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

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 behaviour of field correlators in free field theory, thereby demonstrating the viability of the technique. However, we also show that the na\"ive way of calculating the noise term by introducing a cut-off at the horizon does not give the correct answer unless the cut-off is chosen specifically to give the required result. We discuss the implications of this for interacting theories.