The time-dependent Tsallis statistical distribution describing anomalous diffusion is usually obtained in the literature as the solution of a nonlinear Fokker-Planck (FP) equation [A.R. Plastino and A. Plastino, Physica A 222, 347 (1995)]. The scope of the present paper is twofold. First, we show that this distribution can be obtained also as a solution of the nonlinear porous media equation. Second, we prove that the time-dependent Tsallis distribution can be obtained also as a solution of a linear FP equation [G. Kaniadakis and P. Quarati, Physica A 237, 229 (1997)] with coefficients depending on the velocity, which describes a generalized Brownian motion. This linear FP equation is shown to arise from a microscopic dynamics governed by a standard Langevin equation in the presence of multiplicative noise.