Stochastic inflation at all order in slow-roll parameters: Foundations

In this paper we develop the formalism for stochastic approach to inflation at all order in slow-roll parameters. This is done by including momentum and Hamiltonian constraints into equations. We then specialise widely used Starobinski approximation where interactions between IR UV modes are neglected. show that, whenever holds, no significant deviations observed when comparing two-point correlation functions (power spectrum) calculated with methods, ones QFT linear theory. As a byproduct, argue that: a) approaches based on approximation, generically, do not capture any loop effects of quantum scalar-gravity system; b) correlations can only be theory regimes, thus, non-perturbative statistics extracted within as commonly claimed.