Recent Developments in Stochastic Inflation

This article is dedicated to the memory of Alexei Starobinsky. I begin with some recollections of him and then review the generalization of his wonderful stochastic formalism from scalar potential models to theories which interact with fermions and photons, and finally to theories with derivative interactions such as nonlinear sigma models and gravity. This entails effective potentials generated by the usual field-dependent masses, as well as by field-dependent field strengths, and by field-dependent Hubble parameters. I also discuss secular loop corrections which cannot be captured by stochastic techniques.