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 $f(z,t)=e^{int_0^t A(tau){rm d}tau}z+cdots$‎, ‎where‎ ‎$A:[0,infty]rightarrow L(mathbb{C}^n,mathbb{C}^n)$ is a locally‎ ‎Lebesgue integrable mapping and satisfying the condition‎ ‎$$sup_{sgeq0}int_0^inftyleft|expleft{int_s^t‎ ‎[A(tau)-2m(A(tau))I_n]rm {d}tauright}right|{rm d}t0$ for $tgeq0$‎, ‎where‎ ‎$m(A)=min{mathfrak{Re}leftlangle‎ ‎A(z),zrightrangle:|z|=1}$‎. ‎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.‎