Abstract: This lecture is divided in three main stages:
I-
In the first part, we resume an interesting feature of inflationary
cosmology, which is the slow-roll regime, and its attractor behaviour in
the phase-space;
II- In the second part, we present the stochastic
formalism, which is a non-perturbative approach interesting for
computing large and rare fluctuations in the primordial density field,
showing how backreaction from quantum fluctuations can be incorporated
in this new approach;
III- In the third part, as a first example on
how use the stochastic formalism, we provide the explicit solution for a
case of a massless field in a de Sitter background, showing that indeed
quantum diffusion takes place exactly along the slow-roll attractor
without modifying its classical behaviour;
We will also discuss other scenarios beyond the slow-row regime and the current open questions in this field.