The static problem minimizes endpoint displacement. The Benamou--Brenier formula gives the same distance by minimizing kinetic energy over every mass-preserving flow joining the endpoints. It is the bridge from transport plans to fluid mechanics and the source of the formal Riemannian structure of \(\mathcal P_2\).
Dynamic action
Definition. An admissible dynamic pair between \(\mu_0,\mu_1\in\mathcal P_2(\mathbb R^d)\) consists of a narrowly continuous curve \(\mu_t\) and a velocity field \(v_t\in L^2(\mu_t)\) satisfying \[ \partial_t\mu_t+\nabla\cdot(v_t\mu_t)=0, \qquad \mu_{t=0}=\mu_0,\quad\mu_{t=1}=\mu_1. \] Its action is \[ \mathcal A(\mu,v)=\int_0^1\int_{\mathbb R^d}|v_t(x)|^2\,\mathrm{d}\mu_t(x)\,\mathrm{d}t. \]
The dynamic formula
Benamou--Brenier formula. For \(\mu_0,\mu_1\in\mathcal P_2(\mathbb R^d)\), \[ \boxed{ W_2^2(\mu_0,\mu_1) = \inf_{(\mu_t,v_t)} \int_0^1\int|v_t|^2\,\mathrm{d}\mu_t\,\mathrm{d}t }, \] where the infimum is over all admissible dynamic pairs.
Proof. For any admissible pair, page 13 gives \[ W_2(\mu_0,\mu_1) \le\int_0^1|\dot\mu_t|\,\mathrm{d}t \le\int_0^1\|v_t\|_{L^2(\mu_t)}\,\mathrm{d}t. \] Cauchy--Schwarz on the unit time interval yields \[ W_2^2(\mu_0,\mu_1) \le\int_0^1\|v_t\|_{L^2(\mu_t)}^2\,\mathrm{d}t =\mathcal A(\mu,v). \] Hence \(W_2^2\) is a lower bound on every action.
For the reverse inequality, take the displacement geodesic generated by an optimal plan. It is absolutely continuous with constant metric speed \(W_2(\mu_0,\mu_1)\). Its minimal velocity field satisfies \[ \|v_t\|_{L^2(\mu_t)}=|\dot\mu_t|=W_2(\mu_0,\mu_1) \] for almost every \(t\). Its action is therefore exactly \(W_2^2(\mu_0,\mu_1)\), attaining the lower bound.
Convex momentum formulation
If \(\mu_t=\rho_t\,\mathrm{d}x\), introduce momentum \(m_t=\rho_tv_t\). The constraints become linear: \[ \partial_t\rho+\nabla\cdot m=0. \] The action becomes \[ \int_0^1\int\frac{|m_t(x)|^2}{\rho_t(x)}\,\mathrm{d}x\,\mathrm{d}t, \] with the lower-semicontinuous convention \(0/0=0\) and \(|m|^2/0=+\infty\) for \(m\ne0\). The perspective function \((\rho,m)\mapsto|m|^2/\rho\) is convex, so the dynamic transport problem is a convex optimization problem in \((\rho,m)\).
Convexity. Since \(q(v)=|v|^2\) is convex, its perspective \(\rho q(m/\rho)=|m|^2/\rho\) is convex for \(\rho>0\). The lower-semicontinuous extension preserves convexity at \(\rho=0\).
Equations for a smooth geodesic
A minimal velocity belongs to the tangent space and is therefore a gradient, \(v_t=\nabla\phi_t\), under smoothness assumptions. First variations of the action subject to the continuity equation give \[ \partial_t\phi_t+\frac12|\nabla\phi_t|^2=0. \] Taking a gradient yields the inviscid Burgers equation \[ \partial_tv_t+(v_t\cdot\nabla)v_t=0. \] Together with \[ \partial_t\rho_t+\nabla\cdot(\rho_tv_t)=0, \] this is pressureless Euler flow: particles keep constant velocity along straight characteristics.
Derivation. Add a Lagrange multiplier \(\phi\) for the continuity equation and integrate by parts: \[ \int\left\{\frac12\rho|v|^2+\phi(\partial_t\rho+\nabla\cdot(\rho v))\right\} = \int\rho\left\{\frac12|v|^2-\partial_t\phi-v\cdot\nabla\phi\right\}, \] up to endpoint terms. Variation in \(v\) gives \(v=\nabla\phi\) after the corresponding sign convention; variation in \(\rho\) then gives the Hamilton--Jacobi equation.
Otto's tangent metric
Definition. For a sufficiently regular \(\mu\), \[ T_\mu\mathcal P_2 =\overline{\{\nabla\phi:\phi\in C_c^\infty(\mathbb R^d)\}}^{L^2(\mu)}. \] The formal Riemannian inner product is \[ \langle\nabla\phi,\nabla\psi\rangle_{T_\mu} =\int\nabla\phi\cdot\nabla\psi\,\mathrm{d}\mu. \]
A tangent vector is represented by the distribution \(-\nabla\cdot(\mu\nabla\phi)\), but the gradient field is the useful minimal representative. The metric norm of the tangent vector is \(\|\nabla\phi\|_{L^2(\mu)}\). The Benamou--Brenier formula is therefore the usual energy formula for Riemannian distance, interpreted on an infinite-dimensional space.
First variations and Wasserstein gradients
Formal gradient formula. Suppose a functional \(\mathcal F\) has first variation \(\delta\mathcal F/\delta\rho\), meaning \[ \frac{\mathrm{d}}{\mathrm{d}\varepsilon}\mathcal F(\rho+\varepsilon\sigma)\bigg|_{\varepsilon=0} =\int\frac{\delta\mathcal F}{\delta\rho}\,\sigma\,\mathrm{d}x. \] Along \(\partial_t\rho+\nabla\cdot(\rho v)=0\), \[ \frac{\mathrm{d}}{\mathrm{d}t}\mathcal F(\rho_t) =\int\nabla\!\left(\frac{\delta\mathcal F}{\delta\rho}\right)\cdot v_t\,\rho_t\,\mathrm{d}x. \] Hence \[ \operatorname{grad}_{W_2}\mathcal F(\rho) =\nabla\!\left(\frac{\delta\mathcal F}{\delta\rho}\right). \]
Proof. Substitute \(\partial_t\rho=-\nabla\cdot(\rho v)\) into the first-variation formula and integrate by parts. The resulting \(L^2(\rho)\) pairing identifies the Riemannian gradient.
Examples of the action
Translation. For \(\mu_t=(x\mapsto x+ta)_\#\mu_0\), \(v_t=a\), so \[ \mathcal A=\int_0^1|a|^2\,\mathrm{d}t=|a|^2=W_2^2(\mu_0,\mu_1). \]
General \(p\). The same argument gives \[ W_p^p(\mu_0,\mu_1) =\inf\int_0^1\int|v_t|^p\,\mathrm{d}\mu_t\,\mathrm{d}t \] for \(1<p<\infty\). Only \(p=2\) produces a quadratic inner-product geometry.