Late time solution for interacting scalar in accelerating spaces

We consider stochastic inflation in an interacting scalar field in spatially homogeneous accelerating space-times with a constant principal slow roll parameter ε. We show that, if the scalar potential is scale invariant (which is the case when scalar contains quartic self-interaction and couples non-minimally to gravity), the late-time solution on accelerating FLRW spaces can be described by a probability distribution function (PDF) ρ which is a function of φ/H only, where φ=φ( x-vector ) is the scalar field and H=H(t) denotes the Hubble parameter. We give explicit late-time solutions for ρarrow ρ_∞(φ/H), and thereby find the order ε corrections to the Starobinsky-Yokoyama result. This PDF can then be used to calculate e.g. various n-point functions of the (self-interacting) scalar field, which are valid at late times in arbitrary accelerating space-times with ε= constant