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  "body": "\n## Definition\n\nLet $\\mathbb{H} = \\set{z \\in \\C : \\operatorname{Im} z \u003e 0}$ be the [[upper half-plane|X6XGGH5FPW]], and let $U \\colon [0, \\infty) \\to \\R$ be a [[continuous function|0X3WGX95FE]], the *driving function*.\nThe *chordal Loewner differential equation* with driving function $U$ is the initial value problem\n$$\\frac{\\partial}{\\partial t} g_t(z) = \\frac{2}{g_t(z) - U(t)}, \\qquad g_0(z) = z,$$\nfor each $z \\in \\overline{\\mathbb{H}}$.\nFor each $z$, let $T_z \\in (0, \\infty]$ be the supremum of the times up to which the solution exists with $g_t(z) - U(t) \\ne 0$; for $z = U(0)$ set $T_z = 0$.\nFor $t \\ge 0$, the *hull* at time $t$ is $K_t = \\set{z \\in \\mathbb{H} : T_z \\le t}$.\n\n## Remarks\n\nFor each $t$, the map $g_t$ is a [[conformal map|0VHNKP1N3K]] of $\\mathbb{H} \\setminus K_t$ onto $\\mathbb{H}$ with the normalisation $g_t(z) = z + \\frac{2t}{z} + O(\\abs{z}^{-2})$ as $z \\to \\infty$.\nThe hulls $K_t$ increase with $t$, and the number $2t$ is their half-plane capacity.\nExample: for $U = 0$, the equation gives $\\partial_t (g_t(z)^2) = 4$, so $g_t(z) = \\sqrt{z^2 + 4t}$ with the branch that maps $\\mathbb{H}$ into $\\mathbb{H}$.\nA point $iy$ with $y \u003e 0$ has $g_t(iy)^2 = 4t - y^2$, which reaches $0 = U(t)$ at $t = y^2/4$; so $K_t$ is the vertical segment from $0$ to $2i\\sqrt{t}$.\nNon-example: the translations $f_t(z) = z + t$ are conformal self-maps of $\\mathbb{H}$, but they do not come from the equation, since $f_t(z) - z = t$ does not tend to $0$ at infinity.\nThe *radial* Loewner equation, which Loewner introduced first, is the analogue for the unit disc, with a driving function with values on the unit circle and maps normalised at $0$.\n\n## Explanations\n\n### L1 human\n\nEach point of the half-plane flows according to a simple rule: it is pushed away from the moving point $U(t)$ on the real line, faster when it is closer.\nA point that is pushed into $U(t)$ is swallowed and becomes part of the hull.\nThe equation turns a real function of time into a growing set in the plane and a family of maps that straighten it out.\n",
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