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  "body": "\n## Definition\n\nLet $\\mathcal{P}_2(\\R^n)$ be the set of Borel [[probability measures|059TJN0NZ1]] on $\\R^n$ with finite second moment, with the [[Wasserstein distance|VEY29CS26X]] $W_2$.\nLet $F \\colon \\mathcal{P}_2(\\R^n) \\to (-\\infty, +\\infty]$ be a functional, let $\\tau \u003e 0$ be a time step, and let $\\rho_0 \\in \\mathcal{P}_2(\\R^n)$ with $F(\\rho_0) \u003c \\infty$.\nThe *Jordan–Kinderlehrer–Otto scheme* for $F$ with step $\\tau$ is a sequence $(\\rho_k^\\tau)_{k \\ge 0}$ in $\\mathcal{P}_2(\\R^n)$ with $\\rho_0^\\tau = \\rho_0$.\nFor each $k \\ge 0$, the measure $\\rho_{k+1}^\\tau$ is required to minimize the functional\n$$\\rho \\mapsto F(\\rho) + \\frac{1}{2\\tau} W_2(\\rho, \\rho_k^\\tau)^2$$\nover $\\mathcal{P}_2(\\R^n)$.\nThe associated discrete flow is the piecewise constant curve $\\rho^\\tau(t) = \\rho_k^\\tau$ for $t \\in [k\\tau, (k + 1)\\tau)$.\n\n## Remarks\n\nExample: let $\\beta \u003e 0$, let $\\Psi \\colon \\R^n \\to [0, \\infty)$ be smooth with $\\abs{\\nabla \\Psi(x)} \\le C(\\Psi(x) + 1)$ for a constant $C$, and let\n$$F(\\rho) = \\int \\Psi \\rho \\, dx + \\beta^{-1} \\int \\rho \\log \\rho \\, dx$$\nfor $\\rho$ with a density, and $F(\\rho) = +\\infty$ otherwise.\nJordan, Kinderlehrer and Otto proved that minimizers exist at each step, and that $\\rho^\\tau$ converges, as $\\tau \\to 0$, to the solution of the Fokker–Planck equation $\\partial_t \\rho = \\operatorname{div}(\\rho \\nabla \\Psi) + \\beta^{-1} \\Delta \\rho$ with initial datum $\\rho_0$.\nWith $\\Psi = 0$ and $\\beta = 1$ this gives the heat equation $\\partial_t \\rho = \\Delta \\rho$.\n\nNon-example: replacing $W_2$ by the $L^2$ distance between densities gives a different scheme, the implicit Euler scheme for the $L^2$ gradient flow.\nFor the entropy $\\int \\rho \\log \\rho$ this flow is $\\partial_t \\rho = -(\\log \\rho + 1)$, which is not the heat equation.\n\nThe step is the Wasserstein analogue of the implicit Euler step $x_{k+1} \\in \\operatorname{argmin}_x \\left( f(x) + \\frac{1}{2\\tau} \\abs{x - x_k}^2 \\right)$ for the gradient flow $\\dot x = -\\nabla f(x)$ in $\\R^n$.\n",
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