Sub-Leading Logarithms for Scalar Potential Models on de Sitter

The continual production of long wavelength scalars and gravitons during inflation injects secular growth into loop corrections which would be constant in flat space. One typically finds that each additional factor of the loop counting parameter can induce up to a certain number of logarithms of the scale factor. Loop corrections that attain this number are known as ``leading logarithms’'; those with fewer are sub-leading. Starobinsky’s stochastic formalism has long been known to reproduce the leading logarithms of scalar potential models. We show that the first sub-leading logarithm is captured by applying the stochastic formalism to a certain part of the 1-loop effective potential. This is checked at 2-loops for a massless, minimally coupled scalar with a quartic self-interaction on de Sitter background.