The temporal increase of the moments in diffusion on a fractal with variable hopping range and lower cut-off is given. The essential parameters are the growth ratio, the length scaling and, as a new feature, the time scaling along the hierarchy. We find algebraical or exponential increase, logarithmic corrections, or trapping if the cut-off is removed. For the first time anomalous enhancement of the variance increase σ'0 tθ, θ larger than 2, is obtained as observed in turbulence On etudie la croissance temporelle des moments de la distribution de particules diffusant sur un fractal a portee de saut variable avec une coupure inferieure. Les parametres essentiels sont: le taux de croissance, le facteur d'echelle de la longueur et celui du temps le long de la hierarchie, ce dernier critere est nouveau. On trouve des lois de croissance algebriques et exponentielles et des corrections logarithmiques, ou un piegeage si la coupure est eliminee. Une augmentation anormale du taux de croissance de la variance σαtθ, θ>2, comme cela a deja ete observe pour la turbulence, est obtenue pour la 1ere fois
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