Abstract: Quantum fluctuations during cosmological inflation are usually assumed to be small enough such that they are well described as small perturbations over a homogeneous and isotropic space-time background. However rare large quantum fluctuations can also be randomly generated during inflation. These large inhomogeneities are of great interest because they can lead to the formation of Primordial Black Holes (PBH), the probability of its generation is related with the amplitude of the power spectrum. Both the amplitude of the power spectrum and the non-gaussianities measured at the Cosmic Microwave Background (CMB) are too small to generate a relevant amount of large fluctuations. To form enough PBH that could represent for example a significant fraction of the dark matter, we need to exponentially enhance the amplitude of the power spectrum on scales which are not probed by the CMB, for which a violation of SR is needed. Although the growth of the power spectrum can be described at leading order in perturbation theory, the description of the tail of the Probability Distribution Function (PDF), where the large inhomogeneities are located, must be done beyond linear perturbation theory, hopefully in a non-perturbative way.
In this thesis, we explore the foundations of some of the mathematical frameworks that aim to describe the inhomogeneities generated during inflation in a non-perturbative way such as the stochastic approach or the \deltaN formalism, we highlight some limitations usually overlooked in the literature, particularly during inflationary regimes beyond SR. Finally, we propose for the first time a stochastic formalism which is in principle able to describe inflation in a non-perturbative way and during any inflationary regime with high precision.