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

LAMN in a class of parametric models for null recurrent diffusion