Dynamic Benamou-Brenier Formulation

Updated 25 November 2025

Dynamic Benamou-Brenier formulation is a variational, time-continuous framework for optimal transport that minimizes kinetic energy subject to mass conservation.
It establishes an equivalence with the static Kantorovich problem, validating the approach for quadratic and convex costs in diverse geometric settings.
The formulation extends to sub-Riemannian, stochastic, and unbalanced transport scenarios, enabling efficient numerical discretizations for imaging, control, and measure space geometry.

The dynamic Benamou–Brenier formulation provides a variational, time-continuous characterization of optimal transport between probability measures via a kinetic energy minimization posed under the constraint of mass conservation. Initially proposed as a fluid-mechanics reformulation for the quadratic Wasserstein ( $W_2$ ) distance, this framework underpins modern approaches to computational optimal transport, metric geometry in measure spaces, and has been generalized to a wide array of settings, including sub-Riemannian manifolds, metric graphs, multi-marginal problems, control systems, unbalanced transport, and stochastic/martingale constraints.

1. Mathematical Formulation and Principles

The classical Benamou–Brenier problem seeks a time-parameterized density/velocity pair $(\rho_t, v_t)$ , $t \in [0,1]$ , joining prescribed initial and terminal measures $\rho_0$ , $\rho_1$ on a smooth domain $\Omega \subset \mathbb{R}^d$ . The variational principle is:

$W_2^2(\rho_0,\rho_1) = \inf_{(\rho_t,v_t)} \int_0^1 \int_\Omega |\!|v_t(x)|\!|^2 \rho_t(x) \,dx\,dt$

subject to the continuity equation

$\partial_t \rho_t + \nabla \cdot (\rho_t v_t) = 0,$

and fixed endpoints $\rho_{t=0} = \rho_0$ , $\rho_{t=1} = \rho_1$ (Papadakis et al., 2013).

The integrand is the kinetic energy density, and the infimum is taken over all narrowly continuous curves $\rho_t$ of probability measures and Borel vector fields $v_t$ such that the total action is finite. The continuity equation enforces mass conservation during transport.

This formulation extends naturally to general cost functions $L(v)$ (with $L$ convex), as in the multi-marginal and control-theoretic or sub-Riemannian settings (Pass et al., 26 Sep 2025, Elamvazhuthi, 22 Jul 2024, Citti et al., 28 Jul 2025).

2. Equivalence to Static Kantorovich Problems

A foundational principle is the equivalence between the dynamic Benamou–Brenier formulation and the static Kantorovich problem. For the quadratic cost $c(x,y) = |x-y|^2$ (Euclidean case), or its sub-Riemannian analogue $c(x,y) = d^2(x,y)$ (where $d$ is the sub-Riemannian or control-theoretic distance), optimal dynamic transport and its static counterpart yield the same value and minimizers correspond.

Sub-Riemannian case: Let $(M,H,g_H)$ be a connected, complete, boundaryless sub-Riemannian manifold with no abnormal geodesics and compactly supported $\mu_0,\mu_1 \in P_2(M)$ . Then

$C_{\Kan}(\mu_0, \mu_1) = C_{\BB}(\mu_0, \mu_1)$

where $C_{\Kan}$ is the Kantorovich infimum over plans for cost $c(x,y)=d^2(x,y)$ and $C_{\BB}$ is the dynamic optimum (Citti et al., 28 Jul 2025). For control-affine systems, the equivalence persists under Tonelli’s convexity and regularity conditions (Elamvazhuthi, 22 Jul 2024).

Mechanism of equivalence: The proof exploits relaxation to Young measures (transport plans on path space), tightness and superposition (Ambrosio-Gigli-Savaré theory), measurable selection of geodesics, and convexity/Jensen's arguments. Minimizers in one formulation yield minimizers in the other via (i) pushforward with geodesic flow maps, and (ii) disintegration and projection.

Multi-marginal case: An analogous principle holds for convex $L$ and translation-invariant or semi-convex costs, via flows of couplings on $X^k$ and convexification arguments (Pass et al., 26 Sep 2025).

3. Variational Structures and Duality

The dynamic formulation is convex due to the quadratic (or, more generally, convex) structure of the kinetic-energy functional in $(\rho_t, v_t)$ . The associated dual is a time-dependent Hamilton–Jacobi inequality:

$\partial_t \varphi + L^*(\nabla \varphi) \leq 0$

with the Legendre transform $L^*$ of $L$ , and suitable terminal conditions (Pass et al., 26 Sep 2025). For the quadratic case, this reduces to the Hamilton–Jacobi–Bellman equation.

In the unbalanced case (e.g., Wasserstein–Fisher–Rao/entropy-transport), the dynamic action incorporates non-conservative source terms and the duality involves entropy-regularized potentials and an additional Fisher information term (Gallouët et al., 2021, Gigli et al., 2018). The variational structure is preserved in regularized frameworks and in time-discretizations, with convergence unconditionally under mesh refinement (Lavenant, 2019).

