displacement interpolation

The displacement interpolation between ${\mu }_0,{\mu }_1\in {\mathrm{P}}_2(\mathcal{X})$ is the path $[{\mu }_0,{\mu }_1]=({\mu }_t{)}_{0\le t\le 1}$ defined by the time-marginal flow of the solution to the dynamic Monge-Kantorovich problem $(M{K}_{\text{dyn}})$. Concretely, if $\hat{P}$ solves $(M{K}_{\text{dyn}})$, then ${\mu }_t:=\hat{P}(X_t\in \cdot )\in {\mathrm{P}}_2(\mathcal{X})$.

For each pair $(x,y)$, the optimal path is the constant-speed geodesic ${\gamma }_t^{xy}=(1-t)x+ty$ (in Euclidean space). The displacement interpolation lifts these pointwise geodesics to a geodesic in the Wasserstein space: $[{\mu }_0,{\mu }_1]$ is a minimizing constant-speed geodesic in $({\mathrm{P}}_2(\mathcal{X}),W_2)$, meaning $W_2({\mu }_s,{\mu }_t)=(t-s)W_2({\mu }_0,{\mu }_1)$ for $0\le s\le t\le 1$.

This concept, introduced by McCann (1997) and developed by Otto (2001), encodes geometric properties of the underlying manifold $\mathcal{X}$: Jordan-Kinderlehrer-Otto (1998) showed that many dissipative PDEs are gradient flows in Wasserstein space, and the convexity of functionals along displacement interpolations characterizes Ricci curvature lower bounds (Lott-Sturm-Villani theory).

Key Details

• Benamou-Brenier formula: $W_2^2({\mu }_0,{\mu }_1)/2={\inf }_{(\nu ,v)}{\int }_{[0,1]\times \mathcal{X}}|v_t(x){|}^2/2{\nu }_t(dx)dt$ subject to ${\partial }_t\nu +\nabla \cdot (\nu v)=0$, ${\nu }_0={\mu }_0$, ${\nu }_1={\mu }_1$. The optimal velocity field is $v=\nabla \psi$ (gradient of the Kantorovich potential)
• Forward-backward system: The optimal $(\mu ,\nabla \psi )$ solves the coupled system (1.8): continuity equation + Hamilton-Jacobi equation
• Entropic regularization: The entropic interpolation $[{\mu }_0,{\mu }_1{]}^k$ (from the Schrödinger problem of order $k$) converges to the displacement interpolation as $k\to \infty$, providing a smooth approximation

Textbook References

Optimal Transport Old and New (Villani, 2009)

• Theorem 7.21 (p. 139): Three equivalent characterisations of displacement interpolation: (i) law of random optimal geodesic, (ii) additive cost-splitting $C^{t_1,t_2}({\mu }_{t_1},{\mu }_{t_2})+C^{t_2,t_3}({\mu }_{t_2},{\mu }_{t_3})=C^{t_1,t_3}({\mu }_{t_1},{\mu }_{t_3})$, (iii) action-minimising curve in $P(\mathcal{X})$
• Corollary 7.22 (pp. 139-140): Displacement interpolation $=$ constant-speed geodesic in $(P_p,W_p)$: $W_p({\mu }_s,{\mu }_t)=|t-s|W_p({\mu }_0,{\mu }_1)$
• Corollary 7.23 (p. 140): Uniqueness when OT plan and minimising curves are both unique
• Theorem 7.30(iv) (p. 151): Non-crossing property — $[{\gamma }_t={\tilde{\gamma }}_t]⟹[\gamma =\tilde{\gamma }]$ a.s. Optimal paths cannot cross at intermediate times
• Example 7.2 (pp. 127-128): For strictly convex $L(v)$ in ${\mathbb{R}}^n$, Jensen’s inequality forces minimisers to be constant-velocity straight lines ${\gamma }_t=(1-t)x_0+t{x}_1$. This is the fundamental “straightness” argument
• Aphorism (p. 139): “A geodesic in the space of laws is the law of a geodesic”

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concept