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

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