Benamou–Brenier Formulation in Optimal Transport

• Benamou–Brenier formulation is a dynamic approach to optimal transport that minimizes kinetic energy over evolving densities governed by a continuity equation.
• It extends the classical Monge–Kantorovich problem to encompass unbalanced, martingale, and control-based transport scenarios, enriching the geometry of Wasserstein spaces.
• The formulation underpins robust numerical methods and duality frameworks, facilitating applications in analysis, stochastic processes, and machine learning.

The Benamou–Brenier formulation is a dynamic (Eulerian) perspective on optimal transport that expresses the cost of transporting one measure into another as the minimum kinetic energy of admissible evolutions, subject to a continuity equation linking the time-dependent densities and velocities. This framework complements and generalizes the classical (static) Monge–Kantorovich problem, underpins the geometry of Wasserstein spaces, and has been significantly extended to nonstandard settings including martingale transport, unbalanced transport, sub-Riemannian manifolds, mean-field games, and stochastic and quantum contexts.

1. Classical Benamou–Brenier Formulation: Definition and Structure

The Benamou–Brenier formulation expresses, for $p>1$, the $p$-Wasserstein distance between two probability measures $\mu_0$ and $\mu_1$ as the minimal action over evolving pairs $(\rho_t, v_t)$ governed by the continuity equation: $$W_p^p(\mu_0, \mu_1) = \inf \left\{ \int_0^1 \int_{\mathbb{R}^d} |v_t(x)|^p d\rho_t(x) dt : \partial_t \rho_t + \nabla \cdot (\rho_t v_t) = 0,\,\, \rho_0 = \mu_0,\,\, \rho_1 = \mu_1 \right\}$$ For $p=2$, the "kinetic energy" interpretation of the integrand appears and the formulation admits a formal Riemannian structure on the space of probability measures with finite second moment (Huesmann et al., 7 Feb 2024). The variables $\rho_t$ and $v_t$ represent the intermediate densities and velocities, respectively, under constraint of mass conservation.

This dynamic (Eulerian) view contrasts with the classical Monge problem, which is Lagrangian. The equivalence of the Kantorovich and Benamou–Brenier formulations in Euclidean and (under suitable regularity/controllability assumptions) on manifolds underscores the foundational role of this perspective (Citti et al., 28 Jul 2025, Elamvazhuthi, 22 Jul 2024).

2. Methodological Extensions: Unbalanced, Martingale, and Control-Based Formulations

2.1. Unbalanced and Generalized Mass Transport

To address situations where the source and target measures are not equally normalized, the Benamou–Brenier formulation is extended by relaxing mass conservation via an additional source (or growth) term $\alpha$ in the continuity equation: $$\partial_t \rho + \nabla \cdot (\rho v) = \alpha \rho$$ The dynamic formulation for the Wasserstein–Fisher–Rao (WFR) metric or Hellinger–Kantorovich distance thus becomes

$$(WFR(\rho_0, \rho_1))^2 = \inf_{(\rho, v, \alpha)} \frac{1}{2} \int_0^1 \int_M [a^2 |v(x, t)|^2 + b^2 \alpha(x, t)^2]\, d\rho_t(x) dt$$

subject to the above constraint and prescribed endpoints. This framework allows for creation or annihilation of mass during transport and reduces to the classical setting when $\alpha=0$ (Gallouët et al., 2021, Zhu et al., 2020, Gangbo et al., 2019).

Related genetic-mass transport problems can be encoded as vector-valued Benamou–Brenier formulations, where source terms correspond to fluxes between layers ("channels") in a multi-component system (Zhu et al., 2020).

2.2. Martingale Optimal Transport

The Benamou–Brenier philosophy extends to stochastic settings, particularly for martingale optimal transport (MOT). Here, the goal is to minimize an action over admissible martingales interpolating two laws in convex order, subject to the martingale property, resulting in a continuous-time dynamic cost functional—typically via Fokker–Planck dynamics (Huesmann et al., 2017, Backhoff-Veraguas et al., 2017, Backhoff et al., 6 Jun 2024). The dynamic representation enables precise geodesic and interpolation structures in stochastic transport, and explicit optimizer characterization in terms of stretched Brownian or geometric Bass martingales connects the theory to problems in finance and stochastic control.

2.3. Nonlinear Control-Affine and Sub-Riemannian Structures

For kinetic costs adapted to general controlled dynamics—e.g., $$\dot \omega(t) = f_0(\omega(t)) + \sum_{i=1}^n u_i(t) f_i(\omega(t))$$—the Benamou–Brenier formulation, subject to a generalized continuity equation, provides a convex or relaxable dynamic formulation for the transport cost (Elamvazhuthi, 22 Jul 2024, Citti et al., 28 Jul 2025). On sub-Riemannian manifolds without abnormal geodesics, the formulation in the subspace of horizontal (admissible) velocities combined with a suitable Young measure relaxation is shown to be equivalent to the static (Kantorovich) problem, ensuring existence and consistency of minimizers (Citti et al., 28 Jul 2025).

