Review on stochastic approach to inflation

TL;DR

This review discusses the mathematical framework of stochastic inflation, focusing on its derivation, key differences from other approaches, and applications to various inflationary scenarios, including primordial black hole formation.

Contribution

It provides a comprehensive overview of stochastic inflation, highlighting the derivation process, differences from the separate universe approach, and the non-Markovian nature of the stochastic noises.

Findings

Clarifies the derivation of stochastic inflation formalism.

Highlights the non-Markovian nature of stochastic noises.

Connects stochastic inflation to primordial black hole formation scenarios.

Abstract

We present a review on the state-of-the-art of the mathematical framework known as stochastic inflation, paying special attention to its derivation and giving references for the readers interested on results coming from the application of the stochastic framework to different inflationary scenarios, especially to those of interest for primordial black hole formation. During the derivation of the stochastic formalism, we will emphasise two aspects in particular: the difference between the separate universe approach and the true long wavelength limit of scalar inhomogeneities and the generically non-Markovian nature of the noises that appear in the stochastic equations.