  We study statistical models for one-dimensional diffusions which are
recurrent null. A first parameter in the drift is the principal one, and
determines regular varying rates of convergence for the score and the
information process. A finite number of other parameters, of secondary
importance, introduces additional flexibility for the modelization of the
drift, and does not perturb the null recurrent behaviour. Under time-continuous
observation we obtain local asymptotic mixed normality (LAMN), state a local
asymptotic minimax bound, and specify asymptotically optimal estimators.
