a ω - Y a ω) ^ 2 ∂μ := by
rw [dist_eq_norm, ← MemLp.toLp_sub (hX a) (hY a)]
exact PredictableL2.norm_toLp_sq ((hX a).sub (hY a))
have hs := Real.continuous_sqrt.continuousAt.tendsto.comp hclose
simpa only [Function.comp_def, ← heq, Real.sqrt_sq dist_nonneg,
Real.sqrt_zero] using hs
theorem tendsto_bounded_multiplier_energy
{K : ℕ → Ω → ℝ} {L X : Ω → ℝ}
(hK : ∀ n, AEStronglyMeasurable (K n) μ)
(hL : AEStronglyMeasurable L μ) (hX : MemLp X 2 μ)
{C : ℝ} (hC : 0 ≤ C)
(hboundK : ∀ n, ∀ᵐ ω ∂μ, |K n ω| ≤ C)
(hboundL : ∀ᵐ ω ∂μ, |L ω| ≤ C)
(hlim : ∀ᵐ ω ∂μ, Tendsto (fun n => K n ω) atTop (𝓝 (L ω))) :
Tendsto (fun n => ∫ ω, ((K n ω - L ω) * X ω) ^ 2 ∂μ) atTop (𝓝 0) := by
have hb (n : ℕ) : ∀ᵐ ω ∂μ,
‖((K n ω - L ω) * X ω) ^ 2‖ ≤ (2 * C) ^ 2 * X ω ^ 2 := by
filter_upwards [hboundK n, hboundL] with ω hn hl
have hd : |K n ω - L ω| ≤ 2 * C :=
(abs_sub _ _).trans (by linarith)
have hs : (K n ω - L ω) ^ 2 ≤ (2 * C) ^ 2 := by
simpa only [sq_abs] using
(sq_le_sq₀ (abs_nonneg _) (by positivity : 0 ≤ 2 * C)).2 hd
rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _), mul_pow]
exact mul_le_mul_of_nonneg_right hs (sq_nonneg (X ω))
have hi := tendsto_integral_of_dominated_convergence (μ := μ)
(fun ω => (2 * C) ^ 2 * X ω ^ 2)
(fun n => (((hK n).sub hL).mul hX.aestronglyMeasurable).pow 2)
(hX.integrable_sq.const_mul ((2 * C) ^ 2)) hb
(hlim.mono (fun ω hω => by
simpa using ((hω.sub tendsto_const_nhds).mul_const (X ω)).pow 2))
simpa using hi
def coarseTransform (K M : ℕ → Ω → ℝ) (n : ℕ) (ω : Ω) : ℝ :=
∑ j ∈ Finset.range n, K j ω * (M (j + 1) ω - M j ω)
theorem coarseTransform_memLp (K : ℕ → Ω → ℝ)
(hK : ∀ j, AEStronglyMeasurable (K j) μ) (C : ℕ → ℝ)
(hbound : ∀ j, ∀ᵐ ω ∂μ, |K j ω| ≤ C j)
{M : ℕ → Ω → ℝ} (hM : ∀ j, MemLp (M j) 2 μ) (n : ℕ) :
MemLp (coarseTransform K M n) 2 μ := by
exact memLp_finsetSum _ fun j _ =>
bounded_mul_memLp (hK j) (hbound j) ((hM (j + 1)).sub (hM j))
theorem toLp_coarseTransform (K : ℕ → Ω → ℝ)
(hK : ∀ j, AEStronglyMeasurable (K j) μ) (C : ℕ → ℝ)
(hbound : ∀ j, ∀ᵐ ω ∂μ, |K j ω| ≤ C j)
{M : ℕ → Ω → ℝ} (hM : ∀ j, MemLp (M j) 2 μ) (n : ℕ) :
(coarseTransform_memLp K hK C hbound hM n).toLp (coarseTransform K M n) =
∑ j ∈ Finset.range n, boundedMulL2 (hK j) (hbound j)
((hM (j + 1)).toLp (M (j + 1)) - (hM j).toLp (M j)) := by
have hterm (j : ℕ) : MemLp (fun ω => K j ω * (M (j + 1) ω - M j ω)) 2 μ :=
bounded_mul_memLp (hK j) (hbound j) ((hM (j + 1)).sub (hM j))
calc
_ = ∑ j ∈ Finset.range n, (hterm j).toLp
(fun ω => K j ω * (M (j + 1) ω - M j ω)) :=
PredictableL2.toLp_finsetSum _ hterm _
_ = _ := by
apply Finset.sum_congr rfl
intro j hj
rw [← MemLp.toLp_sub (hM (j + 1)) (hM j), boundedMulL2_toLp]
rfl
theorem tendsto_coarseTransform {α : Type*} {l : Filter α}
(K : ℕ → Ω → ℝ) (hK : ∀ j, AEStronglyMeasurable (K j) μ)
(C : ℕ → ℝ) (hbound : ∀ j, ∀ᵐ ω ∂μ, |K j ω| ≤ C j)
{X : α → ℕ → Ω → ℝ} (hX : ∀ a j, MemLp (X a j) 2 μ)
