Limit theorems for asymmetric transportation networks

We consider a model of an asymmetric transportation network. The transportation network is described by the Markov process $U_N(t)$. This process has values in a compact subset of the finite-dimensional real vector space $\mathbb R^{\alpha}$. We prove that $U_N(t)$ converges in distribution to a non-linear dynamical system $\mathbf g\to \mathbf u(t,\mathbf g)$ (assuming convergence of initial distributions $U_N(0)\to\mathbf g$), where $\mathbf g\in\mathbb R^{\alpha}$. The dynamical system has the only invariant measure to which the invariant measures of processes $U_N(t)$ converge as $N\to\infty$.