Why perturbation theory in Fourier space works?

Question

I am trying to learn some statistical field theory topics having pure math background and I cannot understand some things. I would formulate the question for time-dependent fields but I guess it makes sense to ask it even when one has spatial coordinates only.

Suppose we have a time dependent field $\phi(t)$ (e.g. a solution to an SDE) that is known to be stationary in time. I would like to write a perturbation expansion (loop expansion) for $<\phi> \equiv <\phi(t)>,\ \forall t$.

As far as I understand, after splitting action in free and interacting parts and applying Wick theorem, there are two ways of doing it:

1. Write the expansion directly in time domain, representing how higher order cumulants (involving previous time points) impact the mean, by diagrams. Then one can truncate the expansion and hope that the thrown away part has much lower impact, than the terms that we did not truncate
2. Apply Fourier transform to the free action cumulants, do expansion in frequency domain, do truncation at some level and then apply inverse Fourier transform to the resulting expression

Although it is still mysterious to me, I got used to the fact that in physics one believes that when noise term in the action is small, it is possible to truncate expansion at finite loop level and have a reasonable approximation. But I can only make myself believe in it when one does the case 1), the time domain expansion.

Question: what are the heuristics that explain why truncation of the Fourier domain loop expansion makes sense?

More explicit setup I am thinking about is here, but I am asking more about the general idea.

UPD: one of the reasons for the question is that from time domain 1-loop expansion and from frequency domain 1-loop expansion (converted back to time domain) one gets different expressions and it is not clear which one is supposed to be more correct.

Answer 1

Both the time and the frequency domains are continuums. For any continuous domain, perturbing a point will not alter the relationships between the dimensions, nor will it hit an abrupt change (because its a continuum). For example, if I have a 3D point in an RGB color space, a slight change in position causes a slight change in color. If I then transform that color space to HSV (hue, saturation, and brightness), I will see the same degree of change with a similar change in (relative) point position. I could then transform it back to RGB and see that change in the RGB space. You can even make the changes the same amount as long as the change in the point was the same relative to the size of the space. Of course, one requirement of perturbation is that the change isn't so much that any non-linearity has an effect.