Benamou–Brenier Formulation

• Benamou–Brenier formulation is a dynamical variational approach for optimal transport that recasts static problems into convex optimization over time-dependent densities and velocities.
• It characterizes geodesics in transport spaces by minimizing kinetic action, linking the squared 2-Wasserstein distance with displacement interpolation and numerical schemes.
• The formulation extends to unbalanced, multi-marginal, stochastic, and quantum settings, bridging optimal transport with control theory, PDEs, and modern computational methods.

The Benamou–Brenier formulation provides a dynamical (Eulerian) variational characterization of the optimal transport problem, replacing the classical Monge and Kantorovich (static) approaches by a convex optimization for time-dependent densities and velocities subject to a continuity equation. In its canonical quadratic cost setting, the minimal kinetic action coincides with the squared 2-Wasserstein distance between probability distributions. This formulation admits generalizations to non-quadratic costs, unbalanced transport, martingale constraints, multi-marginal problems, and discrete geometric structures. It facilitates the study of geodesics in transport spaces, enables efficient numerical schemes, and connects optimal transport to control theory, physical PDEs, and quantum analogues.

1. Classical Dynamical Formulation and Equivalence with Kantorovich

Given two probability densities $\rho_0,\rho_1$ of equal total mass on a domain (Euclidean, Riemannian, or sub-Riemannian), the Benamou–Brenier action is

$$\inf_{(\rho, v)} \int_0^1 \int \tfrac12 |v(t,x)|^2 \rho(t,x) \, dx\, dt$$

subject to the continuity equation

$$\partial_t \rho + \nabla\cdot(\rho v) = 0, \qquad \rho(0,\cdot) = \rho_0,\;\rho(1,\cdot)=\rho_1.$$

Equivalently, in momentum variables $j = \rho v$: $$\inf_{(\rho, j)} \int_0^1 \int \frac{|j|^2}{2\rho} \, dx\,dt \quad \text{with } \partial_t\rho + \nabla\cdot j = 0.$$ The infimum coincides with the quadratic Kantorovich cost:

$$W_2^2(\rho_0,\rho_1) = \min_\pi \int |x-y|^2\, d\pi(x,y)$$

where $\pi$ is a coupling with prescribed marginals. The minimizer $(\rho, v)$ recovers the displacement interpolation between $\rho_0$ and $\rho_1$ via the Brenier map $T$, written as $T_t(x) = (1-t)x + tT(x)$, with $\rho(t) = T_t \# \rho_0$ and velocity field $v = d j / d \rho$ (Goldman et al., 2017, Citti et al., 28 Jul 2025).

In sub-Riemannian and control-affine settings, the formulation generalizes to

$$\partial_t \mu_t + \operatorname{div}(\mu_t v_t) = 0,\quad J_\mathrm{BB}(\mu,v) = \int_0^1 \int_M \tfrac12 |v_t(x)|^2 d\mu_t(x)\, dt$$

with equivalence to the Kantorovich cost $c(x,y)=d_{SR}^2(x,y)$ under suitable regularity (no abnormal geodesics, completeness, compactness or moment/tightness bounds) (Citti et al., 28 Jul 2025, Elamvazhuthi, 2024).

2. Extensions: Unbalanced, Multi-Marginal, and General Costs

For measures of unequal mass, the continuity equation is supplemented by a source term $s(t,x)$: $\partial_t \rho + \nabla\cdot m = s$ with action

$$\int_0^1 \int \frac{|m|^2 + \alpha s^2}{2\rho}\,dx\,dt$$

defining the Wasserstein–Fisher–Rao (WFR) metric (Nishino et al., 10 Dec 2025, Gangbo et al., 2019, Zhu et al., 2020). The dual incorporates mass creation/destruction penalties, leading to new generalized Monge–Ampère equations and Kantorovich dual formulas.

Multi-marginal Benamou–Brenier formulations leverage barycenter decompositions or flow the full coupling $\pi(t)$ over $(\mathbb{R}^d)^N$. For infimal-convolution costs $c(x_1,\dots,x_N)=\inf_z \sum_i |x_i-z|^p$, the dynamical problem is convex and admits a unique minimizer tied to the Wasserstein barycenter of $\mu_1,\dots,\mu_N$, with explicit velocity interpolants and flow decompositions (Krannich, 14 Dec 2025, Pass et al., 26 Sep 2025).

Generalizations to non-quadratic, semi-convex, or control-theoretic Lagrangians in the action

$\int_0^1 \int L(x, v) \rho(t,x) dx\,dt$

are governed by convex analysis and coercivity assumptions. These include non-Euclidean or nonlinear control systems (Elamvazhuthi, 2024).