{M : ℕ → Ω → ℝ} (hM : ∀ j, MemLp (M j) 2 μ) (n : ℕ)
(hlim : ∀ j ≤ n, Tendsto (fun a => (hX a j).toLp (X a j)) l
(𝓝 ((hM j).toLp (M j)))) :
Tendsto (fun a => (coarseTransform_memLp K hK C hbound (hX a) n).toLp
(coarseTransform K (X a) n)) l
(𝓝 ((coarseTransform_memLp K hK C hbound hM n).toLp
(coarseTransform K M n))) := by
rw [toLp_coarseTransform K hK C hbound hM n]
have ht : Tendsto (fun a => ∑ j ∈ Finset.range n,
boundedMulL2 (hK j) (hbound j)
((hX a (j + 1)).toLp (X a (j + 1)) - (hX a j).toLp (X a j))) l
(𝓝 (∑ j ∈ Finset.range n, boundedMulL2 (hK j) (hbound j)
((hM (j + 1)).toLp (M (j + 1)) - (hM j).toLp (M j)))) := by
apply tendsto_finsetSum
intro j hj
have hjn : j < n := Finset.mem_range.mp hj
exact ((boundedMulL2 (hK j) (hbound j)).continuous.tendsto _).comp
((hlim (j + 1) hjn).sub (hlim j hjn.le))
exact ht.congr (fun a => (toLp_coarseTransform K hK C hbound (hX a) n).symm)
theorem integralSum_compose_refined_tendsto {α : Type*} {l : Filter α}
(B : ℝ≥0 → Ω → ℝ) (t : α → ℕ → ℝ≥0) (H : α → ℕ → Ω → ℝ)
(k : α → ℕ) (hk : ∀ a, 0 < k a)
(K : ℕ → Ω → ℝ) (hK : ∀ j, AEStronglyMeasurable (K j) μ)
(C : ℕ → ℝ) (hbound : ∀ j, ∀ᵐ ω ∂μ, |K j ω| ≤ C j)
(hI : ∀ a j, MemLp (BrownianGrid.integralSum B (t a) (H a) (j * k a)) 2 μ)
{M : ℕ → Ω → ℝ} (hM : ∀ j, MemLp (M j) 2 μ) (n : ℕ)
(hlim : ∀ j ≤ n, Tendsto (fun a => (hI a j).toLp
(BrownianGrid.integralSum B (t a) (H a) (j * k a))) l
(𝓝 ((hM j).toLp (M j)))) :
Tendsto (fun a => ∫ ω,
(BrownianGrid.integralSum B (t a)
(fun i ω => K (i / k a) ω * H a i ω) (n * k a) ω -
coarseTransform K M n ω) ^ 2 ∂μ) l (𝓝 0) := by
have h := (tendsto_toLp_iff_meanSquare
(fun a => coarseTransform_memLp K hK C hbound (hI a) n)
(coarseTransform_memLp K hK C hbound hM n)).mp
(tendsto_coarseTransform K hK C hbound hI hM n hlim)
refine h.congr (fun a => ?_)
congr 1
funext ω
rw [coarseTransform, BrownianGrid.integralSum_compose_refined B (t a) (H a) K n (hk a)]
theorem integralSum_ae_coarseTransform
(B : ℝ≥0 → Ω → ℝ) (t : ℕ → ℝ≥0) (H K : ℕ → Ω → ℝ)
{k : ℕ} (hk : 0 < k) (I : ℕ → Ω → ℝ)
(hrep : ∀ j, I j =ᵐ[μ] BrownianGrid.integralSum B t H (j * k)) (n : ℕ) :
BrownianGrid.integralSum B t (fun i ω => K (i / k) ω * H i ω) (n * k) =ᵐ[μ]
coarseTransform K I n := by
filter_upwards [ae_all_iff.mpr hrep] with ω hω
rw [← BrownianGrid.integralSum_compose_refined B t H K n hk]
simp only [coarseTransform]
simp_rw [hω]
theorem integralSum_compose_varying_grid_tendsto [IsFiniteMeasure μ]
{F : Filtration ℝ≥0 mΩ}
{M : ℝ≥0 → Ω → ℝ} {I : ℕ → ℝ≥0 → Ω → ℝ}
(hM : Martingale M F μ) (hI : ∀ n, Martingale (I n) F μ)
(hM2 : ∀ t, MemLp (M t) 2 μ) (hI2 : ∀ n t, MemLp (I n t) 2 μ)
(B : ℝ≥0 → Ω → ℝ) (t : ℕ → ℕ → ℝ≥0) (ht : ∀ n, Monotone (t n))
(H K : ℕ → ℕ → Ω → ℝ) (k N : ℕ → ℕ) (hk : ∀ n, 0 < k n)
(hK : ∀ n j, StronglyMeasurable[F (t n (j * k n))] (K n j))
