Benamou–Brenier Optimal Transport

• Benamou–Brenier dynamical characterization is a framework that recasts quadratic optimal transport as a kinetic energy minimization problem subject to the continuity equation.
• It establishes equivalence between static Kantorovich and dynamic fluid-mechanical formulations, enabling geometric interpretation via 2-Wasserstein geodesics.
• Its convex formulation and numerical discretizations have advanced analysis in optimal transport, mean field games, and machine learning applications.

The Benamou–Brenier dynamical characterization is a foundational framework in the theory of optimal transport, establishing the equivalence between static (Kantorovich) and dynamic (fluid-mechanical) formulations of the quadratic optimal transport problem. It realizes the 2-Wasserstein distance as the minimum action for compressible Eulerian flows governed by a PDE constraint, and serves as the geometric basis for Riemannian structure on the space of probability measures.

1. Fluid-Mechanical Formulation of Optimal Transport

The Benamou–Brenier dynamical formulation recasts quadratic optimal transport in terms of kinetic action integrals subject to the continuity equation. Given a smooth, compact Riemannian manifold $(M,g)$, and two probability measures $\mu, \nu \in \mathcal P(M)$ (or, more generally, two positive measures of equal mass), the 2-Wasserstein squared distance is expressed as:

$$W_2^2(\mu,\nu) = \inf_{\rho, v} \int_0^1\! \int_M \frac{1}{2} |v(t,x)|_g^2\,\rho(t,x)\,dx\,dt$$

subject to

$$\partial_t \rho + \nabla \cdot (\rho v) = 0,\quad \rho(0,\,\cdot) = \mu,\quad \rho(1,\,\cdot) = \nu$$

(Lavenant, 2019).

Here, $\rho_t$ is a time-dependent density curve, and $v_t$ is a velocity field; the variational problem seeks flows which transport $\mu$ to $\nu$ with minimal kinetic energy respecting conservation of mass.

In local coordinates or Euclidean space, this reduces to the flat case, where the formulation corresponds to classical compressible Euler equations for potential flows with the aforementioned endpoint constraints.

2. Convexity and Function-Space Interpretation

The Benamou–Brenier formulation is convex in the variables $(\rho, m)$, with $m := \rho v$ being the momentum measure:

$$\inf_{(\rho, m)} \int_{[0,1] \times M} \frac{|m|_g^2}{2\rho}\,dx\,dt,$$

subject to the linear PDE constraint

$$\partial_t \rho + \nabla \cdot m = 0, \qquad \rho(0) = \mu, \quad \rho(1) = \nu$$

The action is declared $+\infty$ when $\rho=0$ but $m \neq 0$, ensuring joint convexity and lower semicontinuity (Lavenant, 2019).

The system lives in the space of positive Radon measures, and the solution curve $\rho_t$ belongs to $C^{1/2}([0,1];(M_+(M),W_2))$ by suitable energy bounds; $v_t$ is such that $t \mapsto \int |v_t|^2 \rho_t(dx)$ is integrable. Primal minimizers exist by classical direct methods in the weak-* topology of measures.

3. Geometric Structure and Wasserstein Geodesics

On both flat and Riemannian manifolds, the minimizers $(\rho^*, v^*)$ describe constant-speed geodesics in the 2-Wasserstein space $(\mathcal P_2(M), W_2)$. The Riemannian metric tensor on tangent spaces at a density $\rho$ is: $$\langle v, w \rangle_\rho = \int_M \rho\, g(v, w)\,dx$$ Thus, $W_2(\mu, \nu)$ represents the length of the shortest path joining $\mu$ to $\nu$ in this infinite-dimensional Riemannian manifold (Lavenant, 2019). The kinetic action $\int_0^1\!\!\int \frac{1}{2} \rho |v|^2\,dx\,dt$ is the energy of a compressible fluid flow and its minimizers correspond to “displacement interpolants” along the optimal transport plan.

On Riemannian manifolds $(M,g)$, all differential operators and volume forms are taken with respect to the underlying geometry, and analytical techniques such as the Bakry–Émery gradient contractivity estimate

$$|\nabla (\Phi_s f)|^2 \le e^{K s} \Phi_s (|\nabla f|^2)$$

(where $K$ is a lower Ricci curvature bound and $\Phi_s$ the Neumann heat semigroup) play a role in convergence and stability arguments for numerical discretizations.

