| import OAI.Probability.InvariantIsing.Fields.FieldCanonicalEndpoint |
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| namespace OAI |
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| /-! Exponential endpoint bounds compose additively over Markov kernels. |
| The proof includes integrability, so no undefined Bochner integral is |
| used in the recursive Gaussian moment estimates. -/ |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory |
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| namespace InvariantIsing |
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| theorem kernel_exp_bound_comp (κ η : Kernel ℝ ℝ) [IsMarkovKernel κ] [IsMarkovKernel η] |
| (t A B : ℝ) |
| (hκ : ∀ z, Integrable (fun y => Real.exp (t * y)) (κ z) ∧ |
| (∫ y, Real.exp (t * y) ∂κ z) ≤ Real.exp (t * z + A)) |
| (hη : ∀ z, Integrable (fun y => Real.exp (t * y)) (η z) ∧ |
| (∫ y, Real.exp (t * y) ∂η z) ≤ Real.exp (t * z + B)) (z : ℝ) : |
| Integrable (fun y => Real.exp (t * y)) ((η ∘ₖ κ) z) ∧ |
| (∫ y, Real.exp (t * y) ∂(η ∘ₖ κ) z) ≤ Real.exp (t * z + A + B) := by |
| have hm : StronglyMeasurable (fun y : ℝ => Real.exp (t * y)) := by fun_prop |
| have hH : Integrable (fun x => ∫ y, Real.exp (t * y) ∂η x) (κ z) := by |
| apply ((hκ z).1.const_mul (Real.exp B)).mono' hm.integral_kernel.aestronglyMeasurable |
| apply ae_of_all |
| intro x |
| rw [Real.norm_eq_abs, abs_of_nonneg (integral_nonneg (fun _ => (Real.exp_pos _).le))] |
| simpa only [Real.exp_add, mul_comm] using (hη x).2 |
| have hI : Integrable (fun y => Real.exp (t * y)) ((η ∘ₖ κ) z) := by |
| apply (ProbabilityTheory.integrable_comp_iff hm.aestronglyMeasurable).mpr |
| refine ⟨ae_of_all _ (fun x => (hη x).1), ?_⟩ |
| simpa only [Real.norm_eq_abs, abs_of_pos (Real.exp_pos _)] using hH |
| refine ⟨hI, ?_⟩ |
| rw [Kernel.integral_comp hI] |
| calc |
| _ ≤ ∫ x, Real.exp B * Real.exp (t * x) ∂κ z := |
| integral_mono hH ((hκ z).1.const_mul (Real.exp B)) |
| (fun x => by simpa only [Real.exp_add, mul_comm] using (hη x).2) |
| _ = Real.exp B * (∫ x, Real.exp (t * x) ∂κ z) := integral_const_mul _ _ |
| _ ≤ Real.exp B * Real.exp (t * z + A) := |
| mul_le_mul_of_nonneg_left (hκ z).2 (Real.exp_pos B).le |
| _ = _ := by rw [← Real.exp_add]; congr 1; ring |
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| end InvariantIsing |
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| end |
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| end OAI |
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