{C : ℝ} (hC : 0 ≤ C) (hbound : ∀ n j, ∀ᵐ ω ∂μ, |K n j ω| ≤ C)
{T : ℝ≥0} (hterminal : ∀ n, t n (N n * k n) = T)
(hrep : ∀ n j, I n (t n (j * k n)) =ᵐ[μ]
BrownianGrid.integralSum B (t n) (H n) (j * k n))
(hlim : Tendsto (fun n => ∫ ω, (I n T ω - M T ω) ^ 2 ∂μ) atTop (𝓝 0)) :
Tendsto (fun n => ∫ ω,
(BrownianGrid.integralSum B (t n)
(fun i ω => K n (i / k n) ω * H n i ω) (N n * k n) ω -
coarseTransform (K n) (fun j => M (t n (j * k n))) (N n) ω) ^ 2 ∂μ)
atTop (𝓝 0) := by
have herr := MartingaleTransformL2.tendsto_integral_sq_grid_transform
(fun n => (hI n).sub hM) (fun n s => (hI2 n s).sub (hM2 s))
(fun n j => t n (j * k n))
(fun n i j hij => ht n (Nat.mul_le_mul_right (k n) hij)) N hterminal K hK hC hbound hlim
refine herr.congr (fun n => ?_)
apply integral_congr_ae
filter_upwards [ae_all_iff.mpr (hrep n)] with ω hω
rw [← BrownianGrid.integralSum_compose_refined B (t n) (H n) (K n) (N n) (hk n)]
congr 1
simp only [Pi.sub_apply, coarseTransform]
simp_rw [hω]
simp only [mul_sub, Finset.sum_sub_distrib]
ring
theorem integralSum_compose_limit [IsFiniteMeasure μ]
{F : Filtration ℝ≥0 mΩ}
{M : ℝ≥0 → Ω → ℝ} {I : ℕ → ℝ≥0 → Ω → ℝ}
(hM : Martingale M F μ) (hI : ∀ n, Martingale (I n) F μ)
(hM2 : ∀ t, MemLp (M t) 2 μ) (hI2 : ∀ n t, MemLp (I n t) 2 μ)
(B : ℝ≥0 → Ω → ℝ) (t : ℕ → ℕ → ℝ≥0) (ht : ∀ n, Monotone (t n))
(H K : ℕ → ℕ → Ω → ℝ) (k N : ℕ → ℕ) (hk : ∀ n, 0 < k n)
(hK : ∀ n j, StronglyMeasurable[F (t n (j * k n))] (K n j))
{C : ℝ} (hC : 0 ≤ C) (hbound : ∀ n j, ∀ᵐ ω ∂μ, |K n j ω| ≤ C)
{T : ℝ≥0} (hterminal : ∀ n, t n (N n * k n) = T)
(hrep : ∀ n j, I n (t n (j * k n)) =ᵐ[μ]
BrownianGrid.integralSum B (t n) (H n) (j * k n))
(hlim : Tendsto (fun n => ∫ ω, (I n T ω - M T ω) ^ 2 ∂μ) atTop (𝓝 0))
{Z : Ω → ℝ} (hZ : MemLp Z 2 μ)
(hproduct : Tendsto (fun n => ∫ ω,
(BrownianGrid.integralSum B (t n)
(fun i ω => K n (i / k n) ω * H n i ω) (N n * k n) ω - Z ω) ^ 2 ∂μ)
atTop (𝓝 0)) :
Tendsto (fun n => ∫ ω,
(coarseTransform (K n) (fun j => M (t n (j * k n))) (N n) ω - Z ω) ^ 2 ∂μ)
atTop (𝓝 0) := by
have hKm (n j : ℕ) : AEStronglyMeasurable (K n j) μ :=
((hK n j).mono (F.le _)).aestronglyMeasurable
have houter (n : ℕ) := coarseTransform_memLp (K n) (hKm n) (fun _ => C)
(hbound n) (fun j => hM2 (t n (j * k n))) (N n)
have hfine (n : ℕ) : MemLp (BrownianGrid.integralSum B (t n)
(fun i ω => K n (i / k n) ω * H n i ω) (N n * k n)) 2 μ := by
apply MemLp.ae_eq
(integralSum_ae_coarseTransform B (t n) (H n) (K n) (hk n)
(fun j => I n (t n (j * k n))) (hrep n) (N n)).symm
exact coarseTransform_memLp (K n) (hKm n) (fun _ => C) (hbound n)
(fun j => hI2 n (t n (j * k n))) (N n)
exact meanSquare_limit_of_close hfine houter hZ
(integralSum_compose_varying_grid_tendsto hM hI hM2 hI2 B t ht H K k N hk
hK hC hbound hterminal hrep hlim) hproduct
end Limits
end Problem367.SimpleIntegral
end
end OAI
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