On averaging principle for diffusion processes with null-recurrent fast component

An averaging principle is proved for diffusion processes of type (Xε(t),Yε(t)) with null-recurrent fast component Xε(t). In contrast with positive recurrent setting, the slow component Yε(t) alone cannot be approximated by diffusion processes. However, one can approximate the pair (Xε(t),Yε(t)) by a Markov diffusion with coefficients averaged in some sense.