This paper examines particle diffusion in N-dimensional Euclidean space with traps of the return type. Under the assumption that the random continuous-diffusion time has a finite mean value, it is established that subdiffusion (which is characterized by an increase in the width of the diffusion packet with time according to the t~(#alpha#)-law, where #alpha# < 1; for normal diffusion #alpha# = 1) emerges if and only if the distribution density of the random time a particle spends in a trap has a tail of the power-law type propor. to t~(#alpha#-1). In these conditions the asymptotic expression for the distribution density of a diffusing particle is found in terms of the density of a one-sided stable law with a characteristic exponent #alpha#. It is shown that the density is a solution of subdiffusion equations in fractional derivatives. The physical meaning of the solution is discussed, and so are the properties of the solution and its relation to the results of other researchers in the field of anomalous-diffusion theory.
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