4. Generalizations: Geometry, Constraints, Multi-marginal, and Stochastic Settings

a. Sub-Riemannian, Control-Theoretic, and Metric Spaces: The Benamou–Brenier framework extends to sub-Riemannian geometries, control-affine systems, and metric graphs. In each case, the continuity equation and action functional are adapted to the geometric structure: restriction to horizontal vector fields, control-affine velocities, Kirchhoff-type node conditions at vertices of graphs (Citti et al., 28 Jul 2025, Elamvazhuthi, 22 Jul 2024, Erbar et al., 2021).

b. Martingale Optimal Transport: The martingale Benamou–Brenier formulation replaces standard velocities by predictable quadratic-variation processes, leading to Fokker–Planck equations associated with purely diffusive dynamics and a dual Hamilton–Jacobi–Bellman equation (Huesmann et al., 2017, Backhoff-Veraguas et al., 2017).

c. Unbalanced and Regularized Optimal Transport: The WFR and entropy-transport extend the dynamic principle to account for creation/destruction of mass, introducing growth/decay terms and coupling to Hamilton–Jacobi equations with quadratic, entropy, and Fisher-information terms (Gallouët et al., 2021, Gigli et al., 2018). Regularization by diffusion (Schrödinger bridge/rOMT) interpolates between deterministic and entropic transport (Chen et al., 2022).

d. Multi-marginal and Conditional OT: The framework generalizes naturally to flows of multi-marginal couplings and conditional transports, replacing the conventional pairwise continuity equation with higher-dimensional mass balances and "triangular" velocity fields (Pass et al., 26 Sep 2025, Kerrigan et al., 5 Apr 2024).

e. Nonlinear Costs and Sparse Solutions: The formulation accommodates general convex cost densities, including penalizations enforcing sparsity or mixed boundary conditions, as well as extensions to color/image spaces via periodic boundary conditions in auxiliary dimensions (Bredies et al., 2019, Fitschen et al., 2015).

5. Numerical Discretization and Optimization Schemes

Time-space discretizations of the Benamou–Brenier problem are standard, and convergence of discrete to continuum solutions is guaranteed under mild mesh constraints (Lavenant, 2019). Efficient first-order convex optimization algorithms—proximal splitting, primal-dual, alternating direction methods—allow for scalable solutions in high dimensions and complex domains (Papadakis et al., 2013).

Discretizations are typically performed on staggered or centered grids, with kinetic energy evaluated cell-wise, and mass conservation enforced via sparse linear systems or Poisson equations. In color image transport, periodic boundary conditions in color or auxiliary coordinates are implemented via mixed spectral/Fourier solvers (Fitschen et al., 2015).

In multi-marginal and generalized settings, similar splitting methods apply, and coordinate-wise proximal updates yield tractable subproblems. Dynamic inverse problems are often regularized using the Benamou–Brenier ball, and optimal solutions are sparse, concentrated on "atomic" Dirac flows along absolutely continuous curves (Bredies et al., 2020, Bredies et al., 2019).

6. Applications and Theoretical Insights

The dynamic Benamou–Brenier formulation has been foundational for multiple contemporary research directions:

Numerical OT and Imaging: Benamou–Brenier is central to scalable computation of Wasserstein metrics for image morphing, color processing, and spatiotemporal imaging; see applications in diffeomorphic mapping frameworks and rOMT for neuroscience imaging (Chen, 2020, Chen et al., 2022).
Geometry of Measure Spaces: The metric induced by the kinetic formulation underpins the structure of the Wasserstein space, leading to gradient flow interpretations for evolution equations, as developed by Ambrosio–Gigli–Savaré and realized on graphs and non-smooth spaces (Erbar et al., 2021, Gigli et al., 2018).
Stochastic Control and Mean-Field Games: The proximal OT divergence, defined via Benamou–Brenier-type dynamic infima plus a terminal divergence penalty, yields a dynamic first-order mean-field game system, coupling a backward Hamilton–Jacobi equation to a forward continuity equation (Baptista et al., 17 May 2025).
Unbalanced and Martingale Transport: The framework handles entropy-regularized and stochastic constraints, interpolating between the purely deterministic and the fully “diffusive” regimes, connecting with Schrödinger bridges and martingale transport (Gallouët et al., 2021, Huesmann et al., 2017).
Optimal Control and Sub-Riemannian Transport: Recent advances establish existence and equivalence of the Benamou–Brenier and Kantorovich formulations in control-affine and sub-Riemannian regimes even without non-degeneracy or absence of abnormal minimizers, via Young measures and measurable selection (Citti et al., 28 Jul 2025, Elamvazhuthi, 22 Jul 2024).
Multi-marginal and Conditional Flows: The dynamical paradigm enables convex formulations for multi-marginal problems, efficient numerical solution by proximal splitting, and provides structure for conditional generative models (Pass et al., 26 Sep 2025, Kerrigan et al., 5 Apr 2024).

7. Extensions, Open Problems, and Key Conditions

Sufficient conditions for existence and equivalence generally require convexity of the kinetic cost $L$ , coercivity, suitable compactness of measures or spaces (e.g., completeness, absence of abnormal geodesics for sub-Riemannian structure), and boundary/marginal support constraints.

Open directions include fine regularity of geodesics and potentials, uniqueness beyond the quadratic and convex cases, handling of branching/geometric singularities (e.g., graphs or manifolds with boundary), and robust numerics for high-dimensional, multi-marginal, or infinite-dimensional problems (Pass et al., 26 Sep 2025, Elamvazhuthi, 22 Jul 2024, Lavenant, 2019).

Key technical tools include relaxation to measures on path space, superposition principles (Ambrosio–Gigli–Savaré), convexification, measurable selection of geodesics in non-smooth geometries, and functional-analytic characterization of extremal points.

In summary, the dynamic Benamou–Brenier formulation provides a canonical bridge between kinetic variational principles and static optimal transport, supports extensive geometric, analytic, and computational developments, and continues to serve as the archetype for dynamic generalizations and numerical methods in optimal transport theory (Citti et al., 28 Jul 2025, Pass et al., 26 Sep 2025, Elamvazhuthi, 22 Jul 2024, Papadakis et al., 2013, Gallouët et al., 2021, Bredies et al., 2020).