In most papers dealing with random motions and diffusion of small magnetic elements in the photosphere, the convective flows and in particular the granulation are considered as drivers of these motions. The results of these works have been discussed in terms of the fractal dimension of the granulation as seen in intensity pictures. So far neither a fractal dimension associated with the granular velocity field nor the nature of the random walks in the granular intergranular space have been determined. Using spectrograms of high spatial resolution taken with the VTT at Izana (Tenerife, Spain) we investigated the granular velocity field in terms of its fractal nature and its diffusion properties. We applied the rescaled range analysis to both the velocity and intensity fields, thus enabling us to calculate a fractal dimension as well as a "diffusion" exponent which together characterize the diffusion properties of the granulation layers. We found a fractal dimension of the granular velocity of the same order as the fractal dimensions of the distribution of the magnetic elements in the photosphere, and the fractal dimension corresponding to the diffusion of the magnetic elements in a fractal geometry. The diffusion processes in the granulation layers show a subdiffusive nature characteristic of anomalous diffusion rather than the classical Fickian diffusion. Anomalous diffusion is often found in stochastic transport in spatially heterogeneous media. The velocity field of the granulation can be thought of as a heterogeneous turbulent medium: the granules show less turbulence than the intergranular space.