| import Mathlib |
| import OAI.Analysis.HardSphere.Bridges.BridgeResidual |
|
|
| namespace OAI |
|
|
| section |
|
|
| section |
| noncomputable section |
| open MeasureTheory Set Filter ProbabilityTheory |
| open scoped ENNReal NNReal Topology BigOperators |
| namespace HardSphere.Wiener |
|
|
| def bridgePrefix (u : Time) : C(Time,ℝ) →L[ℝ] C(Time,ℝ) := centerMap.comp (prefixMap u) |
| @[simp] lemma bridgePrefix_apply (u : Time) (f : C(Time,ℝ)) (s : Time) : |
| bridgePrefix u f s=f (prefixTime u s)-(s:ℝ)*f u := by |
| simp only [bridgePrefix,ContinuousLinearMap.comp_apply,centerMap_apply,prefixMap_apply] |
| congr 2 |
| congr 1 |
| apply Subtype.ext |
| simp [prefixTime] |
|
|
| lemma bridgePrefix_eq (u : Time) (ω : Noise) : |
| bridgePrefix u (bridge ω)=centerMap (prefixMap u (path ω)) := by |
| ext s |
| simp only [bridgePrefix_apply,bridge_apply,centerMap_apply,prefixMap_apply] |
| have he : prefixTime u 1=u := by ext; simp [prefixTime] |
| rw [he] |
| change path ω (prefixTime u s)-((u:ℝ)*(s:ℝ))*path ω 1- |
| (s:ℝ)*(path ω u-(u:ℝ)*path ω 1)=_ |
| ring |
|
|
| lemma bridgePrefix_law (u : Time) : |
| law.map (fun ω ↦ bridgePrefix u (bridge ω))= |
| bridgeLaw.map (fun β ↦ Real.sqrt (u:ℝ) • β) := by |
| calc |
| _ = (law.map (fun ω ↦ prefixMap u (path ω))).map centerMap := by |
| rw [Measure.map_map (f:=fun ω ↦ prefixMap u (path ω)) centerMap.continuous.measurable |
| ((prefixMap u).continuous.measurable.comp measurable_path)] |
| congr 1 |
| funext ω |
| exact bridgePrefix_eq u ω |
| _ = _ := by |
| rw [prefix_law,Measure.map_map (f:=fun ω ↦ Real.sqrt (u:ℝ) • path ω) centerMap.continuous.measurable ((show Continuous (fun w : C(Time,ℝ) ↦ Real.sqrt (u:ℝ) • w) from by fun_prop).measurable.comp measurable_path), |
| bridgeLaw,Measure.map_map (by fun_prop) measurable_bridge] |
| congr 1 |
| funext ω |
| exact (centerMap.map_smul _ _).trans (congrArg (fun β ↦ Real.sqrt (u:ℝ) • β) (centerMap_path ω)) |
|
|
| lemma prefixRemainder_gaussian (u : Time) : |
| IsGaussianProcess (Sum.elim (fun s ω ↦ prefixMap u (bridge ω) s) |
| (fun s ω ↦ bridgeRemainder u (bridge ω) s)) law := by |
| classical |
| constructor |
| intro I |
| let loc : Option I → Time := fun i ↦ i.elim u |
| (fun j ↦ j.val.elim (prefixTime u) (suffixTime u)) |
| let ev (i : Option I) : (Option I → ℝ) →L[ℝ] ℝ := ContinuousLinearMap.proj i |
| let L : (Option I → ℝ) →L[ℝ] (I → ℝ) := ContinuousLinearMap.pi |
| (fun j ↦ match j.val with |
| | .inl _ => ev (some j) |
| | .inr s => ev (some j) - (1-(s:ℝ)) • ev none) |
| convert! (finite_bridge_gaussian loc).map_fun L using 1 |
| funext ω j |
| cases hj : j.val <;> simp [L,loc,ev,hj] |
|
|
| lemma prefixRemainder_independent (u : Time) : |
| IndepFun (fun ω ↦ prefixMap u (bridge ω)) (fun ω ↦ bridgeRemainder u (bridge ω)) law := by |
| have hc (a s : Time) : |
| cov[fun ω ↦ prefixMap u (bridge ω) a,fun ω ↦ bridgeRemainder u (bridge ω) s;law]=0 := by |
| simp only [prefixMap_apply,bridgeRemainder_apply] |
| rw [covariance_fun_sub_right (bridge_eval_law _).hasGaussianLaw.memLp_two |
| (bridge_eval_law _).hasGaussianLaw.memLp_two ((bridge_eval_law u).hasGaussianLaw.memLp_two.const_mul _), |
| covariance_const_mul_right,bridge_covariance,bridge_covariance, |
| min_eq_left ((prefixTime_le u a).trans (le_suffixTime u s)),min_eq_left (prefixTime_le u a)] |
| change (u:ℝ)*(a:ℝ)-((u:ℝ)*(a:ℝ))*((u:ℝ)+(1-(u:ℝ))*(s:ℝ))- |
| (1-(s:ℝ))*((u:ℝ)*(a:ℝ)-((u:ℝ)*(a:ℝ))*(u:ℝ))=0 |
| ring |
| have hi := (prefixRemainder_gaussian u).indepFun_of_covariance_eq_zero |
| (fun s ↦ ((continuous_eval_const s).measurable.comp ((prefixMap u).continuous.measurable.comp measurable_bridge)).aemeasurable) |
| (fun s ↦ ((continuous_eval_const s).measurable.comp ((bridgeRemainder u).continuous.measurable.comp measurable_bridge)).aemeasurable) hc |
| have hid := hi.comp |
| (Measurable.of_eval fun n ↦ measurable_pi_apply (TopologicalSpace.denseSeq Time n)) |
