File size: 3,332 Bytes
c4cdbbd | 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 | import OAI.Probability.InvariantIsing.Fields.SimpleFieldTransport
namespace OAI
/-! The extended magnetic value is one-Lipschitz for first-moment transport. -/
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped Topology ENNReal
namespace InvariantIsing
lemma magneticFieldFunctional_coupling_le
(ν μ ξ : ProbabilityMeasure ℝ) (a b : ℝ)
(hcompact : IsCompact (ν : Measure ℝ).support)
(hbound : (ν : Measure ℝ).support ⊆ Icc a b)
(ha : a∈(ν : Measure ℝ).support) (hb : b∈(ν : Measure ℝ).support)
(hμ : Integrable (fun x : ℝ => x) (μ : Measure ℝ))
(hξ : Integrable (fun x : ℝ => x) (ξ : Measure ℝ)) (p : FieldLawCoupling μ ξ) :
|magneticFieldFunctional (ν : Measure ℝ) b μ-magneticFieldFunctional (ν : Measure ℝ) b ξ| ≤
∫ xy, |xy.1-xy.2| ∂(p.law : Measure (ℝ × ℝ)) := by
have hl := ((magneticFieldFunctional_approximation ν μ a b hcompact hbound ha hb hμ).sub
(magneticFieldFunctional_approximation ν ξ a b hcompact hbound ha hb hξ)).abs
have hr := (((fieldSimpleApproximation_tendsto (μ : Measure ℝ) hμ).add_const
(∫ xy, |xy.1-xy.2| ∂(p.law : Measure (ℝ × ℝ)))).add
(fieldSimpleApproximation_tendsto (ξ : Measure ℝ) hξ))
simp only [zero_add,add_zero] at hr
exact le_of_tendsto_of_tendsto hl hr (Eventually.of_forall fun n =>
fieldCoupling_simple_error ν a b hcompact hbound ha hb p hμ hξ
(fieldSimpleApproximation n) (fieldSimpleApproximation n))
lemma magneticFieldFunctional_wasserstein_le
(ν μ ξ : ProbabilityMeasure ℝ) (a b : ℝ)
(hcompact : IsCompact (ν : Measure ℝ).support)
(hbound : (ν : Measure ℝ).support ⊆ Icc a b)
(ha : a∈(ν : Measure ℝ).support) (hb : b∈(ν : Measure ℝ).support)
(hμ : Integrable (fun x : ℝ => x) (μ : Measure ℝ))
(hξ : Integrable (fun x : ℝ => x) (ξ : Measure ℝ)) :
ENNReal.ofReal |magneticFieldFunctional (ν : Measure ℝ) b μ-
magneticFieldFunctional (ν : Measure ℝ) b ξ| ≤ fieldWassersteinOne μ ξ := by
apply le_iInf
intro p
rw [p.cost_eq_ofReal hμ hξ]
exact ENNReal.ofReal_le_ofReal (magneticFieldFunctional_coupling_le ν μ ξ a b hcompact hbound ha hb hμ hξ p)
lemma magneticFieldFunctional_tendsto_wasserstein
(ν μ : ProbabilityMeasure ℝ) (μs : ℕ → ProbabilityMeasure ℝ) (a b : ℝ)
(hcompact : IsCompact (ν : Measure ℝ).support)
(hbound : (ν : Measure ℝ).support ⊆ Icc a b)
(ha : a∈(ν : Measure ℝ).support) (hb : b∈(ν : Measure ℝ).support)
(hμ : Integrable (fun x : ℝ => x) (μ : Measure ℝ))
(hμs : ∀ n, Integrable (fun x : ℝ => x) (μs n : Measure ℝ))
(hw : Tendsto (fun n => fieldWassersteinOne (μs n) μ) atTop (𝓝 0)) :
Tendsto (fun n => magneticFieldFunctional (ν : Measure ℝ) b (μs n))
atTop (𝓝 (magneticFieldFunctional (ν : Measure ℝ) b μ)) := by
apply Metric.tendsto_nhds.mpr
intro ε hε
have hh := hw.eventually (Iio_mem_nhds (ENNReal.ofReal_pos.mpr hε))
filter_upwards [hh] with n hn
rw [Real.dist_eq]
have hc := (magneticFieldFunctional_wasserstein_le ν (μs n) μ a b hcompact hbound ha hb (hμs n) hμ).trans_lt hn
exact (ENNReal.ofReal_lt_ofReal_iff hε).mp hc
end InvariantIsing
end
end OAI
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