Resonances in Loewner equations

We prove that given a Herglotz vector field on the unit ball of Cn of the form H(z,t)=(a1z1,…,anzn)+O(2|z|) with for all 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.