The stochastic formalism known as an effective theory of superHubble coarse-grained fields is a powerful tool for a non-perturbative analysis of the inflationary universe. It views the coarse-grained inflaton field as a stochastic Brownian motion and its random behaviour leaves an imprint for the later universe as the primordial curvature perturbation.
In this talk, I will first review the basic derivation of the stochastic formalism and then describe in detail the so-called stochastic-deltaN approach to calculate the curvature perturbation in the stochastic formalism.
I will formulate the power spectrum and the probability density function of the coarse-grained curvature perturbation and show its numerical implementation.
As practical examples, the hybrid-inflation and flat-inflection models are discussed, related to the recent hot topic: primordial black holes.
I may mention its further application to gauge field production in axion inflation beyond the dynamics of the scalar inflaton fields.
