Generalized geodesics in the Wasserstein space #

The squared quadratic Wasserstein distance ν ↦ W₂ (μ, ν) ^ 2 from a fixed law μ is in general not convex along the Wasserstein geodesics of ℝ²; along them it satisfies the reverse, semiconcavity inequality instead. It is, however, convex along a different family of curves joining two laws ν₀ and ν₁, the generalized geodesics based at μ. These are built from a single measure γ on X × X × X, read as the joint law of a triple (x, y₀, y₁), whose projection to (x, y₀) is an optimal coupling of μ and ν₀ and whose projection to (x, y₁) is an optimal coupling of μ and ν₁ (TauCeti.IsGeneralizedGeodesicPlan). Such a three-plan exists on every Polish metric space, by gluing two optimal couplings along their common first marginal (TauCeti.exists_isGeneralizedGeodesicPlan). In a real inner product space the generalized geodesic of γ is the curve of laws of (1 - t) • y₀ + t • y₁ under γ (TauCeti.generalizedGeodesic), which runs from ν₀ at t = 0 to ν₁ at t = 1.

Along it, for t ∈ [0, 1],

W₂ (μ, ν_t) ^ 2 + t (1 - t) ∫ ‖y₀ - y₁‖ ^ 2 dγ ≤ (1 - t) W₂ (μ, ν₀) ^ 2 + t W₂ (μ, ν₁) ^ 2,

and since γ projects to a coupling of ν₀ and ν₁, the integral may be replaced by the smaller W₂ (ν₀, ν₁) ^ 2: the function ν ↦ W₂ (μ, ν) ^ 2 / 2 is 1-convex along generalized geodesics based at μ. The reason is the pointwise Hilbert identity ‖x - ((1 - t) a + t b)‖ ^ 2 + t (1 - t) ‖a - b‖ ^ 2 = (1 - t) ‖x - a‖ ^ 2 + t ‖x - b‖ ^ 2 (TauCeti.edist_smul_add_smul_sq_add), integrated against γ, together with the coupling of μ and ν_t that γ provides. All statements are inequalities in [0, ∞] and need no moment hypotheses.

This convexity is the estimate on which the theory of minimizing movements in the quadratic Wasserstein space rests: for an energy that is λ-convex along the generalized geodesics based at the previous step, it makes the penalized functional of an implicit Euler step of size τ (λ + 1 / τ)-convex along them, hence strictly convex when λ + 1 / τ > 0, which is what yields uniqueness of the step and the discrete energy estimates.

Main definitions #

TauCeti.IsGeneralizedGeodesicPlan p γ μ ν₀ ν₁: the measure γ on X × X × X projects to p-optimal couplings of μ with ν₀ and of μ with ν₁.
TauCeti.generalizedGeodesic γ t: the law of (1 - t) • y₀ + t • y₁ under γ.

Main statements #

TauCeti.exists_isGeneralizedGeodesicPlan: on a Polish metric space, every finite base measure μ and two measures coupled with it carry such a three-plan, for every nonzero exponent.
TauCeti.IsGeneralizedGeodesicPlan.generalizedGeodesic_zero and TauCeti.IsGeneralizedGeodesicPlan.generalizedGeodesic_one: the generalized geodesic runs from ν₀ to ν₁.
TauCeti.hasFiniteMoment_generalizedGeodesic and TauCeti.exists_coe_eq_generalizedGeodesic: generalized geodesics between laws of finite q-th moment stay in laws of finite q-th moment, so those between laws of P₂(E) run in P₂(E).
TauCeti.wassersteinEDist_generalizedGeodesic_sq_add_le: the convexity inequality for an arbitrary three-plan with first marginal μ, against the three transport integrals of the plan.
TauCeti.IsGeneralizedGeodesicPlan.wassersteinEDist_sq_add_le and TauCeti.IsGeneralizedGeodesicPlan.wassersteinEDist_sq_add_wassersteinEDist_sq_le: the convexity of ν ↦ W₂ (μ, ν) ^ 2 along generalized geodesics based at μ.

References #

L. Ambrosio, N. Gigli, G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd ed., Birkhäuser 2008, §7.3 for the semiconcavity along geodesics, §9.1 for the failure of convexity, and §9.2, Lemma 9.2.1 and Definition 9.2.2.
F. Santambrogio, Optimal Transport for Applied Mathematicians, Birkhäuser 2015, §7.3.