$Z[J,\tilde{J}] = \int DxD\tilde{x} \exp(-S[x,\tilde{x}] + \int J x dt + \tilde{J}\tilde{x} dt) $ is the moment generating function for a stochastic path integral. We may approximate this function by splitting the action $S[x,\tilde{x}]$ into a "Free Action", $S_F$, and an "interaction action", $S_I$, and then taylor expanding the exponential around the free action, for example
$Z[J,\tilde{J}] = \int Dx D\tilde{x} e^{-S_F} (1 + (\int \tilde{x}x^2 dt + \int \tilde{x}^2 x dt + \tilde{x}(0) + \int J x dt + \int \tilde{J}\tilde{x} dt) + \text{higher order terms}) $
where $S_F =\int \tilde{x} (\dot{x}+x) dt$.
However, I become confused by the next claim: if the Free action is bilinear in $x$ and $\tilde{x}$ then only terms with an equal number of $x$ and $\tilde{x}$ factors survive? Would anyone be able to expand on this maybe with an example?