In order to provide a probability framework for the time-fractional diffusion equation, neither classical random walk models, based on finite difference scheme for the fractional derivative, nor continuous time random walks seem satisfying. This is because they do not provide stochastic processes with stationary increments. The Grey noise theory provides a general stochastic framework that leads up to a class of H-sssi processes that have as one-dimensional probability density the fundamental solutions of the time-fractional diffusion equation. This class is extended in order to contain the fractional Brownian motion as particularly case. The related stochastic processes can be adopted to model anomalous (slow and fast) diffusion in statistical physics