Stochastic Approach to a non de Sitter Inflation

Abstract

We develop a stochastic approach to a non de Sitter Universe in a gauge-invariant way and obtain a system of Langevin-type equations which may be considered to be renormalization group equations for the long wave parts of the scalar fields and metric. We investigate in detail the case of generalized power-law inflation that appears in the model of the deflationary exit from the inflationary stage. For this case the above system is simplified greatly, we derive a Fokker-Planck equation for the scalar field driving inflation, find its stationary solutions and solve the FP equation for a delta-function initial distribution of the field.