Feynman Rules for Stochastic Inflationary Correlators

We elaborate on the functional integral describing the stochastic dynamics of a spectator field during inflation, comparing its diagrammatic expansion to that obtained directly from a perturbative solution of the corresponding Langevin equation. We state Feynman rules for computing arbitrary temporal $n$-point functions and perform some illustrative computations for a $\lambda\phi^4$ interaction, paying attention to the role played by a functional Jacobian determinant in the path integral. We also briefly consider the case when the field contributes to the expansion rate, making the noise multiplicative, which introduces additional vertices.