Free scalar correlators in de Sitter space via the stochastic approach beyond the slow-roll approximation

The stochastic approach to calculating scalar correlation functions in de Sitter spacetime is extended beyond the overdamped ``slow-roll'' approximation. We show that with the correct noise term, it reproduces the exact asymptotic long-distance behavior of field correlators in free field theory, thereby demonstrating the viability of the technique. However, we also show that the na\"{\i}ve way of calculating the noise term by introducing a cutoff at the horizon does not give the correct answer unless the cutoff is chosen specifically to give the required result. We discuss the implications of this for interacting theories.