The dynamics of the Higgs and other light scalar fields during inflation can have important cosmological consequences, but because of the infrared problem, they cannot be computed using perturbation theory. A powerful alternative is the stochastic Starobinsky-Yokoyama approach, which is based on the observation that on superhorizon distances the field behaves classically, with a noise term produced by subhorizon quantum modes. It has been mostly used to calculate the one-point probability distribution of the field, but its real power lies in describing the asymptotic long-distance dynamics through a spectral expansion. I demonstrate this by calculating correlation functions and decay rates of metastable vacua. I use them to determine the parameters of the theory at one-loop order in perturbation theory, showing that they do not suffer from the same infrared problems as a direct perturbative computation of observables, and providing evidence that they are given by the constraint effective potential.