Benamou–Brenier formula

Remarks

Jean-David Benamou and Yann Brenier introduced the formula in 2000, as a basis for numerical computation of optimal transport. To see that the infimum is at most $W_2(\mu_0, \mu_1)^2$, take an optimal coupling $\pi$ and move mass along straight lines. So $\mu_t$ is the image of $\pi$ under $(x, y) \mapsto (1 - t) x + t y$, and the velocity of each particle is $y - x$. The energy is then $\int \abs{x - y}^2 \, d\pi = W_2(\mu_0, \mu_1)^2$. For the other inequality, note that a curve with velocity field $v$ is the image of the flow of $v$. Jensen's inequality then bounds the cost of the induced coupling by the kinetic energy. Example: for $\mu_0 = \delta_a$ and $\mu_1 = \delta_b$, the curve $\mu_t = \delta_{(1 - t)a + tb}$ with constant velocity $b - a$ has energy $\abs{b - a}^2 = W_2(\delta_a, \delta_b)^2$. In the substitution $m_t = v_t \mu_t$ the problem becomes convex in $(\mu, m)$, which is what makes it useful for computation.

Going deeper

For a student

The formula shows that $(\mathcal{P}_2(\R^n), W_2)$ behaves like a Riemannian manifold, in Otto's formal calculus: tangent vectors are velocity fields, and the squared length of a tangent vector is $\int \abs{v}^2 \, d\mu$. Optimal curves are the displacement interpolations, along which displacement convexity is measured. The same structure underlies the interpretation of the heat and Fokker–Planck equations as gradient flows, as in the Jordan–Kinderlehrer–Otto scheme.