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

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