Speaker
Description
Stochastic inflation allows the study of large inflationary fluctuations, casting their evolution into the form of a Fokker-Planck equation. I discuss solving this equation using the spectral decomposition method, a technique underutilized in modern stochastic inflation studies. The method gives easy access to the late-time distributions of the inflaton field and its first-passage times through the eigenvalues and eigenfunctions of the Fokker-Planck operator. The lowest eigenvalues control the late-time and diffusion-dominated regimes associated with tunneling and eternal inflation; higher eigenvalues are needed to describe classical motion. I demonstrate this in the case of constant-roll inflation in a hilltop potential, relevant for primordial black hole models. I compare the stochastic computation with the classical $\Delta N$ formalism, illustrating, in particular, the difficulties that arise from trajectories that pass `beyond the hilltop,' into the diffusion-dominated regime.