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  "body": "\n## Statement\n\nLet $\\mathcal{P}_2(\\R^n)$ be the set of Borel [[probability measures|059TJN0NZ1]] on $\\R^n$ with finite second moment.\nLet $W_2$ be the [[Wasserstein distance|VEY29CS26X]] of order $2$ on it.\nLet $\\mu_0, \\mu_1 \\in \\mathcal{P}_2(\\R^n)$.\nConsider pairs $(\\mu_t, v_t)_{t \\in [0, 1]}$ such that:\n\n1. $t \\mapsto \\mu_t$ is a weakly continuous curve in $\\mathcal{P}_2(\\R^n)$ from $\\mu_0$ to $\\mu_1$;\n2. $(t, x) \\mapsto v_t(x) \\in \\R^n$ is a Borel vector field with $\\int_0^1 \\int \\abs{v_t}^2 \\, d\\mu_t \\, dt \u003c \\infty$;\n3. the continuity equation $\\partial_t \\mu_t + \\operatorname{div}(v_t \\mu_t) = 0$ holds in the sense of distributions on $(0, 1) \\times \\R^n$.\n\nThen\n$$W_2(\\mu_0, \\mu_1)^2 = \\inf \\int_0^1 \\int_{\\R^n} \\abs{v_t(x)}^2 \\, d\\mu_t(x) \\, dt,$$\nwhere the infimum runs over all such pairs, and it is attained.\n\n## Remarks\n\nJean-David Benamou and Yann Brenier introduced the formula in 2000, as a basis for numerical computation of optimal transport.\nTo 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.\nSo $\\mu_t$ is the image of $\\pi$ under $(x, y) \\mapsto (1 - t) x + t y$, and the velocity of each particle is $y - x$.\nThe energy is then $\\int \\abs{x - y}^2 \\, d\\pi = W_2(\\mu_0, \\mu_1)^2$.\nFor the other inequality, note that a curve with velocity field $v$ is the image of the flow of $v$.\nJensen's inequality then bounds the cost of the induced coupling by the kinetic energy.\nExample: 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 $\\abs{b - a}^2 = W_2(\\delta_a, \\delta_b)^2$.\nIn the substitution $m_t = v_t \\mu_t$ the problem becomes convex in $(\\mu, m)$, which is what makes it useful for computation.\n\n## Explanations\n\n### L1 human\n\nThe Wasserstein distance measures the cost of moving one distribution of mass onto another.\nThe Benamou–Brenier formula says that this cost is the least kinetic energy of a flow of mass that carries the first distribution to the second in unit time.\nSo the static problem of matching points becomes a dynamic problem of moving a fluid.\n\n### L2 human\n\nThe 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 $\\int \\abs{v}^2 \\, d\\mu$.\nOptimal curves are the displacement interpolations, along which displacement convexity is measured.\nThe same structure underlies the interpretation of the heat and Fokker–Planck equations as gradient flows, as in the Jordan–Kinderlehrer–Otto scheme.\n",
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