A holomorphic vector field \(H\) on a domain \(D\subset\mathbb C^N\) is semicomplete if the Cauchy problem \(\dot x(t)=H(x(t))\), \(x(0)=z_0\), has a solution defined for all \(t\in\mathbb R^+:=[0,\infty)\) and \(z_0\in D\). A Herglotz vector field of order \(d\in[1,\infty]\) on \(D\) is a mapping \(G: D\times\mathbb R^+\to\mathbb C^N\) such that \(G(z,\cdot)\) is measurable on \(\mathbb R^+\) for all \(z\in D\), \(G(\cdot,t)\) is a semicomplete holomorphic vector field on \(D\) for all \(t\in\mathbb R^+\), and, for any compact set \(K\subset D\) and all \(T>0\), there exists a function \(C_{T,K}\in L^d([0,T],\mathbb R^+)\) for which \(\|G(z,t)\|\leq C_{T,K}(t)\), \(z\in K\), \(a.a.t\in[0,T]\). A family \((\varphi_{s,t})_{0\leq s\leq t}\) of holomorphic self-mappings of \(D\) is an evolution family of order \(d\) if \(\varphi_{s,s}=\mathrm{id}\), \(\varphi_{s,t}=\varphi_{u,t}\circ\varphi_{s,u}\), \(0\leq s\leq u\leq t\), and if for any \(T>0\) and any compact set \(K\subset\subset D\) there exists a function \(c_{T,K}\in L^d([0,T],\mathbb R^+)\) such that \[ \|\varphi_{s,t}(z)-\varphi_{s,u}(z)\|\leq\int_u^tc_{T,K}(\xi)d\xi, \;\;z\in K,\;\;0\leq s\leq u\leq t\leq T. \] For a non-empty open subset \(M\) of a complex manifold \(\tilde M\), the pair \((M,\tilde M)\) is a Runge pair if \(\mathcal O(\tilde M)\) is dense in \(\mathcal O(M)\). A domain \(D\subset\mathbb C^N\) is Runge if \((D,\mathbb C^N)\) is a Runge pair.
The main result of the paper can be stated in the following way.
Theorem 1.1. For a complete hyperbolic starlike domain \(D\subset\mathbb C^N\), let \(G:D\times\mathbb R^+\to\mathbb C^N\) be a Herglotz vector field of order \(d\in[1,\infty]\). Then there exists a family of univalent mappings \((f_t:D\to\mathbb C^N)\) of order \(d\) which solves the Loewner PDE \[ \frac{\partial f_t}{\partial t}(z)=-df_t(z)G(z,t),\;\;\text{ a. a. }t\geq0 \text{ and for all } z\in D. \] Moreover, \(R:=\bigcup_{t\geq0}f_t(D)\) is a Runge and Stein domain in \(\mathbb C^N\) and any other solution to the Loewner PDE is of the form \((\Phi\circ f_t)\) for a suitable holomorphic map \(\Phi:R\to\mathbb C^N\).
It is also proved that, for a univalent self-mapping \(\varphi\) of the unit ball \(\mathbb B^N\), without assuming that \(\varphi(\mathbb B^N)\) is Runge, the question whether there exists an evolution family \((\varphi_{s,t}:\mathbb B^n\to\mathbb B^N)\) of order \(d\in[1,\infty]\) such that \(\varphi_{0,1}=\varphi\) has a negative answer.