File size: 3,648 Bytes
9b6e014 | 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 | import OAI.Probability.ContinuumTransition.IdealSlopes
namespace OAI
universe u_1 u_2
section
open scoped Topology
open Set Filter Metric
noncomputable section
namespace ContinuumTemperature
lemma locallyUniform_reindex {ι : Type u_1} {κ : Type u_2} {l : Filter ι} {k : Filter κ}
{Q : ι → Parameter → ℝ} {q : Parameter → ℝ}
(hQ : TendstoLocallyUniformlyOn Q q l pressureDomain)
{θ : κ → ι} (hθ : Tendsto θ k l) :
TendstoLocallyUniformlyOn (fun n => Q (θ n)) q k pressureDomain := by
intro u hu x hx
obtain ⟨t,ht,h⟩ := hQ u hu x hx
exact ⟨t,ht,hθ.eventually h⟩
lemma quadraticCost_tendsto {Q : ℕ → Parameter → ℝ} {q : Parameter → ℝ}
(hQ : TendstoLocallyUniformlyOn Q q atTop pressureDomain)
(hq : ContinuousOn q pressureDomain)
{a d : ℕ → ℝ} {a₀ d₀ : ℝ} (ha : Tendsto a atTop (𝓝 a₀))
(hd : Tendsto d atTop (𝓝 d₀)) {X : ℕ → Parameter} {x : Parameter}
(hx : x ∈ pressureDomain) (hX : Tendsto X atTop (𝓝 x)) :
Tendsto (fun n => quadraticCost (Q n) (a n) (X n) (d n)) atTop
(𝓝 (quadraticCost q a₀ x d₀)) := by
have hX1 := (continuous_fst.tendsto x).comp hX
have hX2 := (continuous_snd.tendsto x).comp hX
have hf : Tendsto (fun n => shiftedField (a n) (X n) (d n)) atTop
(𝓝 (shiftedField a₀ x d₀)) :=
hX1.prodMk_nhds (hX2.add ((hX1.mul ha).mul hd))
have he : ∀ᶠ n in atTop, shiftedField (a n) (X n) (d n) ∈ pressureDomain :=
(hX.eventually (isOpen_pressureDomain.mem_nhds hx))
have hval := hQ.tendsto_comp (hq _ (show shiftedField a₀ x d₀ ∈ pressureDomain from hx))
(show shiftedField a₀ x d₀ ∈ pressureDomain from hx) (tendsto_nhdsWithin_iff.2 ⟨hf,he⟩)
exact hval.sub (((hX1.mul ha).mul (hd.pow 2)).div_const 2)
lemma meanField_tendsto_of_bounded_maximizers {Q : ℕ → Parameter → ℝ} {q : Parameter → ℝ}
{P : ℕ → Parameter → ℝ} {p : Parameter → ℝ} {a : ℕ → ℝ} {a₀ : ℝ}
(hQ : TendstoLocallyUniformlyOn Q q atTop pressureDomain) (hq : ContinuousOn q pressureDomain)
(ha : Tendsto a atTop (𝓝 a₀)) (hP : ∀ n, IsMeanField (Q n) (a n) (P n))
(hp : IsMeanField q a₀ p) {X : ℕ → Parameter} {x : Parameter}
(hx : x ∈ pressureDomain) (hX : Tendsto X atTop (𝓝 x))
(hXS : ∀ n, X n ∈ pressureDomain) {D : ℝ}
(hatt : ∀ n, ∃ d ∈ Icc 0 D, quadraticCost (Q n) (a n) (X n) d = P n (X n)) :
Tendsto (fun n => P n (X n)) atTop (𝓝 (p x)) := by
choose d hd he using hatt
apply metric_tendsto_of_subseq
intro φ hφ
obtain ⟨r,_,ψ,hψ,hr⟩ := (isCompact_Icc : IsCompact (Icc (0:ℝ) D)).tendsto_subseq
(fun n => hd (φ n))
let θ := φ ∘ ψ
have hθ := (hφ.comp hψ).tendsto_atTop
have hlim := quadraticCost_tendsto (locallyUniform_reindex hQ hθ) hq (ha.comp hθ) hr hx (hX.comp hθ)
have hle : quadraticCost q a₀ x r ≤ p x := (hp x hx).1 (mem_range_self r)
have hge : p x ≤ quadraticCost q a₀ x r := by
apply (hp x hx).2
rintro _ ⟨s,rfl⟩
have hs := quadraticCost_tendsto (locallyUniform_reindex hQ hθ) hq (ha.comp hθ)
(tendsto_const_nhds (x := s)) hx (hX.comp hθ)
apply le_of_tendsto_of_tendsto hs hlim
exact Eventually.of_forall fun n => by
change quadraticCost (Q (θ n)) (a (θ n)) (X (θ n)) s ≤
quadraticCost (Q (θ n)) (a (θ n)) (X (θ n)) (d (θ n))
rw [he]
exact (hP (θ n) _ (hXS (θ n))).1 (mem_range_self s)
have heq := le_antisymm hle hge
refine ⟨ψ,hψ,?_⟩
rw [heq] at hlim
exact hlim.congr (fun n => he (θ n))
end ContinuumTemperature
end
end
end OAI
|