Solution
= Solution
For a continuous real driving function $U$, solve the <Chordal Loewner equation>
$$
\partial_tg_t(z)=\frac2{g_t(z)-U_t},
\qquad g_0(z)=z,
$$
up to the first time $T_z$ at which $g_t(z)-U_t$ tends to zero. The generated hulls are
$$
K_t=\{z\in\mathbb H:T_z\leq t\}.
$$
They form an increasing <Loewner chain> of compact H-hulls, satisfy $\operatorname{hcap}(K_t)=2t$, and $g_t$ is the mapping-out function of $K_t$.