3. Analytical and Structural Properties

The Benamou–Brenier action minimization is convex in $(\rho, v)$ for $p=2$. Its dual formulations often link to Hamilton–Jacobi–BeLLMan (HJB) equations in both classical and quantum contexts (Wirth, 2021), providing insight into geodesicity, convexity, and regularity properties.

A significant result in regularity theory is the variational approach using the Benamou–Brenier formulation to obtain $\epsilon$-regularity and $C^{1,\alpha}$ partial regularity of optimal transport maps, relying on precise comparison between the minimal kinetic energy flows and interpolating harmonic functions (Goldman et al., 2017). The dynamic nature facilitates Campanato-type excess decay, underlining the connection between minimal energy and local regularity without invoking the maximum principle.

In extensions involving additional phenomena (e.g., bulk-interface transfer (Monsaingeon, 2020), diffusion (Chen et al., 2022), entropic effects (Wirth, 2021), or mean-field game coupling (Baptista et al., 17 May 2025)), the Benamou–Brenier structure enables both new analytical results and meaningful geometric interpolation between disparate transport regimes.

4. Numerical and Algorithmic Developments

The dynamic (Eulerian) structure of the Benamou–Brenier formulation is highly amenable to numerical discretization. Practical computational strategies include finite volume schemes for gradient flows (Cancès et al., 2019), space-time discretizations for convergence guarantees even on manifolds (Lavenant, 2019), and the use of advanced preconditioners for saddle-point systems arising from discretized convex optimization formulations (Facca et al., 2022).

For unbalanced and generalized OT, primal-dual and proximal algorithms are facilitated by explicit convexity and separability properties in the Benamou–Brenier framework (Gangbo et al., 2019, Baptista et al., 17 May 2025). Proximal OT divergences, viewed as an infimal convolution of OT cost and information divergence, yield mean-field game structures where optimality conditions are governed by coupled backward HJB and forward continuity PDEs (Baptista et al., 17 May 2025).

The Eulerian approach also underlies state-of-the-art methodologies for spatiotemporal imaging, motion estimation, and tomographic reconstruction, where diffeomorphic constraints and optimal transport costs are naturally encoded and efficiently optimized (Chen, 2020).

5. Variants, Dualities, and Generalizations

The duality theory associated with the Benamou–Brenier formulation recurs across numerous extensions. In the noncommutative (quantum) transport setting, a dual formula emerges with Hamilton–Jacobi–BeLLMan subsolutions as noncommutative analogues of classical test functions, confirming a parallel to the classical Wasserstein geometry at the level of density matrices (Wirth, 2021).

A significant geometric insight is that on closed manifolds (or suitable cones), the equivalence between dynamic (Benamou–Brenier-type) and static (semi-coupling, cone lift) OT formulations can be rigorously established, with the dynamic approach supporting polar factorization theorems and transfer of regularity conditions such as the Ma–Trudinger–Wang property (Gallouët et al., 2021). On metric graphs and random measure spaces, weak formulations via regularized continuity equations and Palm probability techniques extend the Benamou–Brenier framework to singular, stochastic, or infinite-mass settings (Erbar et al., 2021, Huesmann et al., 7 Feb 2024).

6. Applications and Impact Across Disciplines

The Benamou–Brenier formulation is influential in analysis, geometry, statistical physics, machine learning, stochastic processes, and quantum information. Key applications include:

• Wasserstein gradient flows and diffusion equations on complex domains (Erbar et al., 2021, Cancès et al., 2019)
• Martingale and geometric transport for financial modeling (Backhoff-Veraguas et al., 2017, Backhoff et al., 6 Jun 2024)
• Bulk-interface coupled systems and diffusion-regularized transport in computational neurosciences (Monsaingeon, 2020, Chen et al., 2022)
• Proximal optimization and generative modeling in probabilistic machine learning (Baptista et al., 17 May 2025)
• Control-theoretic and sub-Riemannian transport (Elamvazhuthi, 22 Jul 2024, Citti et al., 28 Jul 2025)
• Noncommutative and quantum transport (Wirth, 2021)

The dynamic action-minimization paradigm enables geometric, analytic, and computational approaches that are flexible to generalizations, robust to singularities, and compatible with domain-specific constraints, supporting both theoretical insight and algorithmic tractability.

7. Limitations, Open Problems, and Future Directions

A principal limitation in the general theory arises in situations with abnormal geodesics or when compactness fails. For general sub-Riemannian manifolds with abnormal minimizers, the equivalence of dynamic and static optimal transport is nontrivial and remains an active area of research (Citti et al., 28 Jul 2025). The robustness of dynamic and dual formulations under weak regularity, nonconvex costs, or measure-valued controls also presents technical challenges, requiring further development of relaxation and compactness arguments.

Ongoing extensions include:

• Couplings with mean-field game systems and learning in high-dimensional or singular spaces (Baptista et al., 17 May 2025)
• Further exploration of unbalanced and entropy-transport settings, transferring regularity and geometric properties from static to dynamic formulations (Gallouët et al., 2021)
• Synergizing with data-driven and stochastic approaches, including within-network optimization and dynamic generative models

The continued unification and generalization of the Benamou–Brenier formulation with stochastic, quantum, unbalanced, and control-theoretic optimal transport is likely to drive new developments in analysis, geometry, and their applications.