Plain-English statement
A displacement interpolation is the curve obtained by linearly interpolating an optimally coupled endpoint pair and taking its law at each time.
Mathematical statement
For an optimal coupling gamma of mu0 and mu1, mu_t is the pushforward of gamma by (x0,x1) mapped to (1-t)x0+t x1 on [0,1].
Intuition
The same optimally paired particles move along straight lines at constant individual velocity; their time-t distribution is the interpolation curve.
Conditions
- mu0 and mu1 belong to P2,ac
- gamma has these two marginals and attains the quadratic transport cost
- time lies in the unit interval
Why these conditions cannot be dropped
- P2,ac is the source domain for the surrounding Wasserstein calculus
- optimality distinguishes displacement interpolation from an arbitrary coupling interpolation
- the endpoint weights have the intended geodesic meaning on [0,1]
Proof route
- define the affine endpoint map
- push the coupling measure forward by that measurable map
- package the optimal coupling and curve identity
- verify that times zero and one recover the two marginals
Lean interface notes
- Measure.map represents the law of the interpolated endpoint pair
- IsQuadraticOptimalCoupling separates attainment from existence
- the constant-speed and uniqueness theorem is intentionally not part of this definition
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Lean statement
def IsWassersteinGeodesic
(μ₀ μ₁ : Measure E) (curve : ℝ → Measure E) : Prop :=
WassersteinSpace.IsAbsolutelyContinuousFiniteSecondMoment μ₀ ∧
WassersteinSpace.IsAbsolutelyContinuousFiniteSecondMoment μ₁ ∧
∃ γ : Measure (E × E),
IsQuadraticOptimalCoupling γ μ₀ μ₁ ∧
∀ t ∈ Icc (0 : ℝ) 1,
curve t = displacementInterpolation γ t
/-- An optimal coupling and two `P₂,ac` endpoints generate the source
displacement-interpolation predicate. -/Open AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementInterpolation.lean:57published source at 7bcd37294df1