Inflation is currently the leading paradigm describing the physical conditions that prevail in the very early universe. During this epoch, quantum fluctuations are extracted out of the vacuum and amplified to cosmological perturbations at astronomical scales, later seeded CMB anisotropies and large-scale structures. As inflation proceeds, this inflow of super-horizon perturbations may impact on the large-scale dynamics of the universe, potentially perturbing the inflationary trajectory. The stochastic inflation formalism is an effective approach to account for such modifications of the large-scale dynamics where the amplified quantum fluctuations act as a source of stochastic noise. In this talk, I will present in a pedestrian way such a formalism, and advocate why such an approach should be extended to the full phase space. One of the appealing features of inflation is the presence of a phase-space attractor, dubbed slow-roll. Considering first the case of a test and free scalar field, I will show that quantum noise (hence the quantum diffusion) is aligned with the slow-roll attractor. For non-test and non-free scalar field, some misalignment appears which is however small enough for letting slow-roll to remain an attractor of the stochastic dynamics at large scales. If time permits, I will briefly discuss how such a formalism could be used to study bouncing cosmologies as well as the stochastic generation of super-horizon anisotropies.