| import Mathlib |
| import OAI.Probability.LogConcave.Dynamics.TriangularEquiv |
| import OAI.Probability.LogConcave.Sampling.MeasureSmoothLipschitz |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| namespace LogConcaveSampling.GlobalODE |
| open Set Function Filter MeasureTheory |
| open scoped Topology NNReal |
|
|
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
|
|
| theorem stationary_integral {f : E → E} (hf : ContDiff ℝ 1 f) |
| {K : ℝ≥0} (hL : LipschitzWith K f) {μ : Measure E} [IsFiniteMeasure μ] |
| (hfi : Integrable (fun x => ‖f x‖) μ) |
| (hflux : ∀ψ : E → ℝ,ContDiff ℝ 1 ψ → |
| (∃L : ℝ≥0,LipschitzWith L ψ) → (∃B : ℝ,∀x,‖ψ x‖≤B) → |
| (∫x,(fderiv ℝ ψ x) (f x) ∂μ)=0) |
| {a b : ℝ} (hab : a<b) {φ : E → ℝ} (hφ : ContDiff ℝ 1 φ) |
| {L : ℝ≥0} (hφL : LipschitzWith L φ) {B : ℝ} (hφB : ∀x,‖φ x‖≤B) : |
| (∫y,φ (flow (fun _ _ => hL) (hf.continuous.comp continuous_snd) |
| ⟨a,le_rfl,hab.le⟩ y b) ∂μ)=∫y,φ y ∂μ := by |
| let hc : Continuous (uncurry (fun (_ : ℝ) => f)) := hf.continuous.comp continuous_snd |
| let hD : Continuous (uncurry (fun (_ : ℝ) => fderiv ℝ f)) := |
| (hf.continuous_fderiv (by norm_num)).comp continuous_snd |
| let hd : ∀ (_ : ℝ) (z : E),HasFDerivAt f (fderiv ℝ f z) z := |
| fun _ z => (hf.differentiable (by norm_num) z).hasFDerivAt |
| let T : Icc a b := ⟨b,hab.le,le_rfl⟩ |
| let ψ := backwardTest hab.le hc (fun _ _ => hL) T φ |
| let S := fun t => fderiv ℝ (fun y => ψ (t,y)) |
| let C : ℝ := (L:ℝ)*Real.exp ((K:ℝ)*(b-a)) |
| have hC : 0≤C := by dsimp [C]; positivity |
| have hψ : Continuous ψ := backwardTest_continuous hab.le hc (fun _ _ => hL) T hφ.continuous |
| have hψc (t : ℝ) : ContDiff ℝ 1 (fun y => ψ (t,y)) := by |
| have he : (fun y => ψ (t,y))=φ ∘ (fun y => |
| flow (fun _ _ => hL) hc (projIcc a b hab.le t) y T) := by |
| funext y |
| exact backwardTest_eq hab.le hc (fun _ _ => hL) T φ (t,y) |
| rw [he] |
| exact hφ.comp (flow_contDiff_one hab.le hc (fun _ _ => hL) hD hd _ T) |
| have hψL (t : ℝ) := backwardTest_lipschitz hab.le hc (fun _ _ => hL) T hφL t |
| have hψB (t : ℝ) (y : E) : ‖ψ (t,y)‖≤B := hφB _ |
| have hSb (t : ℝ) (y : E) : ‖S t y‖≤C := norm_fderiv_le_of_lipschitz ℝ (hψL t) |
| have him (t : ℝ) : Integrable (fun y => ψ (t,y)) μ := |
| (integrable_const B).mono' (hψ.measurable.comp (measurable_const.prodMk measurable_id)).aestronglyMeasurable |
| (Eventually.of_forall (hψB t)) |
| have hdm (t : ℝ) : Measurable (fun y => -(S t y (f y))) := |
| ((continuous_fst.clm_apply continuous_snd).measurable.comp |
| ((measurable_fderiv ℝ (fun y => ψ (t,y))).prodMk hf.continuous.measurable)).neg |
| have hdb (t : ℝ) (y : E) : ‖-(S t y (f y))‖≤C*‖f y‖ := by |
| rw [norm_neg] |
| exact (ContinuousLinearMap.le_opNorm _ _).trans (mul_le_mul_of_nonneg_right (hSb t y) (norm_nonneg _)) |
| have hpd (t : ℝ) (ht : t∈Ioo a b) (y : E) : |
| HasDerivAt (fun t => ψ (t,y)) (-(S t y (f y))) t := by |
