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 $epsilon$. 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) $rho$ which is a function of $varphi/H$ only, where $varphi=varphi(vec x)$ is the scalar field and $H=H(t)$ denotes the Hubble parameter. We give explicit late-time solutions for $rhorightarrow rho_infty(varphi/H)$, and thereby find the order $epsilon$ 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 $epsilon=$ constant.