| (Measurable.of_eval fun n ↦ measurable_pi_apply (TopologicalSpace.denseSeq Time n)) |
| have hh := hid.comp denseEval_embedding.measurable_invFun denseEval_embedding.measurable_invFun |
| convert hh using 1 <;> funext ω <;> exact (denseEval_embedding.leftInverse_invFun _).symm |
|
|
| lemma prefixValue_gaussian (u : Time) : |
| IsGaussianProcess (Sum.elim (fun _ : Unit ↦ fun ω ↦ bridge ω u) |
| (fun s ω ↦ bridgePrefix u (bridge ω) s)) law := by |
| classical |
| constructor |
| intro I |
| let loc : Option I → Time := fun i ↦ i.elim u |
| (fun j ↦ j.val.elim (fun _ ↦ u) (prefixTime u)) |
| let ev (i : Option I) : (Option I → ℝ) →L[ℝ] ℝ := ContinuousLinearMap.proj i |
| let L : (Option I → ℝ) →L[ℝ] (I → ℝ) := ContinuousLinearMap.pi |
| (fun j ↦ match j.val with |
| | .inl _ => ev (some j) |
| | .inr s => ev (some j) - (s:ℝ) • ev none) |
| convert! (finite_bridge_gaussian loc).map_fun L using 1 |
| funext ω j |
| cases hj : j.val <;> simp [L,loc,ev,hj] |
|
|
| lemma prefixValue_independent (u : Time) : |
| IndepFun (fun ω ↦ bridge ω u) (fun ω ↦ bridgePrefix u (bridge ω)) law := by |
| have hc (_ : Unit) (s : Time) : |
| cov[fun ω ↦ bridge ω u,fun ω ↦ bridgePrefix u (bridge ω) s;law]=0 := by |
| simp only [bridgePrefix_apply] |
| rw [covariance_fun_sub_right (bridge_eval_law _).hasGaussianLaw.memLp_two |
| (bridge_eval_law _).hasGaussianLaw.memLp_two ((bridge_eval_law u).hasGaussianLaw.memLp_two.const_mul _), |
| covariance_const_mul_right,bridge_covariance,bridge_covariance, |
| min_eq_right (prefixTime_le u s),min_self] |
| change (u:ℝ)*(s:ℝ)-(u:ℝ)*((u:ℝ)*(s:ℝ))- |
| (s:ℝ)*((u:ℝ)-(u:ℝ)*(u:ℝ))=0 |
| ring |
| have hi := (prefixValue_gaussian u).indepFun_of_covariance_eq_zero |
| (fun _ ↦ ((continuous_eval_const u).measurable.comp measurable_bridge).aemeasurable) |
| (fun s ↦ ((continuous_eval_const s).measurable.comp ((bridgePrefix u).continuous.measurable.comp measurable_bridge)).aemeasurable) hc |
| have hid := hi.comp (measurable_pi_apply ()) |
| (Measurable.of_eval fun n ↦ measurable_pi_apply (TopologicalSpace.denseSeq Time n)) |
| have hh := hid.comp measurable_id denseEval_embedding.measurable_invFun |
| convert hh using 1 |
| · rfl |
| · funext ω |
| exact (denseEval_embedding.leftInverse_invFun _).symm |
|
|
| lemma bridge_three_law (u : Time) : |
| bridgeLaw.map (fun β ↦ ((β u,bridgePrefix u β),bridgeRemainder u β))= |
| ((gaussianReal 0 (Real.toNNReal ((u:ℝ)-(u:ℝ)^2))).prod |
| (bridgeLaw.map (fun β ↦ Real.sqrt (u:ℝ) • β))).prod |
| (bridgeLaw.map (fun β ↦ Real.sqrt (1-(u:ℝ)) • β)) := by |
| have hi := (prefixRemainder_independent u).comp |
| (show Measurable (fun f : C(Time,ℝ) ↦ (f 1,centerMap f)) from by fun_prop) measurable_id |
| have he : (fun ω ↦ (prefixMap u (bridge ω) 1,centerMap (prefixMap u (bridge ω))))= |
| (fun ω ↦ (bridge ω u,bridgePrefix u (bridge ω))) := by |
| funext ω |
| apply Prod.ext |
| · simp only [prefixMap_apply] |
| congr 1 |
| apply Subtype.ext |
| simp [prefixTime] |
| · rfl |
| simp only [Function.comp_def,id_eq] at hi |
| rw [he] at hi |
| rw [bridgeLaw,Measure.map_map (by fun_prop) measurable_bridge] |
| simp only [Function.comp_def] |
| have hh := hi.map_prod_eq_prod_map_map |
| ((((continuous_eval_const u).measurable.comp measurable_bridge).prodMk |
| ((bridgePrefix u).continuous.measurable.comp measurable_bridge)).aemeasurable) |
| (((bridgeRemainder u).continuous.measurable.comp measurable_bridge).aemeasurable) |
| simp only [Function.comp_def] at hh |
| rw [hh] |
| have hp := (prefixValue_independent u).map_prod_eq_prod_map_map |
| ((continuous_eval_const u).measurable.comp measurable_bridge).aemeasurable |
| ((bridgePrefix u).continuous.measurable.comp measurable_bridge).aemeasurable |
| simp only [Function.comp_def] at hp |
| rw [hp,(bridge_eval_law u).map_eq,bridgePrefix_law,bridgeRemainder_law] |
| rfl |
|
|
| end HardSphere.Wiener |
|
|
| end |
| end |
|
|
| end |
|
|
| end OAI |
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|
