We consider the kinetics of the diffusion process of an assembly of N o initially aggregated particles. The particles are non-interacting except that the double occupancy is forbidden. A computer simulation provides the number of distinct visited sites after a given number of steps performed by the particles on a 2D square grid. The particles are supposed to diffuse according to a random-walk process on the lattice where all jumps to the nearestneighbour sites are allowed. As the system exhibits a centrifugal diffusion the simulation shows that the number of distinct visited sites obeys three distinct successive processes. The first one which occurs right at the start of the diffusion involves a classic diffusion in the 2D space and correspondingly displays the classic Euclidean laws. Then the system exhibits a crossover to fractal-like kinetics and extends over at least three time decades where the exponent characteristic of the power law is determined by the number of distinct visited sites as a function of time. The third process takes places after a fractal to Euclidean space crossover has taken place when the particles are far enough from each other so as not to interfere any more, thereby leading to a classic diffusion process. The various singular points of these three processes have been determined analytically as functions of N o and of the exponent d, whereas the N o dependence of d is derived from the computer simulation. These results which are relevant to several significant physical situations shed a new light on the processes where the fractal nature of the kinetics depends on the initial conditions of the problem rather than on the mere geometry of the diffusion paths.