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 o. 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 phi/H only, where phi = phi((x) over right arrow) is the scalar field and H = H(t) denotes the Hubble parameter. We give explicit late-time solutions for rho -> rho(infinity)(phi/H), and thereby find the order o 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.