3. Discrete and Geometric Formulations

On graphs, the Benamou–Brenier principle is discretized via incidence matrices $\Omega$, vertex densities $f_x(t)$, edge velocities $v_k(t)$, and edge-mass $g_k(t)$. The continuity equation and edge-action yield

$$\partial_t f = \Omega \cdot (v\,g),\quad \mathcal{I}_q(v,g) = \left( \int_0^1 \sum_k g_k(t) |v_k(t)|^q dt \right)^{1/q}$$

with minimizers recovering the Wasserstein-1 distance and geodesic paths classified by time-constant Beckmann flows (Morris et al., 7 Jan 2026). In metric graphs, the formulation involves kinetic actions on edges with Kirchhoff conditions at vertices, and regularization via spatial convolution provides technical control of geodesics and action minimizers (Erbar et al., 2021).

Numerical schemes exploit variational finite-volume discretizations, staggered grids, implicit time-stepping, and first-discretize–then-optimize paradigms, ensuring nonnegativity, energy decay, and unconditional convergence under mesh refinement for a variety of cost structures and topologies (Cancès et al., 2019, Lavenant, 2019, Facca et al., 2022).

4. Martingale, Stochastic, and Quantum Generalizations

Martingale optimal transport admits a Benamou–Brenier framework via action minimization over diffusive processes constrained to be martingales, yielding equivalence with Fokker–Planck PDEs and Hamilton–Jacobi–Bellman duality: $$BB(\mu,\nu) = \inf_{X\in AC^p} \int_0^1 \mathbb{E}[c(t, X_t, \dot{X}_t)]dt, \quad \text{subject to}~\text{Martingale}~X_0\sim\mu, X_1\sim\nu$$ and the associated geodesic equations for density and optimal potential (Huesmann et al., 2017, Backhoff-Veraguas et al., 2017, Backhoff et al., 2024). These structures incorporate links with porous medium equations in dimension one, explicit construction of Bass and geometric martingales, and convex duals via maximal covariance and HJB PDEs.

Quantum analogues replace densities by noncommutative states and velocity fields by operator-valued flows in finite *-algebras, with continuity equations, entropic regularization, and subsolution Hamilton–Jacobi–Bellman inequalities leading to saddle-point duality theorems (Wirth, 2021).

Transport for stationary random measures utilizes Palm probabilities to define joint laws, continuity equations on the product space, and geodesic minimizers with actions integrating kinetic moments under the Palm measure, connecting to applications in stochastic geometry and ergodic theory (Huesmann et al., 2024).

5. Regularity, Geodesic Characterization, and Structural Insights

Regularity of minimizers in the Benamou–Brenier framework is governed by variational and affine-invariant arguments. Harmonic approximation yields local $C^{1,\alpha}$ regularity of the Brenier map, with excess decay controlled via Campanato iteration, boundary layer analysis, and quasi-orthogonality identities. The velocity field $v$ approximates gradients of harmonic functions when the kinetic cost is small, demonstrating the hidden ellipticity in the continuity–kinetic-energy system (Goldman et al., 2017).

On graphs, geodesics are classified by Beckmann flows, with uniqueness on trees and cycle-space parameterization of non-uniqueness in cyclic graphs (Morris et al., 7 Jan 2026). In multi-marginal, unbalanced, and vector-valued settings, geodesics arise from barycentric minimization, source-layer embedding, and explicit velocity decomposition (Krannich, 14 Dec 2025, Zhu et al., 2020, Nishino et al., 10 Dec 2025).

Geometric and martingale transport problems employ PDE approaches for regularity, such as uniform parabolicity in HJB equations and feedback control for Kolmogorov flows (Backhoff et al., 2024, Elamvazhuthi, 2024). Quantum and stochastic analogues inherit operator regularity and ergodic Palm conditioning.

6. Computational Methods and Applications

The Benamou–Brenier formulation underpins scalable algorithms for computing optimal transport, including primal-dual proximal splitting, staggered grid schemes, and global FFT-based solvers. These methods are employed for Wasserstein gradient flows, imaging and tomography (with diffeomorphic constraints), population density evolution, constrained mass transport with affine and inequality constraints, and dynamical PDE control (e.g., porous medium, Burgers equations) (Chen, 2020, Jean-Marie et al., 4 Nov 2025, Nishino et al., 10 Dec 2025).

Efficient solution of the KKT saddle-point system in dynamical OT leverages BB-preconditioners, exploiting operator commutation identities for nearly linear scaling with discretization size (Facca et al., 2022).

The scope of applications includes economics, statistics of barycenters, multi-source interpolation, shape analysis, stochastic geometry, quantum information, and control systems.

7. Generalizations, Open Problems, and Outlook

Benamou–Brenier theory extends to discrete, geometric, stochastic, multi-marginal, unbalanced, vector-valued, quantum, and nonlinear control domains. Key challenges and open problems reside in the treatment of abnormal geodesics in sub-Riemannian settings, regularity and uniqueness in presence of singular minimizers or weak data, and adaptation to high-dimensional grids, non-Euclidean geometries, and coupled PDE/optimal control frameworks (Citti et al., 28 Jul 2025, Elamvazhuthi, 2024, Erbar et al., 2021).

Ongoing research addresses unconditional convergence, scalability of solvers, embedding of reaction and source terms, and structural characterization of geodesic flows under general cost and system constraints. The central role of the Benamou–Brenier dynamical approach continues to deepen its impact across mathematical, computational, and applied disciplines.