Polynomially bounded solutions of the Loewner‎ ‎differential equation in several complex variables

‎We determine the‎ ‎form of polynomially bounded solutions to the Loewner differential ‎equation that is satisfied by univalent subordination chains of the‎ ‎form f(z,t)=e∫t0A(τ)dτz+⋯‎, ‎where‎ ‎A:[0,∞]→L(Cn,Cn) is a locally‎ ‎Lebesgue integrable mapping and satisfying the condition‎ ‎sups≥0∫∞0 exp{∫ts‎‎[A(τ)−2m(A(τ))In]dτ}dt ∞,‎ ‎and m(A(t)) 0 for t≥0‎, ‎where‎ ‎m(A)=min{Re⟨‎‎A(z),z⟩:∥z∥=1}‎. ‎We also give sufficient conditions‎ ‎for g(z,t)=M(f(z,t)) to be polynomially bounded‎, ‎where f(z,t) is‎ ‎an A(t)-normalized polynomially bounded Loewner chain solution to‎ ‎the Loewner differential equation and M is an entire function‎. ‎On ‎the other hand‎, ‎we show that all A(t)-normalized polynomially‎ ‎bounded solutions to the Loewner differential equation are Loewner‎ ‎chains.‎