 Solution

For a continuous real driving function $U$, solve the Chordal Loewner equationup to the first time $T_z$ at which $g_t(z)-U_t$ tends to zero. The generated hulls areThey form an increasing Loewner chain of compact H-hulls, satisfy $\mathrm{hcap}(K_t)=2t$, and $g_t$ is the mapping-out function of $K_t$.