Resonances in Loewner equations

We prove that given a Herglotz vector field on the unit ball of ${\mathbb{C}}^n$ of the form $H(z,t)=(a_1{z}_1,\mathrm{.}\mathrm{.}\mathrm{.},a_n{z}_n)+O(\mathrm{\mid }z{\mathrm{\mid }}^2)$ with $\Re a_j<0$forall$j$, its evolution family admits an associated Loewner chain, which is normal if no real resonances occur. Hence the Loewner-Kufarev PDE admits a solution defined for all positive times.