| have hh := backwardTest_chain hab.le hc (fun _ _ => hL) hD hd T |
| (hφ.differentiable (by norm_num)) ht (hasDerivAt_const t y) |
| simpa only [zero_sub,map_neg] using hh |
| have hQd (t : ℝ) (ht : t∈Ioo a b) : HasDerivAt (fun t => ∫y,ψ (t,y) ∂μ) 0 t := by |
| have hh := hasDerivAt_integral_of_dominated_loc_of_deriv_le |
| (F:=fun t y => ψ (t,y)) (F':=fun t y => -(S t y (f y))) |
| (Ioo_mem_nhds ht.1 ht.2) |
| (Eventually.of_forall fun _ => |
| (hψ.measurable.comp (measurable_const.prodMk measurable_id)).aestronglyMeasurable) |
| (him t) (hdm t).aestronglyMeasurable |
| (Eventually.of_forall fun y s _ => hdb s y) (hfi.const_mul C) |
| (Eventually.of_forall fun y s hs => hpd s hs y) |
| have hz := hflux (fun y => ψ (t,y)) (hψc t) ⟨_,hψL t⟩ ⟨B,hψB t⟩ |
| have he : (∫y,-(S t y (f y)) ∂μ)=0 := by rw [integral_neg]; exact neg_eq_zero.mpr hz |
| rw [he] at hh |
| exact hh.2 |
| have hQc : Continuous (fun t => ∫y,ψ (t,y) ∂μ) := |
| continuous_of_dominated |
| (fun t => (hψ.measurable.comp (measurable_const.prodMk measurable_id)).aestronglyMeasurable) |
| (fun t => Eventually.of_forall (hψB t)) (integrable_const B) |
| (Eventually.of_forall fun y => hψ.comp (continuous_id.prodMk continuous_const)) |
| obtain ⟨t,ht,he⟩ := exists_deriv_eq_slope (fun t => ∫y,ψ (t,y) ∂μ) hab hQc.continuousOn |
| (fun t ht => (hQd t ht).differentiableAt.differentiableWithinAt) |
| rw [(hQd t ht).deriv] at he |
| have heq : (∫y,ψ (b,y) ∂μ)=(∫y,ψ (a,y) ∂μ) := |
| sub_eq_zero.mp ((div_eq_zero_iff).mp he.symm |>.resolve_right (sub_ne_zero.mpr hab.ne')) |
| have ha : (fun y => ψ (a,y))=(fun y => φ (flow (fun _ _ => hL) hc ⟨a,le_rfl,hab.le⟩ y b)) := by |
| funext y |
| dsimp [ψ] |
| rw [backwardTest_eq,projIcc_of_mem _ (show a∈Icc a b from ⟨le_rfl,hab.le⟩)] |
| have hb : (fun y => ψ (b,y))=φ := by |
| funext y |
| dsimp [ψ] |
| rw [backwardTest_eq,projIcc_of_mem _ (show b∈Icc a b from ⟨hab.le,le_rfl⟩)] |
| exact congrArg φ (flow_initial (fun _ _ => hL) hc T y) |
| rw [ha,hb] at heq |
| exact heq.symm |
| |
| theorem stationary_law {f : E → E} (hf : ContDiff ℝ 1 f) |
| {K : ℝ≥0} (hL : LipschitzWith K f) {μ : Measure E} [IsFiniteMeasure μ] |
| (hfi : Integrable (fun x => ‖f x‖) μ) |
| (hflux : ∀ψ : E → ℝ,ContDiff ℝ 1 ψ → |
| (∃L : ℝ≥0,LipschitzWith L ψ) → (∃B : ℝ,∀x,‖ψ x‖≤B) → |
| (∫x,(fderiv ℝ ψ x) (f x) ∂μ)=0) |
| {a b : ℝ} (hab : a<b) : |
| μ.map (fun y => flow (fun _ _ => hL) (hf.continuous.comp continuous_snd) |
| ⟨a,le_rfl,hab.le⟩ y b)=μ := by |
| apply measure_eq_of_smooth_lipschitz |
| intro φ hφ hφL hφB |
| obtain ⟨L,hφL⟩ := hφL |
| obtain ⟨B,hφB⟩ := hφB |
| rw [integral_map (flow_lipschitz (fun _ _ => hL) (hf.continuous.comp continuous_snd) |
| ⟨a,le_rfl,hab.le⟩ ⟨b,hab.le,le_rfl⟩).continuous.aemeasurable hφ.continuous.aestronglyMeasurable] |
| exact stationary_integral hf hL hfi hflux hab hφ hφL hφB |
| end LogConcaveSampling.GlobalODE |
| |
| end |
| |
| end |
| |
| end |
| |
| end OAI |
| |
