= Capacity-parametrized scale invariance of a Loewner chain
{title2=$\lambda^{-1}K_{\lambda^2t}\overset d=K_t$}
For a <Loewner chain> with <half-plane capacity> $2t$, scale invariance means $(\lambda^{-1}K_{\lambda^2t})_{t\geq0}$ has the original law for every $\lambda>0$. Its mapping-out maps are $\lambda^{-1}g_{\lambda^2t}(\lambda z)$ and its <Loewner driving function> is $\lambda^{-1}U_{\lambda^2t}$. This follows by differentiating the <Chordal Loewner equation>. <Brownian scaling> gives this property for <Schramm–Loewner evolution>.
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