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  "body": "\n## Definition\n\nLet $\\kappa \u003e 0$, and let $(B_t)_{t \\ge 0}$ be a standard one-dimensional [[Brownian motion|22735FGJV6]] with $B_0 = 0$.\nLet $\\mathbb{H}$ be the [[upper half-plane|X6XGGH5FPW]].\n*Chordal $\\mathrm{SLE}_\\kappa$* in $\\mathbb{H}$ from $0$ to $\\infty$ is a random family of conformal maps $(g_t)_{t \\ge 0}$ and hulls $(K_t)_{t \\ge 0}$.\nIt is given by the chordal [[Loewner differential equation|51G4QJZ345]]\n$$\\frac{\\partial}{\\partial t} g_t(z) = \\frac{2}{g_t(z) - \\sqrt{\\kappa}\\, B_t}, \\qquad g_0(z) = z,$$\nwith the driving function $U(t) = \\sqrt{\\kappa}\\, B_t$.\nLet $D \\ne \\C$ be a simply connected domain with two distinct boundary points $a$ and $b$.\nLet $\\phi$ be a [[conformal map|0VHNKP1N3K]] of $\\mathbb{H}$ onto $D$ that sends $0$ to $a$ and $\\infty$ to $b$.\nChordal $\\mathrm{SLE}_\\kappa$ in $D$ from $a$ to $b$ is the image under $\\phi$ of chordal $\\mathrm{SLE}_\\kappa$ in $\\mathbb{H}$.\n\n## Remarks\n\nThe definition in $D$ does not depend on the choice of $\\phi$, because two choices differ by a scaling $z \\mapsto cz$ of $\\mathbb{H}$ with $c \u003e 0$.\nBrownian scaling shows that such a scaling only changes the time parameter of the hulls.\nAlmost surely the hulls are generated by a continuous curve $\\gamma$: $\\mathbb{H} \\setminus K_t$ is the unbounded component of $\\mathbb{H} \\setminus \\gamma[0, t]$.\nThis was proved by Rohde and Schramm for $\\kappa \\ne 8$, and by Lawler, Schramm and Werner for $\\kappa = 8$.\nThe curve is simple for $\\kappa \\le 4$, touches itself for $4 \u003c \\kappa \u003c 8$, and fills space for $\\kappa \\ge 8$.\nExample: for $\\kappa = 2$, $\\mathrm{SLE}_2$ is the scaling limit of loop-erased random walk, a theorem of Lawler, Schramm and Werner.\nFor $\\kappa = 0$ the driving function is $0$ and the hulls are deterministic vertical segments; this degenerate case is sometimes included as $\\mathrm{SLE}_0$.\nRadial $\\mathrm{SLE}_\\kappa$ is defined in the same way from the radial Loewner equation.\n",
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