4. Duality: The Hamilton–Jacobi/Kantorovich Formulation

The dynamical formulation admits a dual representation via Hamilton–Jacobi theory: $$W_2^2(\mu, \nu) = \sup_{\phi \in C^1([0,1] \times M)} \Big[ \langle \phi(1), \nu \rangle - \langle \phi(0), \mu \rangle - \int_0^1\! \!\int_M [\partial_t \phi + \frac{1}{2} |\nabla \phi|_g^2]\,dx\,dt \Big]$$ This dual encodes the Hamilton–Jacobi PDE: $$\partial_t \phi + \frac{1}{2} |\nabla \phi|^2 \le 0$$ with $\phi$ acting as a Lagrange multiplier enforcing the continuity equation. The dual formulation connects the BB framework to the Monge–Kantorovich theory, establishing that any static optimal coupling $\pi$ induces $(\rho_t, v_t)$ satisfying the dynamic constraint with matching cost, and vice versa.

5. Discretization, Numerical Schemes, and Extensions

Computational approaches for Benamou–Brenier minimization typically discretize space and time, yielding convex optimization problems which remain faithful to the continuous structure. The convex change of variables—or equivalently, the $(\rho, m)$ parameterization—is vital in ensuring well-posedness and tractability. Grid-based methods, including those on staggered or triangulated domains, preserve conservation properties and respect the PDE constraint (Papadakis et al., 2013, Lavenant et al., 2018).

Extensions have been developed for:

• Arbitrary Riemannian or sub-Riemannian manifolds (under no abnormal geodesics assumptions, the dynamic and static formulations remain equivalent and minimizers exist (Citti et al., 28 Jul 2025)).
• Discrete graphs, where mass transport and flux are formulated in combinatorial terms and limit to BB as the mesh refines (Hillion, 2014).
• Random and stationary settings leveraging Palm calculus for "infinite-mass" measures (Huesmann et al., 2024).
• Models in imaging and shape analysis incorporating diffeomorphic flows under Hilbert-space constraints (Chen, 2020).
• Multi-marginal or infimal-convolution cost settings, where the structure of Wasserstein barycenters and convexity is preserved (Krannich, 14 Dec 2025, Pass et al., 26 Sep 2025).

6. Theoretical Significance and Impact

The Benamou–Brenier dynamical characterization provides a rigorous and flexible foundation for both the analysis and computation of optimal transport. It exposes deep links between gradient flows in Wasserstein spaces, compressible fluid mechanics, and infinite-dimensional Riemannian geometry. It underlies modern developments in mean field games, entropic interpolation (e.g., Schrödinger problems), statistical mechanics, and machine learning applications.

The dynamical view is essential for extending optimal transport theory beyond Euclidean settings (e.g., Riemannian, sub-Riemannian, discrete, or stochastic structures), for generalizing to branched/concave-cost transport (Brasco et al., 2010), and for obtaining tractable algorithms via convex optimization, which are robust to non-smooth data and mesh refinement (Lavenant, 2019).

7. Further Directions and Generalizations

Recent work generalizes the Benamou–Brenier framework to:

• Martingale optimal transport: Dynamical characterization with pure diffusion (Fokker–Planck) constraints, connecting optimal transport with stochastic processes and yielding unique Markov or Bass martingale interpolants (Backhoff-Veraguas et al., 2017, Backhoff-Veraguas et al., 2023).
• Weak and barycentric optimal transport: Dynamic analogues of weak or barycentric cost structures, where only the barycenter of the transport kernel is controlled and martingale components are free (Guo et al., 26 Nov 2025).
• Schrödinger bridges and entropic interpolation: BB-type dynamical variational problems with additional Fisher-information (entropy) penalties, leading to unique entropic interpolants and convergence to $W_2$ geodesics as the entropic parameter vanishes (Gigli et al., 2018).

These generalizations broaden the scope of classical optimal transport, enabling the analysis of more intricate mass movement scenarios, stochastic control problems, and applications in high-dimensional statistical modeling.