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f18e7de | 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 | import OAI.MathematicalPhysics.Boltzmann.Branching.FiveGainPullback
namespace OAI
/-! # A finite parameterized gain chain as one product integral
This is the Fubini assembly used after replacing each physical collision by its
thermal and redundant-plane parameters. The transition maps and densities are
explicit functions, and the conclusion is an exact integral identity.
-/
noncomputable section
open MeasureTheory
open scoped ENNReal
namespace BoltzmannNonuniqueness.FiveGainMap
def parameterChain (next : ℕ → State → StageControls → State)
(density : ℕ → State → StageControls → ℝ≥0∞) (g : State → ℝ≥0∞)
(i : ℕ) : ℕ → State → ℝ≥0∞
| 0, z => g z
| n+1, z => ∫⁻ p : StageControls, density i z p*
parameterChain next density g (i+1) n (next i z p)
def chainIntegrand (next : ℕ → State → StageControls → State)
(density : ℕ → State → StageControls → ℝ≥0∞) (g : State → ℝ≥0∞)
(i : ℕ) : (n : ℕ) → State → (Fin n → StageControls) → ℝ≥0∞
| 0, z, _ => g z
| n+1, z, q => density i z (q 0)*
chainIntegrand next density g (i+1) n (next i z (q 0)) (Fin.tail q)
theorem integrateControls_const_mul {c : ℝ≥0∞} (hc : c ≠ ∞) (n : ℕ)
(F : (Fin n → StageControls) → ℝ≥0∞) :
integrateControls n (fun q => c*F q) = c*integrateControls n F := by
induction n with
| zero => rfl
| succ n ih =>
simp only [integrateControls,ih]
exact lintegral_const_mul' _ _ hc
theorem parameterChain_eq_nested (next : ℕ → State → StageControls → State)
(density : ℕ → State → StageControls → ℝ≥0∞)
(hfinite : ∀ i z p, density i z p ≠ ∞) (g : State → ℝ≥0∞) (i n : ℕ) (z : State) :
parameterChain next density g i n z =
integrateControls n (chainIntegrand next density g i n z) := by
induction n generalizing i z with
| zero => rfl
| succ n ih =>
simp only [parameterChain,integrateControls]
apply lintegral_congr
intro p
rw [ih]
have he := integrateControls_const_mul (hfinite i z p) n
(chainIntegrand next density g (i+1) n (next i z p))
simpa only [chainIntegrand,Fin.cons_zero,Fin.tail_cons] using he.symm
theorem chainIntegrand_measurable (next : ℕ → State → StageControls → State)
(density : ℕ → State → StageControls → ℝ≥0∞)
(hnext : ∀ i, Measurable (fun p : State × StageControls => next i p.1 p.2))
(hdensity : ∀ i, Measurable (fun p : State × StageControls => density i p.1 p.2))
{g : State → ℝ≥0∞} (hg : Measurable g) (i n : ℕ) :
Measurable (fun p : State × (Fin n → StageControls) =>
chainIntegrand next density g i n p.1 p.2) := by
induction n generalizing i with
| zero => exact hg.comp measurable_fst
| succ n ih =>
have hfirst : Measurable (fun p : State × (Fin (n+1) → StageControls) => (p.1,p.2 0)) :=
measurable_fst.prodMk ((measurable_pi_apply 0).comp measurable_snd)
have htail : Measurable (fun p : State × (Fin (n+1) → StageControls) => Fin.tail p.2) := by
exact Measurable.of_eval (fun j => (measurable_pi_apply j.succ).comp measurable_snd)
exact ((hdensity i).comp hfirst).mul
((ih (i+1)).comp (((hnext i).comp hfirst).prodMk htail))
theorem parameterChain_eq_product (next : ℕ → State → StageControls → State)
(density : ℕ → State → StageControls → ℝ≥0∞)
(hnext : ∀ i, Measurable (fun p : State × StageControls => next i p.1 p.2))
(hdensity : ∀ i, Measurable (fun p : State × StageControls => density i p.1 p.2))
(hfinite : ∀ i z p, density i z p ≠ ∞) {g : State → ℝ≥0∞} (hg : Measurable g)
(i n : ℕ) (z : State) :
parameterChain next density g i n z =
∫⁻ q : Fin n → StageControls, chainIntegrand next density g i n z q := by
rw [parameterChain_eq_nested next density hfinite g i n z]
exact integrateControls_eq n ((chainIntegrand_measurable next density hnext hdensity hg i n).comp
measurable_prodMk_left)
end BoltzmannNonuniqueness.FiveGainMap
end
end OAI
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