The Schachermayer--Teichmann theorem

For a finite-valued measurable cost c : X × Y → ℝ≥0∞ and probability measures on Polish spaces, a coupling concentrated on a c-cyclically monotone set is optimal. Combined with TauCeti.IsOptimalCoupling.exists_isCyclicallyMonotone_of_lowerSemicontinuous, this characterises the optimal plans of finite cost for a finite-valued lower semicontinuous cost as exactly the plans concentrated on a measurable c-cyclically monotone set. Optimality can thus be checked on finite families of points, without solving the dual problem.

The cost must be finite everywhere. Ambrosio and Pratelli's example shows why: on the circle ℝ / ℤ with Lebesgue measure as both marginals, fix an irrational α and let c (x, x) = 1, c (x, x + α) = 0, and c = ∞ elsewhere. Every nonidentity permutation of finitely many distinct diagonal points has infinite cost: a finite-cost permutation could only move a point by the translation x ↦ x + α, and a cycle of k such moves would give k • α ∈ ℤ, contradicting irrationality. So the diagonal is c-cyclically monotone. Yet the identity plan costs 1, while the plan induced by the translation by α costs 0.

A plan is strongly c-monotone in the sense of Schachermayer and Teichmann when it is concentrated on the contact set of real potentials φ, ψ that satisfy the dual constraint on a product of sets of full measure. Unlike a dual certificate, the potentials need not be integrable, so they need not qualify as a dual optimizer; strong c-monotonicity is nevertheless an optimality criterion (TauCeti.IsCoupling.isOptimalCoupling_of_ae_mem_dualContactSet). For a finite-valued measurable cost on Polish spaces, every coupling of probability measures concentrated on a c-cyclically monotone set is strongly c-monotone, via Rüschendorf's potentials (TauCeti.IsCoupling.exists_ae_add_eq_of_isCyclicallyMonotone).

Main statements

• TauCeti.IsCoupling.isOptimalCoupling_of_isCyclicallyMonotone — a coupling concentrated on a c-cyclically monotone set is optimal for a finite-valued measurable cost;
• TauCeti.IsCoupling.isOptimalCoupling_iff_exists_isCyclicallyMonotone — the Schachermayer--Teichmann theorem: for a finite-valued lower semicontinuous cost, a coupling of finite cost is optimal if and only if it is concentrated on a measurable c-cyclically monotone set.

References

• W. Schachermayer and J. Teichmann, Characterization of optimal transport plans for the Monge--Kantorovich problem, Proc. Amer. Math. Soc. 137 (2009), 519--529, Theorem 1.
• C. Villani, Optimal Transport: Old and New, Grundlehren 338, Springer 2009, Theorem 5.10 (ii).
• L. Ambrosio and A. Pratelli, Existence and stability results in the L¹ theory of optimal transportation, in Optimal Transportation and Applications, Lecture Notes in Math. 1813, Springer 2003, for the counterexample with an infinite cost.

theorem TauCeti.IsCoupling.isOptimalCoupling_of_isCyclicallyMonotone {X : Type u_1} {Y : Type u_2} [MeasurableSpace X] [TopologicalSpace X] [PolishSpace X] [BorelSpace X] [MeasurableSpace Y] [TopologicalSpace Y] [PolishSpace Y] [BorelSpace Y] {μ : MeasureTheory.Measure X} {ν : MeasureTheory.Measure Y} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] {c : X × Y → ENNReal} {π : MeasureTheory.Measure (X × Y)} (hπ : IsCoupling π μ ν) (hc : Measurable c) (hctop : ∀ (z : X × Y), c z ≠ ⊤) {S : Set (X × Y)} (hS : IsCyclicallyMonotone c S) (hπS : ∀ᵐ (z : X × Y) ∂π, z ∈ S) : IsOptimalCoupling c π μ ν

Cyclically monotone plans are optimal. For a finite-valued measurable cost on a product of Polish spaces, a coupling of two probability measures that is concentrated on a c-cyclically monotone set S is an optimal coupling. The set S need not be measurable.

The finiteness of c cannot be dropped; see the module docstring.

theorem TauCeti.IsCoupling.isOptimalCoupling_iff_exists_isCyclicallyMonotone {X : Type u_1} {Y : Type u_2} [MeasurableSpace X] [TopologicalSpace X] [PolishSpace X] [BorelSpace X] [MeasurableSpace Y] [TopologicalSpace Y] [PolishSpace Y] [BorelSpace Y] {μ : MeasureTheory.Measure X} {ν : MeasureTheory.Measure Y} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] {c : X × Y → ENNReal} {π : MeasureTheory.Measure (X × Y)} (hπ : IsCoupling π μ ν) (hc : LowerSemicontinuous c) (hctop : ∀ (z : X × Y), c z ≠ ⊤) (hfin : ∫⁻ (z : X × Y), c z ∂π ≠ ⊤) : IsOptimalCoupling c π μ ν ↔ ∃ (S : Set (X × Y)), MeasurableSet S ∧ IsCyclicallyMonotone c S ∧ π Sᶜ = 0

The Schachermayer--Teichmann theorem. For a finite-valued lower semicontinuous cost on a product of Polish spaces, a coupling of finite cost between two probability measures is optimal if and only if it is concentrated on a measurable c-cyclically monotone set.

The finite-cost hypothesis is used only for the forward implication: when every coupling has infinite cost, every coupling is optimal.