Linear interpolation of measures creates and destroys mass at fixed locations. Wasserstein interpolation instead follows particles along shortest paths. This produces the intrinsic geodesics of the probability-measure space.
Constant-speed geodesics
Definition. A curve \(t\mapsto\mu_t\) in a metric space is a constant-speed geodesic from \(\mu_0\) to \(\mu_1\) if \[ W_p(\mu_s,\mu_t)=|t-s|W_p(\mu_0,\mu_1) \] for all \(s,t\in[0,1]\).
Assume first that \(X=\mathbb R^d\). If \(\pi\in\Pi(\mu_0,\mu_1)\), define \[ e_t(x,y)=(1-t)x+ty,\qquad \mu_t=(e_t)_\#\pi. \] Each pair \((x,y)\) is moved along the Euclidean segment joining its endpoints.
Displacement interpolation theorem. If \(\pi\) is \(p\)-optimal between \(\mu_0,\mu_1\), then \[ \mu_t=((1-t)\operatorname{pr}_1+t\operatorname{pr}_2)_\#\pi \] is a constant-speed \(W_p\)-geodesic.
Proof. For \(s<t\), the joint law of \[ x_s=(1-s)x+sy,\qquad x_t=(1-t)x+ty \] under \(\pi\) couples \(\mu_s,\mu_t\). Since \(|x_t-x_s|=(t-s)|y-x|\), \[ W_p(\mu_s,\mu_t)\le(t-s)W_p(\mu_0,\mu_1). \] Apply this bound to the intervals \([0,s]\), \([s,t]\), and \([t,1]\). The triangle inequality gives \[ W_p(\mu_0,\mu_1) \le W_p(\mu_0,\mu_s)+W_p(\mu_s,\mu_t)+W_p(\mu_t,\mu_1) \le W_p(\mu_0,\mu_1). \] All inequalities are equalities, proving the formula for every \(s,t\).
Interpolation induced by a map
Corollary. If the optimal plan is induced by \(T\), then \[ \mu_t=T_{t\#}\mu_0,\qquad T_t=(1-t)\operatorname{id}+tT. \] For quadratic transport with \(\mu_0\) absolutely continuous, Brenier's theorem makes this the unique geodesic whenever the optimal plan is unique.
Translation. If \(T(x)=x+a\), then \(\mu_t=(x\mapsto x+ta)_\#\mu_0\). The entire distribution moves rigidly at constant speed.
Geodesic base spaces and dynamical plans
Definition. Let \(\operatorname{Geo}(X)\) be the set of constant-speed minimizing geodesics \(\gamma:[0,1]\to X\), and let \(e_t(\gamma)=\gamma_t\). A probability measure \(\boldsymbol\pi\) on \(\operatorname{Geo}(X)\) is a dynamical optimal plan if \[ (e_0,e_1)_\#\boldsymbol\pi \] is an optimal coupling of its endpoint marginals.
Geodesic-space theorem. If \(X\) is a complete separable geodesic space, then \((\mathcal P_p(X),W_p)\) is geodesic. Every dynamical optimal plan generates a geodesic by \[ \mu_t=(e_t)_\#\boldsymbol\pi. \] Conversely, every \(W_p\)-geodesic admits such a representation under the standard measurable-selection hypotheses.
Proof idea. Start with an optimal endpoint coupling and choose measurably a minimizing geodesic between almost every endpoint pair. The same distance-scaling proof as in Euclidean space gives a Wasserstein geodesic. Conversely, sample the curve at dyadic times, glue optimal couplings between successive samples, and use equality in the triangle inequality to show that almost every resulting path is a constant-speed base-space geodesic. Tightness and consistency pass to a path-space limit.
Displacement versus mixture interpolation
The ordinary convex mixture \[ \bar\mu_t=(1-t)\mu_0+t\mu_1 \] leaves part of the mass at the source and creates the remainder at the target. By contrast, displacement interpolation locates the mass between its endpoints.
Two Dirac masses. For \(\mu_0=\delta_0\), \(\mu_1=\delta_1\), \[ \bar\mu_t=(1-t)\delta_0+t\delta_1, \qquad \mu_t=\delta_t. \] Only the second curve describes one particle moving continuously from \(0\) to \(1\).
Gaussian geodesics
Proposition. For nondegenerate Gaussian measures, let \(T(x)=m_1+A(x-m_0)\) be the Brenier map from page 8. Then the \(W_2\)-geodesic is Gaussian: \[ \mu_t=N(m_t,\Sigma_t), \] where \[ m_t=(1-t)m_0+tm_1,\qquad \Sigma_t=((1-t)I+tA)\Sigma_0((1-t)I+tA)^{\mathsf T}. \]
Proof. The interpolating map \(T_t=(1-t)I+tT\) is affine. The image of a Gaussian under an affine map is Gaussian, with the displayed mean and covariance.
Uniqueness and branching
A unique endpoint coupling and unique base-space geodesics imply a unique displacement interpolation. Nonuniqueness can arise either from several optimal couplings or from branching geodesics in \(X\). In \(\mathbb R^d\) with quadratic cost and absolutely continuous source, Brenier's theorem removes the first ambiguity and straight lines remove the second. On branching metric spaces, the same mass can follow several shortest paths.
Curvature visible in probability space
Although Euclidean particles travel along straight lines, \(\mathcal P_2(\mathbb R^d)\) is not a Hilbert space when \(d\ge2\). Distances between displacement geodesics reflect the interaction of their transport maps. This nonlinear geometry is precisely why ordinary convexity must later be replaced by displacement convexity.